text: we introduce one step, we assume that all pervious steps can not be applied anymore. For the purpose of presentation, we will analyze the correctness and running time of each step after describing it. (cid:0) (cid:1) ≥ Step 1 (Vertices of degree d + 2) ≥ If there is a vertex v of degree d + 2 in the graph, we branch on v into 1 + d(v−) branches according to Lemma 1 by either including v or each set of −d(v) d d(v) d neighbors of v to the solution set. In the branch where v is included to the solution set, we delete v from the (cid:48) ⊆ graph and decrease the parameter k by 1. In the branch where a set N N(V ) − (cid:48) of d(v) d neighbors of v are included to the solution set, we delete N from − the graph and decrease the param(cid:18)eter k by(cid:19)d(v) d. For this operation, we get a recurrence relation ≤ − d(v) · − − C(k) C(k 1) + − C(k (d(v) d)). (1) d(v) d Let γ denote the branching factor of (1). − ≥ Lemma 5. If d(v) d 2, the branching factor γ of (1) satisﬁes that √ ≤ 1 + 2d2 + 6d + 5 γ . (2) 2 A proof of this lemma can be found the full version of this paper. It is easy ≤ ≥ to verify that γ d + 1 for d 2. After Step 1, the graph has maximum degree d + 1. Step 2 (Proper dominations) If a vertex v of degree d + 1 is dominated by a vertex u (or dominates a neighbor u of it), we branch on v into d(v) branches by including each vertex in \ { } N(v) (or N[v] u ) to the solution set. The correctness of this step is based on Lemma 2 and Lemma 3. In each branch, a vertex is included to the solution set and k decreases by 1. Vertex v is of degree d + 1 since the graph has maximum degree at most d + 1 after Step 1. We get a recurrence relation ≤ · − · − C(k) d(v) C(k 1) = (d + 1) C(k 1), the branching factor of which is d + 1. Step 3 (Good pairs of degree-(d + 1) vertices) Recall that a pair of adjacent degree-(d+1) vertices is a good pair if they have − at least one and at most d 2 common neighbors. we use the following branching { } ∩ ∪ { } rule to deal with a good pair v , v . Let N+ = (N(v ) N(v )) v , v , \ 1\ 2 1 2 1 2 N = N(v ) N+ and N = N(v ) N+. Assume that v and v have x common 1 1 2 2 (cid:48)1 (cid:48) 2 neighbors. Note that for any d-degree deletion set S , if S does not contain (cid:48) any vertex in N+, then S contains at least one vertex in N and one vertex 1 7 ##LTLine## ##LTLine##

text: we

text: introduce

text: one

text: step,

text: we

text: assume

text: that

text: all

text: pervious

text: steps

text: can

text: not

text: be

text: applied

text: anymore.

text: For

text: the

text: purpose

text: of

text: presentation,

text: we

text: will

text: analyze

text: the

text: correctness

text: and

text: running

text: time

text: of

text: each

text: step

text: after

text: describing

text: it.

text: Step

text: (Vertices

text: of

text: degree

text: d

text: If

text: there

text: is

text: a

text: vertex

text: v

text: of

text: degree

text: d

text: in

text: the

text: graph,

text: we

text: branch

text: on

text: v

text: into

text: d(v−)

text: branches

text: according

text: to

text: Lemma

text: by

text: either

text: including

text: v

text: or

text: each

text: set

text: of

text: −d(v)

text: d

text: d(v)

text: d

text: neighbors

text: of

text: v

text: to

text: the

text: solution

text: set.

text: In

text: the

text: branch

text: where

text: v

text: is

text: included

text: to

text: the

text: solution

text: set,

text: we

text: delete

text: v

text: from

text: the

text: graph

text: and

text: decrease

text: the

text: parameter

text: k

text: by

text: In

text: the

text: branch

text: where

text: a

text: set

text: N

text: N(V

text: of

text: d(v)

text: d

text: neighbors

text: of

text: v

text: are

text: included

text: to

text: the

text: solution

text: set,

text: we

text: delete

text: N

text: from

text: the

text: graph

text: and

text: decrease

text: the

text: k

text: d.

text: For

text: this

text: operation,

text: we

text: get

text: a

text: recurrence

text: relation

text: d(v)

text: C(k)

text: C(k

text: C(k

text: (d(v)

text: d)).

text: d(v)

text: d

text: Let

text: γ

text: denote

text: the

text: branching

text: factor

text: of

text: Lemma

text: If

text: d(v)

text: d

text: the

text: branching

text: factor

text: γ

text: of

text: satisﬁes

text: that

text: γ

text: A

text: proof

text: of

text: this

text: lemma

text: can

text: be

text: found

text: the

text: full

text: version

text: of

text: this

text: paper.

text: It

text: is

text: easy

text: to

text: verify

text: that

text: γ

text: d

text: for

text: d

text: After

text: Step

text: the

text: graph

text: has

text: maximum

text: degree

text: d

text: Step

text: (Proper

text: dominations)

text: If

text: a

text: vertex

text: v

text: of

text: degree

text: d

text: is

text: dominated

text: by

text: a

text: vertex

text: u

text: (or

text: dominates

text: a

text: neighbor

text: u

text: of

text: it),

text: we

text: branch

text: on

text: v

text: into

text: d(v)

text: branches

text: by

text: including

text: each

text: vertex

text: in

text: N(v)

text: (or

text: N[v]

text: u

text: to

text: the

text: solution

text: set.

text: The

text: correctness

text: of

text: this

text: step

text: is

text: based

text: on

text: Lemma

text: and

text: Lemma

text: In

text: each

text: branch,

text: a

text: vertex

text: is

text: included

text: to

text: the

text: solution

text: set

text: and

text: k

text: decreases

text: by

text: Vertex

text: v

text: is

text: of

text: degree

text: d

text: since

text: the

text: graph

text: has

text: maximum

text: degree

text: at

text: most

text: d

text: after

text: Step

text: We

text: get

text: a

text: recurrence

text: relation

text: C(k)

text: d(v)

text: C(k

text: (d

text: C(k

text: the

text: branching

text: factor

text: of

text: which

text: is

text: d

text: Step

text: (Good

text: pairs

text: of

text: degree-(d

text: vertices)

text: Recall

text: that

text: a

text: pair

text: of

text: adjacent

text: vertices

text: is

text: a

text: good

text: pair

text: if

text: they

text: have

text: at

text: least

text: one

text: and

text: at

text: most

text: d

text: common

text: neighbors.

text: we

text: use

text: the

text: following

text: branching

text: rule

text: to

text: deal

text: with

text: a

text: good

text: pair

text: v

text: v

text: Let

text: N+

text: (N(v

text: N(v

text: v

text: v

text: N

text: N(v

text: N+

text: and

text: N

text: N(v

text: N+.

text: Assume

text: that

text: v

text: and

text: v

text: have

text: x

text: common

text: neighbors.

text: Note

text: that

text: for

text: any

text: d-degree

text: deletion

text: set

text: S

text: if

text: S

text: does

text: not

text: contain

text: any

text: vertex

text: in

text: N+,

text: then

text: S

text: contains

text: at

text: least

text: one

text: vertex

text: in

text: N

text: and

text: one

text: vertex

text: ##LTLine##

text: ##LTLine##


text: | | | || | − in N . We branch into N+ + N N = (x + 2) + (d x)2 branches. In the 2| | 1 2 ﬁrst N+ branches each vertex in N+ is included to the solution set; and in | || | the last N N branches each pair of vertices in N and N is included to the 1 2 1 2 solution set. In each branch, if z vertices are included to the solution set, then the parameter k in this branch decreases by z. This branching operation gives a recurrence relation ≤ · − − · − C(k) (x + 2) C(k 1) + (d x)2 C(k 2), (cid:16) (cid:112) (cid:17) the branching factor of which is 1 − 2 + x + 5x2 8dx + 4d2 + 4x + 4 . 2 ≤ ≤ − It is easy to verify that when 1 x d 2, the branching factor is at most d + 1. Step 4 (Close triples of degree-(d + 1) vertices) Recall that a pair of adjacent degree-(d+1) vertices is a close pair if they have − exactly d 1 common neighbors. The formal deﬁnition of close triple is that: the { } set of three degree-(d + 1) vertices v , v and v is called a close triple if v , v { } 1 2 3 1 2 and v , v are two close pairs and v and v are not adjacent. According 2 3 1 3 ∩ ∩ to the deﬁnition of close triples, we can see that N(v ) N(v ) N(v ) = \ { } { } 1 2 3 N(v ) v , v . For a close triple v , v , v , we observe the following. Vertex 2 1 3 1 2 3 (cid:54)∈ (cid:54)∈ v (resp., v ) is adjacent to a degree -(d+1) vertex v N[v ] (resp., v N[v ]) 1 3 − \ { } 0 2 4 (cid:48) 2 by Lemma 4. Let N = N[v ] v , v . For any d-degree deletion set S , if ∩ − ∅ (cid:48)2 2 1 3 { } (cid:48) S N = , then S contains either v and a vertex in v , v (since S must 2 1 3 4 { } contain a vertex in N[v ] and a vertex in N[v ]) or v and a vertex in v , v (cid:48) 2 3 3 0 1 (since S must contain a vertex in N[v ] and a vertex in N[v ]). Then we can 2 − 1 branch by either including each vertex in N to the solution set or including each { } { } { } 2 of v , v , v , v and v , v to the solution set. This branching operation 1 3 1 4 0 3 gives a recurrence relation ≤ − · − · − C(k) (d 1) C(k 1) + 3 C(k 2), (cid:16) (cid:112) (cid:17) the branching factor of which is 1 − − d 1 + d2 4d + 13 . 2 ≥ It is easy to verify that when d 2, the branching factor is less than d + 1. Step 5 (Type-I close quadruples of degree-(d + 1) vertices) { } A set of four degree-(d + 1) vertices v , v , v , v is called a type-I close { } 1 2 3 4 { } quadruple if v , v , v , v induces a cycle or a path of 4 vertices, and v , v and { } 1 2 3 4 − ∩ − 1∩ 2 v , v are two close pairs. Let N = N(v ) N(v ) and N = N(v ) N(v ). 3 4 12 1 2 34 3 4 When the graph has no proper dominations, good pairs or close triples, it holds − ∩ − ∅ that N N = . 12 (cid:48) 34 Let S be an arbitrary d-degree deletion set. Our branching rule for type-I { } close quadruples is diﬀerent for the cases whether v , v , v , v induces a cycle 1 2 3 4 or a path. 8 ##LTLine## ##LTLine## ##LTLine## ##LTLine##

label: paragraph

text: in

text: N

text: We

text: branch

text: into

text: N+

text: N

text: N

text: (x

text: (d

text: branches.

text: In

text: the

text: ﬁrst

text: N+

text: branches

text: each

text: vertex

text: in

text: N+

text: is

text: included

text: to

text: the

text: solution

text: set;

text: and

text: in

text: the

text: last

text: N

text: N

text: branches

text: each

text: pair

text: of

text: vertices

text: in

text: N

text: and

text: N

text: is

text: included

text: to

text: the

text: solution

text: set.

text: In

text: each

text: branch,

text: if

text: z

text: vertices

text: are

text: included

text: to

text: the

text: solution

text: set,

text: then

text: the

text: parameter

text: k

text: in

text: this

text: branch

text: decreases

text: by

text: z.

text: This

text: branching

text: operation

text: gives

text: a

text: recurrence

text: relation

text: C(k)

text: (x

text: C(k

text: (d

text: C(k

text: the

text: branching

text: factor

text: of

text: which

text: is

text: x

text: It

text: is

text: easy

text: to

text: verify

text: that

text: when

text: x

text: d

text: the

text: branching

text: factor

text: is

text: at

text: most

text: d

text: Step

text: (Close

text: triples

text: of

text: degree-(d

text: vertices)

text: Recall

text: that

text: a

text: pair

text: of

text: adjacent

text: vertices

text: is

text: a

text: close

text: pair

text: if

text: they

text: have

text: exactly

text: d

text: common

text: neighbors.

text: The

text: formal

text: deﬁnition

text: of

text: close

text: triple

text: is

text: that:

text: the

text: set

text: of

text: three

text: degree-(d

text: vertices

text: v

text: v

text: and

text: v

text: is

text: called

text: a

text: close

text: triple

text: if

text: v

text: v

text: and

text: v

text: v

text: are

text: two

text: close

text: pairs

text: and

text: v

text: and

text: v

text: are

text: not

text: adjacent.

text: According

text: to

text: the

text: deﬁnition

text: of

text: close

text: triples,

text: we

text: can

text: see

text: that

text: N(v

text: N(v

text: N(v

text: N(v

text: v

text: v

text: For

text: a

text: close

text: triple

text: v

text: v

text: v

text: we

text: observe

text: the

text: following.

text: Vertex

text: v

text: (resp.,

text: v

text: is

text: adjacent

text: to

text: a

text: degree

text: vertex

text: v

text: N[v

text: (resp.,

text: v

text: N[v

text: by

text: Lemma

text: Let

text: N

text: N[v

text: v

text: v

text: For

text: any

text: d-degree

text: deletion

text: set

text: S

text: if

text: S

text: N

text: then

text: S

text: contains

text: either

text: v

text: and

text: a

text: vertex

text: in

text: v

text: v

text: (since

text: S

text: must

text: contain

text: a

text: vertex

text: in

text: N[v

text: and

text: a

text: vertex

text: in

text: N[v

text: or

text: v

text: and

text: a

text: vertex

text: in

text: v

text: v

text: (since

text: S

text: must

text: contain

text: a

text: vertex

text: in

text: N[v

text: and

text: a

text: vertex

text: in

text: N[v

text: Then

text: we

text: can

text: branch

text: by

text: either

text: including

text: each

text: vertex

text: in

text: N

text: to

text: the

text: solution

text: set

text: or

text: including

text: each

text: of

text: v

text: v

text: v

text: v

text: and

text: v

text: v

text: to

text: the

text: solution

text: set.

text: This

text: branching

text: operation

text: gives

text: a

text: recurrence

text: relation

text: C(k)

text: (d

text: C(k

text: C(k

text: the

text: branching

text: factor

text: of

text: which

text: is

text: d

text: It

text: is

text: easy

text: to

text: verify

text: that

text: when

text: d

text: the

text: branching

text: factor

text: is

text: less

text: than

text: d

text: Step

text: (Type-I

text: close

text: quadruples

text: of

text: degree-(d

text: vertices)

text: A

text: set

text: of

text: four

text: degree-(d

text: vertices

text: v

text: v

text: v

text: v

text: is

text: called

text: a

text: type-I

text: close

text: quadruple

text: if

text: v

text: v

text: v

text: v

text: induces

text: a

text: cycle

text: or

text: a

text: path

text: of

text: vertices,

text: and

text: v

text: v

text: and

text: 1∩

text: v

text: v

text: are

text: two

text: close

text: pairs.

text: Let

text: N

text: N(v

text: N(v

text: and

text: N

text: N(v

text: N(v

text: When

text: the

text: graph

text: has

text: no

text: proper

text: dominations,

text: good

text: pairs

text: or

text: close

text: triples,

text: it

text: holds

text: that

text: N

text: N

text: Let

text: S

text: be

text: an

text: arbitrary

text: d-degree

text: deletion

text: set.

text: Our

text: branching

text: rule

text: for

text: type-I

text: close

text: quadruples

text: is

text: diﬀerent

text: for

text: the

text: cases

text: whether

text: v

text: v

text: v

text: v

text: induces

text: a

text: cycle

text: or

text: a

text: path.

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: Magnetoquantum oscillations in the resistance of metallic point contacts: Inﬂuence of nonequilibrium phonons ∗ N. L. Bobrov1,2, , J. A. Kokkedee1, N. N. Gribov1,2, I. K. Yanson2, A. G. M. Jansen1, P. Wyder1 1High Magnetic Field Laboratory, Max-Planck Institut fur Festkorperforschung and Centre National de la Recherche Scientiﬁque, F-38042, Grenoble, Cedex 09, France 2B.I. Verkin Institute for Low Temperature Physics and Engineering, of the National Academy of Sciences of Ukraine, prospekt Lenina, 47, Kharkov 61103, Ukraine E-mail address: bobrov@ilt.kharkov.ua (Dated: May 28, 2020; Published Physica B 218, 42(1996)) The amplitude of magnetoresistance quantum oscillations of Al and Be point contacts in a mag- netic ﬁeld parallel to the contact axis has been studied as a function of voltage applied over the contact. It was found that for one group of contacts the oscillation amplitude nonmonotonously in- creases with the bias voltage increase, while for the other part of the contacts a decrease of amplitude was observed. The scattering of electrons with nonequilibrium phonons and also phonon-phonon collisions will be discussed as the possible reasons of the observed eﬀects. PACS numbers: 71.38.-k, 73.40.Jn, 74.25.Kc, 74.45.+c I. INTRODUCTION where x = h¯/2πk τ is the Dingle temperature, and τ is B the elastic lifetime of the electrons. As follows from Eqs. The resistance of metallic point contacts at low tem- (1) and (2) the ratio between the ﬁrst and the second peratures shows the oscillations as a function of magnetic harmonic of the oscillations is ﬁeld, which are due to the Landau quantization of the conduction electron energy spectrum [1, 2]. An increase A /A = Cch(2π2k /¯hω ) exp(π/ω τ), (3) in temperature reduces the amplitude of oscillations. The 1 2 B c c corresponding reduction factor is where the constant C has a meaning of the ratio of the harmonics at T = 0 in the absence of scatterers. 2π2nk T/¯hω RT = sh(2π2nkB T/¯hωc ); (1) From the theory for noninteracting particles it follows B c that the Dingle temperature can be aﬀected by any elec- tron scattering process, including electron-phonon colli- where n is the harmonic number in the oscillations and sions [3]. At the same time, with many-body interac- ∗ ωc = eB/m c the cyclotron frequency. Elastic scattering tions taken into account, the scattering of electrons on of the electrons by impurities also causes the decrease phonons seems to have no eﬀect on the Dingle temper- of the oscillation amplitude nearly the same as resulted ature [3]. The study of the inﬂuence of electron-phonon from the temperature increase from actual value T up to collisions on the amplitude of magnetoquantum oscilla- Teff = T + x. The reduction factor (the Dingle factor) tions in traditional experiments is limited to metals with has the form low Debye energy, in which phonons can be excited at a tem, perature low enough to observe the eﬀect. Metal- lic point contacts present a unique opportunity to study − RD = exp( 2π2kBnx/¯hωc), (2) the inﬂuence of nonequilibrium phonons generated in the contact on the quantum oscillations. Just by applying the bias over the contact, the nonequilibrium phonons can be generated in the constriction with any possible energy up to eV [4]. In this work we study the eﬀect ∗ Corresponding author. of electron- phonon scattering on the amplitude of the ##LTLine## ##LTLine##

text: Magnetoquantum

label: title

text: oscillations

text: in

text: the

text: resistance

text: of

text: metallic

text: point

text: contacts:

text: Inﬂuence

text: of

text: nonequilibrium

text: phonons

label: author

text: N.

text: L.

text: J.

text: A.

text: N.

text: N.

text: I.

text: K.

text: A.

text: G.

text: M.

text: P.

text: Magnetic

text: Field

text: Laboratory,

text: Max-Planck

text: Institut

text: fur

text: Festkorperforschung

text: and

text: Centre

text: National

text: de

text: la

text: Recherche

text: Scientiﬁque,

text: Grenoble,

text: Cedex

text: France

text: Verkin

text: Institute

text: for

text: Low

text: Temperature

text: Physics

text: and

text: Engineering,

text: of

text: the

text: National

text: Academy

text: of

text: Sciences

text: of

text: Ukraine,

text: prospekt

text: Lenina,

text: Kharkov

text: Ukraine

text: E-mail

text: address:

text: bobrov@ilt.kharkov.ua

text: (Dated:

text: May

label: date

text: Published

text: Physica

text: B

text: The

label: abstract

text: amplitude

text: of

text: magnetoresistance

text: quantum

text: oscillations

text: of

text: Al

text: and

text: Be

text: point

text: contacts

text: in

text: a

text: mag-

text: netic

text: ﬁeld

text: parallel

text: to

text: the

text: contact

text: axis

text: has

text: been

text: studied

text: as

text: a

text: function

text: of

text: voltage

text: applied

text: over

text: the

text: contact.

text: It

text: was

text: found

text: that

text: for

text: one

text: group

text: of

text: contacts

text: the

text: oscillation

text: amplitude

text: nonmonotonously

text: in-

text: creases

text: with

text: the

text: bias

text: voltage

text: increase,

text: while

text: for

text: the

text: other

text: part

text: of

text: the

text: contacts

text: a

text: decrease

text: of

text: amplitude

text: was

text: observed.

text: The

text: scattering

text: of

text: electrons

text: with

text: nonequilibrium

text: phonons

text: and

text: also

text: phonon-phonon

text: collisions

text: will

text: be

text: discussed

text: as

text: the

text: possible

text: reasons

text: of

text: the

text: observed

text: eﬀects.

text: PACS

text: numbers:

text: I.

text: INTRODUCTION

text: where

text: x

text: τ

text: is

text: the

text: Dingle

text: temperature,

text: and

text: τ

text: is

text: B

text: the

text: elastic

text: lifetime

text: of

text: the

text: electrons.

text: As

text: follows

text: from

text: Eqs.

text: The

text: resistance

text: of

text: metallic

text: point

text: contacts

text: at

text: low

text: tem-

text: and

text: the

text: ratio

text: between

text: the

text: ﬁrst

text: and

text: the

text: second

text: peratures

text: shows

text: the

text: oscillations

text: as

text: a

text: function

text: of

text: magnetic

text: harmonic

text: of

text: the

text: oscillations

text: is

text: ﬁeld,

text: which

text: are

text: due

text: to

text: the

text: Landau

text: quantization

text: of

text: the

text: conduction

text: electron

text: energy

text: spectrum

text: An

text: increase

text: A

text: /A

text: /¯hω

text: exp(π/ω

text: τ),

text: in

text: temperature

text: reduces

text: the

text: amplitude

text: of

text: oscillations.

text: The

text: B

text: c

text: c

text: corresponding

text: reduction

text: factor

text: is

text: where

text: the

text: constant

text: C

text: has

text: a

text: meaning

text: of

text: the

text: ratio

text: of

text: the

text: harmonics

text: at

text: T

text: in

text: the

text: absence

text: of

text: scatterers.

text: T/¯hω

text: RT

text: T/¯hωc

text: From

text: the

text: theory

text: for

text: noninteracting

text: particles

text: it

text: follows

text: B

text: c

text: that

text: the

text: Dingle

text: temperature

text: can

text: be

text: aﬀected

text: by

text: any

text: elec-

text: tron

text: scattering

text: process,

text: including

text: electron-phonon

text: colli-

text: where

text: n

text: is

text: the

text: harmonic

text: number

text: in

text: the

text: oscillations

text: and

text: sions

text: At

text: the

text: same

text: time,

text: with

text: many-body

text: interac-

text: ωc

text: eB/m

text: c

text: the

text: cyclotron

text: frequency.

text: Elastic

text: scattering

text: tions

text: taken

text: into

text: account,

text: the

text: scattering

text: of

text: electrons

text: on

text: of

text: the

text: electrons

text: by

text: impurities

text: also

text: causes

text: the

text: decrease

text: phonons

text: seems

text: to

text: have

text: no

text: eﬀect

text: on

text: the

text: Dingle

text: temper-

text: of

text: the

text: oscillation

text: amplitude

text: nearly

text: the

text: same

text: as

text: resulted

text: ature

text: The

text: study

text: of

text: the

text: inﬂuence

text: of

text: electron-phonon

text: from

text: the

text: temperature

text: increase

text: from

text: actual

text: value

text: T

text: up

text: to

text: collisions

text: on

text: the

text: amplitude

text: of

text: magnetoquantum

text: oscilla-

text: Teff

text: T

text: x.

text: The

text: reduction

text: factor

text: (the

text: Dingle

text: factor)

text: tions

text: in

text: traditional

text: experiments

text: is

text: limited

text: to

text: metals

text: with

text: has

text: the

text: form

text: low

text: Debye

text: energy,

text: in

text: which

text: phonons

text: can

text: be

text: excited

text: at

text: a

text: tem,

text: perature

text: low

text: enough

text: to

text: observe

text: the

text: eﬀect.

text: Metal-

text: lic

text: point

text: contacts

text: present

text: a

text: unique

text: opportunity

text: to

text: study

text: RD

text: exp(

text: the

text: inﬂuence

text: of

text: nonequilibrium

text: phonons

text: generated

text: in

text: the

text: contact

text: on

text: the

text: quantum

text: oscillations.

text: Just

text: by

text: applying

text: the

text: bias

text: over

text: the

text: contact,

text: the

text: nonequilibrium

text: phonons

text: can

text: be

text: generated

text: in

text: the

text: constriction

text: with

text: any

text: possible

text: energy

text: up

text: to

text: eV

text: In

text: this

text: work

text: we

text: study

text: the

text: eﬀect

text: Corresponding

text: author.

text: of

text: electron-

text: phonon

text: scattering

text: on

text: the

text: amplitude

text: of

text: the

text: ##LTLine##

text: ##LTLine##


text: 6 ing on scales far below the operating wavelength. In such a a) b) scenario, the use of quantum resources in a plasmonic system (1) (2) (3) (1) would be the best strategy, although at the cost of the sensing (2) (3) resolution. We also investigate the resolution δnbio as N increases for the NOON state, the optimal state, the SIL and the HL. These more general results are shown in Figs. 5c and d, where the c) d) optimal states remain beyond the SIL (upper dashed line), re- NOON gardless of N. It should be noted that the quantum enhance- NOON ment is reduced with increasing N, i.e. in Figs. 5c and d the optimal optimal gap between the HL and the S√IL decreases with increasing N, δn(SIL) − δn(HL) = √1 (1+√ η − √1 )| ∂φ |−1, while the bio bio N 2 η N ∂nbio optimal state resolution remains at a roughly ﬁxed distance below the SIL. On the other hand, the relative diﬀerence, FIG. 5: Resolutions of optimal states and increasing average num- − deﬁned by (δn(SIL) δn(HL))/δn(SIL), approaches unity in the ber of photons. Input states optimized for a certain amount of loss bio bio bio limit of large N, while the other relative diﬀerence, deﬁned by still beat the SIL over the whole range of nbio for the lossy plas- (δn(SIL) −δn(HL))/δn(HL), diverges in th√e limit of large N. Such monic nanowire and wedge waveguides considered in Fig. 4. a) a bebihoavior nbaioturallybaiorises from the N improvement of the Three points are randomly chosen from Fig. 4c): (1) nbio = 1.13, (2) nbio = 1.19 and (3) nbio = 1.37. For these points, the optimal HL over the SIL. Considering Figs. 5c and d it might seem at states found with the corresponding amount of loss are used as the ﬁrst that quantum plasmonic sensing is not necessary because input states and the resolutions are obtained over the whole range a given resolution can always be achieved by simply increas- of n for the nanowire waveguide. b) Same as panel a, but for the bio ing the intensity of a classical input source to get the SIL, e.g. wedge waveguide. Here the points correspond to those in Fig. 4d): ∼ − δnbio 10 8 RIU is in principle achievable by∼a λ×0 = 810 nm (1) nbio = 1.34392, (2) nbio = 1.36576 and (3) nbio = 1.43128. c) laser with an initial power of 1 mW having N 4 1015 pho- The resolution δnbio with increasing N for the NOON state, optimal tons per second and an appropriate mode volume with a large state, SIL, and HL cases show that the optimal states remain beyond the SIL, regardless of N for the nanowire. d) Same as panel c, but power density [31, 53, 60]. A high-power source, however, for the wedge waveguide. In both panels c and d, the value of n is not commonly desired for biological measurements since bio corresponds to point (2) in Figs. 4c and d, respectively. The quantum it may damage the specimen under investigation [60–62] or enhancement in both plots (c and d) is reduced with increasing N, as cause other unwanted phenomena such as thermal modula- the gap between the optimal state resolution and the SIL decreases tion of the surface plasmon mode [63]. In this case, at the with N. A line within the shaded area shows an improvement over few-photon level, one may then consider the beneﬁts of using classical plasmonic sensing. quantum sensing with either a dielectric or plasmonic waveg- uide. Here, for a ﬁxed value of N, the quantum plasmonic sensor provides a low mode volume and allows one to go be- tum plasmonic sensor would beneﬁt from further investigation yond the SIL in a compact setting below the diﬀraction limit. into diﬀerent excitation platforms such as a prism, grating, lo- This is crucial when only a small biological sample is avail- calized SPR sensor, metamaterials and graphene, as well as able, or one would like a more compact and integrated sens- modulation-based approaches. In addition to the sensitivity ing device than standard dielectric components can achieve. and resolution considered in this work, other ﬁgures of merit Note that while the power density in a plasmonic waveguide such as accuracy, precision, or kinetic analysis will need to be is enhanced compared to a dielectric waveguide due to the low taken into account for practical use in industry. We envisage mode volume, it is the total power that is important in gain- that progress in quantum metrology will reshape the ﬁeld of ing the quantum advantage, as the shot-noise or Heisenberg plasmonic biosensing – a ﬁeld that has already developed into limits are related to the photon number statistics rather than mature technology for at least two decades [31]. Our work the optical power density. Despite all the demanding require- opens up a path between the quantum metrology and plas- ments of quantum measurement with plasmonic systems, in monic sensing ﬁelds, and has the potential to lead to a vari- recent years several experimental studies have demonstrated ety of studies at the level of practical realization. Integrated the feasibility of quantum plasmonic sensing using low inten- quantum plasmonic sensors may also ﬁnd application in on- sity input sources [28–30]. chip nanoscale quantum network devices, with potential uses Discussion—. In this work we studied quantum plasmonic in quantum tasks where precise and compact measurement is sensing in the few photon regime and showed an example required [64, 65]. quantum plasmonic sensor that can beat the SNL in the pres- Acknowledgments. C. L. thanks S.-Y. Lee and K. H. Seol ence of metallic losses. We have demonstrated how the in- for discussions. This research was supported by the Ministry clusion of quantum techniques in a plasmonic system enables of Science, ICT and Future Planning (MISP) Korea, under one to further improve its sensitivity and resolution beyond the the Information Technology Research Center (ITRC) support SNL, while keeping the compactness of the device on scales program IITP-2016-R0992-16-1017 supervised by the Insti- far below the diﬀraction limit. Our analysis is applicable to tute for Information and Communications Technology Pro- any type of plasmonic sensing platform and we leave a vari- motion (IITP), the National Research Foundation of Korea ety of technical issues related to quantum plasmonic sensors (NRF) Grant funded by the Korea government (MSIP; No. for future works. For example, the performance of a quan- 2014R1A2A1A10050117), the South African National Re- ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: ing

text: on

text: scales

text: far

text: below

text: the

text: operating

text: wavelength.

text: In

text: such

text: a

text: a)

font: default

text: b)

text: scenario,

text: the

text: use

text: of

text: quantum

text: resources

text: in

text: a

text: plasmonic

text: system

text: would

text: be

text: the

text: best

text: strategy,

text: although

text: at

text: the

text: cost

text: of

text: the

text: sensing

text: resolution.

text: We

text: also

text: investigate

text: the

text: resolution

text: δnbio

text: as

text: N

text: increases

text: for

text: the

text: NOON

text: state,

text: the

text: optimal

text: state,

text: the

text: SIL

text: and

text: the

text: HL.

text: These

text: more

text: general

text: results

text: are

text: shown

text: in

text: Figs.

text: and

text: d,

text: where

text: the

text: c)

text: d)

text: optimal

text: states

text: remain

text: beyond

text: the

text: SIL

text: (upper

text: dashed

text: line),

text: re-

text: NOON

text: gardless

text: of

text: N.

text: It

text: should

text: be

text: noted

text: that

text: the

text: quantum

text: enhance-

text: NOON

text: ment

text: is

text: reduced

text: with

text: increasing

text: N,

text: i.e.

text: in

text: Figs.

text: and

text: d

text: the

text: optimal

text: optimal

text: gap

text: between

text: the

text: HL

text: and

text: the

text: S√IL

text: decreases

text: with

text: increasing

text: N,

text: δn(SIL)

text: δn(HL)

text: √1

text: η

text: √1

text: ∂φ

text: while

text: the

text: bio

text: bio

text: N

text: η

text: N

text: ∂nbio

text: optimal

text: state

text: resolution

text: remains

text: at

text: a

text: roughly

text: ﬁxed

text: distance

text: below

text: the

text: SIL.

text: On

text: the

text: other

text: hand,

text: the

text: relative

text: diﬀerence,

text: FIG.

text: Resolutions

text: of

text: optimal

text: states

text: and

text: increasing

text: average

text: num-

text: deﬁned

text: by

text: (δn(SIL)

text: δn(HL))/δn(SIL),

text: approaches

text: unity

text: in

text: the

text: ber

text: of

text: photons.

text: Input

text: states

text: optimized

text: for

text: a

text: certain

text: amount

text: of

text: loss

text: bio

text: bio

text: bio

text: limit

text: of

text: large

text: N,

text: while

text: the

text: other

text: relative

text: diﬀerence,

text: deﬁned

text: by

text: still

text: beat

text: the

text: SIL

text: over

text: the

text: whole

text: range

text: of

text: nbio

text: for

text: the

text: lossy

text: plas-

text: (δn(SIL)

text: −δn(HL))/δn(HL),

text: diverges

text: in

text: th√e

text: limit

text: of

text: large

text: N.

text: Such

text: monic

text: nanowire

text: and

text: wedge

text: waveguides

text: considered

text: in

text: Fig.

text: a)

text: a

text: bebihoavior

text: nbaioturallybaiorises

text: from

text: the

text: N

text: improvement

text: of

text: the

text: Three

text: points

text: are

text: randomly

text: chosen

text: from

text: Fig.

text: nbio

text: nbio

text: and

text: nbio

text: For

text: these

text: points,

text: the

text: optimal

text: HL

text: over

text: the

text: SIL.

text: Considering

text: Figs.

text: and

text: d

text: it

text: might

text: seem

text: at

text: states

text: found

text: with

text: the

text: corresponding

text: amount

text: of

text: loss

text: are

text: used

text: as

text: the

text: ﬁrst

text: that

text: quantum

text: plasmonic

text: sensing

text: is

text: not

text: necessary

text: because

text: input

text: states

text: and

text: the

text: resolutions

text: are

text: obtained

text: over

text: the

text: whole

text: range

text: a

text: given

text: resolution

text: can

text: always

text: be

text: achieved

text: by

text: simply

text: increas-

text: of

text: n

text: for

text: the

text: nanowire

text: waveguide.

text: b)

text: Same

text: as

text: panel

text: a,

text: but

text: for

text: the

text: bio

text: ing

text: the

text: intensity

text: of

text: a

text: classical

text: input

text: source

text: to

text: get

text: the

text: SIL,

text: e.g.

text: wedge

text: waveguide.

text: Here

text: the

text: points

text: correspond

text: to

text: those

text: in

text: Fig.

text: δnbio

text: RIU

text: is

text: in

text: principle

text: achievable

text: by∼a

text: λ×0

text: nm

text: nbio

text: nbio

text: and

text: nbio

text: c)

text: laser

text: with

text: an

text: initial

text: power

text: of

text: mW

text: having

text: N

text: pho-

text: The

text: resolution

text: δnbio

text: with

text: increasing

text: N

text: for

text: the

text: NOON

text: state,

text: optimal

text: tons

text: per

text: second

text: and

text: an

text: appropriate

text: mode

text: volume

text: with

text: a

text: large

text: state,

text: SIL,

text: and

text: HL

text: cases

text: show

text: that

text: the

text: optimal

text: states

text: remain

text: beyond

text: the

text: SIL,

text: regardless

text: of

text: N

text: for

text: the

text: nanowire.

text: d)

text: Same

text: as

text: panel

text: c,

text: but

text: power

text: density

text: A

text: high-power

text: source,

text: however,

text: for

text: the

text: wedge

text: waveguide.

text: In

text: both

text: panels

text: c

text: and

text: d,

text: the

text: value

text: of

text: n

text: is

text: not

text: commonly

text: desired

text: for

text: biological

text: measurements

text: since

text: bio

text: corresponds

text: to

text: point

text: in

text: Figs.

text: and

text: d,

text: respectively.

text: The

text: quantum

text: it

text: may

text: damage

text: the

text: specimen

text: under

text: investigation

text: or

text: enhancement

text: in

text: both

text: plots

text: (c

text: and

text: d)

text: is

text: reduced

text: with

text: increasing

text: N,

text: as

text: cause

text: other

text: unwanted

text: phenomena

text: such

text: as

text: thermal

text: modula-

text: the

text: gap

text: between

text: the

text: optimal

text: state

text: resolution

text: and

text: the

text: SIL

text: decreases

text: tion

text: of

text: the

text: surface

text: plasmon

text: mode

text: In

text: this

text: case,

text: at

text: the

text: with

text: N.

text: A

text: line

text: within

text: the

text: shaded

text: area

text: shows

text: an

text: improvement

text: over

text: few-photon

text: level,

text: one

text: may

text: then

text: consider

text: the

text: beneﬁts

text: of

text: using

text: classical

text: plasmonic

text: sensing.

text: quantum

text: sensing

text: with

text: either

text: a

text: dielectric

text: or

text: plasmonic

text: waveg-

text: uide.

text: Here,

text: for

text: a

text: ﬁxed

text: value

text: of

text: N,

text: the

text: quantum

text: plasmonic

text: sensor

text: provides

text: a

text: low

text: mode

text: volume

text: and

text: allows

text: one

text: to

text: go

text: be-

text: tum

text: plasmonic

text: sensor

text: would

text: beneﬁt

text: from

text: further

text: investigation

text: yond

text: the

text: SIL

text: in

text: a

text: compact

text: setting

text: below

text: the

text: diﬀraction

text: limit.

text: into

text: diﬀerent

text: excitation

text: platforms

text: such

text: as

text: a

text: prism,

text: grating,

text: lo-

text: This

text: is

text: crucial

text: when

text: only

text: a

text: small

text: biological

text: sample

text: is

text: avail-

text: calized

text: SPR

text: sensor,

text: metamaterials

text: and

text: graphene,

text: as

text: well

text: as

text: able,

text: or

text: one

text: would

text: like

text: a

text: more

text: compact

text: and

text: integrated

text: sens-

text: modulation-based

text: approaches.

text: In

text: addition

text: to

text: the

text: sensitivity

text: ing

text: device

text: than

text: standard

text: dielectric

text: components

text: can

text: achieve.

text: and

text: resolution

text: considered

text: in

text: this

text: work,

text: other

text: ﬁgures

text: of

text: merit

text: Note

text: that

text: while

text: the

text: power

text: density

text: in

text: a

text: plasmonic

text: waveguide

text: such

text: as

text: accuracy,

text: precision,

text: or

text: kinetic

text: analysis

text: will

text: need

text: to

text: be

text: is

text: enhanced

text: compared

text: to

text: a

text: dielectric

text: waveguide

text: due

text: to

text: the

text: low

text: taken

text: into

text: account

text: for

text: practical

text: use

text: in

text: industry.

text: We

text: envisage

text: mode

text: volume,

text: it

text: is

text: the

text: total

text: power

text: that

text: is

text: important

text: in

text: gain-

text: that

text: progress

text: in

text: quantum

text: metrology

text: will

text: reshape

text: the

text: ﬁeld

text: of

text: ing

text: the

text: quantum

text: advantage,

text: as

text: the

text: shot-noise

text: or

text: Heisenberg

text: plasmonic

text: biosensing

text: a

text: ﬁeld

text: that

text: has

text: already

text: developed

text: into

text: limits

text: are

text: related

text: to

text: the

text: photon

text: number

text: statistics

text: rather

text: than

text: mature

text: technology

text: for

text: at

text: least

text: two

text: decades

text: Our

text: work

text: the

text: optical

text: power

text: density.

text: Despite

text: all

text: the

text: demanding

text: require-

text: opens

text: up

text: a

text: path

text: between

text: the

text: quantum

text: metrology

text: and

text: plas-

text: ments

text: of

text: quantum

text: measurement

text: with

text: plasmonic

text: systems,

text: in

text: monic

text: sensing

text: ﬁelds,

text: and

text: has

text: the

text: potential

text: to

text: lead

text: to

text: a

text: vari-

text: recent

text: years

text: several

text: experimental

text: studies

text: have

text: demonstrated

text: ety

text: of

text: studies

text: at

text: the

text: level

text: of

text: practical

text: realization.

text: Integrated

text: the

text: feasibility

text: of

text: quantum

text: plasmonic

text: sensing

text: using

text: low

text: inten-

text: quantum

text: plasmonic

text: sensors

text: may

text: also

text: ﬁnd

text: application

text: in

text: on-

text: sity

text: input

text: sources

text: chip

text: nanoscale

text: quantum

text: network

text: devices,

text: with

text: potential

text: uses

text: Discussion—.

text: In

text: this

text: work

text: we

text: studied

text: quantum

text: plasmonic

text: in

text: quantum

text: tasks

text: where

text: precise

text: and

text: compact

text: measurement

text: is

text: sensing

text: in

text: the

text: few

text: photon

text: regime

text: and

text: showed

text: an

text: example

text: required

text: quantum

text: plasmonic

text: sensor

text: that

text: can

text: beat

text: the

text: SNL

text: in

text: the

text: pres-

text: Acknowledgments.

text: C.

text: L.

text: thanks

text: S.-Y.

text: Lee

text: and

text: K.

text: H.

text: Seol

text: ence

text: of

text: metallic

text: losses.

text: We

text: have

text: demonstrated

text: how

text: the

text: in-

text: for

text: discussions.

text: This

text: research

text: was

text: supported

text: by

text: the

text: Ministry

text: clusion

text: of

text: quantum

text: techniques

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text: a

text: plasmonic

text: system

text: enables

text: of

text: Science,

text: ICT

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text: Future

text: Planning

text: (MISP)

text: Korea,

text: under

text: one

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text: further

text: improve

text: its

text: sensitivity

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text: resolution

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text: the

text: the

text: Information

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text: Center

text: (ITRC)

text: support

text: SNL,

text: while

text: keeping

text: the

text: compactness

text: of

text: the

text: device

text: on

text: scales

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text: supervised

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text: the

text: diﬀraction

text: limit.

text: Our

text: analysis

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text: applicable

text: to

text: tute

text: for

text: Information

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text: any

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text: plasmonic

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text: motion

text: (IITP),

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text: Korea

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text: of

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text: quantum

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text: sensors

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text: by

text: the

text: Korea

text: government

text: (MSIP;

text: No.

text: for

text: future

text: works.

text: For

text: example,

text: the

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text: T a given , in reasonable time. Thus we opted for a minimalist approach: for each T we generate an example, provided that one is known. We note, however, that some of constructions implemented can generate many examples with the same T ; e.g. we have implemented the construction to generate the point graph of ∗ O O ⊂ the generalized quadrangle T ( ) (see [49]) from any hyperoval PG(2, 2k). T 2 As well, many ’s can be realized by more than one implemented construction, sometimes leading to isomorphic graphs, and sometimes not. Our desire to take on this project was motivated by the following consider- ations. • One wants to double-check that the constructions are correct and their descriptions are complete; indeed, a program is more trustworthy than a proof in some situations, and coding a construction is a good test for completeness of the description provided. • We wanted to see that the Sage combinatorial, graph-theoretic, and group- theoretic primitives to deal with such constructions are mature and ver- satile, so that coding of constructions is relatively easy and quick. • One learns a lot while working on such a project, both the underlying mathematics, and how the toolset can be improved. In particular, one might come along simpliﬁcations of constructions, and this actually hap- pened on couple of occasions, see Sect. . • As time goes by, possible gaps in constructions are harder and harder to ﬁll in. Reconstructing omitted proof details becomes a tricky and time- consuming task. In particular, as far as the latter item is concerned, we seem to have uncov- ered at least one substantial gap in constructions (see Sect. ). Furthermore, a number of constructions needed feedback from their authors or discussions with experts – sometimes quite substantial – to code them. A large part of the constructions use in a nontrivial way another combi- natorial or algebraic object: block design, Hadamard matrix, two-graph, two- distance code, ﬁnite group, etc. In particular, at the start of the project some of these were lacking in Sagemath, we needed to implement constructions of certain block designs, regular symmetric Hadamard matrices with constant di- agonal (where the gap just mentioned was uncovered), skew-Hadamard matrices, and two-graphs. As well, we created a small database of two-distance codes (see Sect. ). The remainder of the paper consists of a short introduction to strongly regu- lar graphs, pointing out particular relevant Sagemath features, and a description of our implementations, few of them seemingly novel. We list the constructions that we implemented, and discuss few gaps we discovered in the literature. 2 Strongly regular graphs and related objects 2

text: T

text: a

text: given

text: in

text: reasonable

text: time.

text: Thus

text: we

text: opted

text: for

text: a

text: minimalist

text: approach:

text: for

text: each

text: T

text: we

text: generate

text: an

text: example,

text: provided

text: that

text: one

text: is

text: known.

text: We

text: note,

text: however,

text: that

text: some

text: of

text: constructions

text: implemented

text: can

text: generate

text: many

text: examples

text: with

text: the

text: same

text: T

text: e.g.

text: we

text: have

text: implemented

text: the

text: construction

text: to

text: generate

text: the

text: point

text: graph

text: of

text: O

text: O

text: the

text: generalized

text: quadrangle

text: T

text: (see

text: from

text: any

text: hyperoval

text: T

text: As

text: well,

text: many

text: ’s

text: can

text: be

text: realized

text: by

text: more

text: than

text: one

text: implemented

text: construction,

text: sometimes

text: leading

text: to

text: isomorphic

text: graphs,

text: and

text: sometimes

text: not.

text: Our

text: desire

text: to

text: take

text: on

text: this

text: project

text: was

text: motivated

text: by

text: the

text: following

text: consider-

text: ations.

text: One

text: wants

text: to

text: double-check

text: that

text: the

text: constructions

text: are

text: correct

text: and

text: their

text: descriptions

text: are

text: complete;

text: indeed,

text: a

text: program

text: is

text: more

text: trustworthy

text: than

text: a

text: proof

text: in

text: some

text: situations,

text: and

text: coding

text: a

text: construction

text: is

text: a

text: good

text: test

text: for

text: completeness

text: of

text: the

text: description

text: provided.

text: We

text: wanted

text: to

text: see

text: that

text: the

text: Sage

text: combinatorial,

text: graph-theoretic,

text: and

text: group-

text: theoretic

text: primitives

text: to

text: deal

text: with

text: such

text: constructions

text: are

text: mature

text: and

text: ver-

text: satile,

text: so

text: that

text: coding

text: of

text: constructions

text: is

text: relatively

text: easy

text: and

text: quick.

text: One

text: learns

text: a

text: lot

text: while

text: working

text: on

text: such

text: a

text: project,

text: both

text: the

text: underlying

text: mathematics,

text: and

text: how

text: the

text: toolset

text: can

text: be

text: improved.

text: In

text: particular,

text: one

text: might

text: come

text: along

text: simpliﬁcations

text: of

text: constructions,

text: and

text: this

text: actually

text: hap-

text: pened

text: on

text: couple

text: of

text: occasions,

text: see

text: Sect.

text: As

text: time

text: goes

text: by,

text: possible

text: gaps

text: in

text: constructions

text: are

text: harder

text: and

text: harder

text: to

text: ﬁll

text: in.

text: Reconstructing

text: omitted

text: proof

text: details

text: becomes

text: a

text: tricky

text: and

text: time-

text: consuming

text: task.

text: In

text: particular,

text: as

text: far

text: as

text: the

text: latter

text: item

text: is

text: concerned,

text: we

text: seem

text: to

text: have

text: uncov-

text: ered

text: at

text: least

text: one

text: substantial

text: gap

text: in

text: constructions

text: (see

text: Sect.

text: Furthermore,

text: a

text: number

text: of

text: constructions

text: needed

text: feedback

text: from

text: their

text: authors

text: or

text: discussions

text: with

text: experts

text: sometimes

text: quite

text: substantial

text: to

text: code

text: them.

text: A

text: large

text: part

text: of

text: the

text: constructions

text: use

text: in

text: a

text: nontrivial

text: way

text: another

text: combi-

text: natorial

text: or

text: algebraic

text: object:

text: block

text: design,

text: Hadamard

text: matrix,

text: two-graph,

text: two-

text: distance

text: code,

text: ﬁnite

text: group,

text: etc.

text: In

text: particular,

text: at

text: the

text: start

text: of

text: the

text: project

text: some

text: of

text: these

text: were

text: lacking

text: in

text: Sagemath,

text: we

text: needed

text: to

text: implement

text: constructions

text: of

text: certain

text: block

text: designs,

text: regular

text: symmetric

text: Hadamard

text: matrices

text: with

text: constant

text: di-

text: agonal

text: (where

text: the

text: gap

text: just

text: mentioned

text: was

text: uncovered),

text: skew-Hadamard

text: matrices,

text: and

text: two-graphs.

text: As

text: well,

text: we

text: created

text: a

text: small

text: database

text: of

text: two-distance

text: codes

text: (see

text: Sect.

text: The

text: remainder

text: of

text: the

text: paper

text: consists

text: of

text: a

text: short

text: introduction

text: to

text: strongly

text: regu-

text: lar

text: graphs,

text: pointing

text: out

text: particular

text: relevant

text: Sagemath

text: features,

text: and

text: a

text: description

text: of

text: our

text: implementations,

text: few

text: of

text: them

text: seemingly

text: novel.

text: We

text: list

text: the

text: constructions

text: that

text: we

text: implemented,

text: and

text: discuss

text: few

text: gaps

text: we

text: discovered

text: in

text: the

text: literature.

text: Strongly

text: regular

text: graphs

text: and

text: related

text: objects


text: [29] W. H. Haemers and Q. Xiang. Strongly regular graphs with parameters (4m4, 2m4 + m2, m4 + m2, m4 + m2) exist for all m > 1. European J. Combin., 31(6):1553–1559, 2010. [30] M. Hall, Jr. Combinatorial theory. Wiley-Interscience Series in Discrete Mathematics. John Wiley & Sons, Inc., New York, second edition, 1986. A Wiley-Interscience Publication. [31] H. D. L. Hollmann. Association schemes, 1982. MSc thesis, Eindhoven University of Technology. [32] T. Huang, L. Huang, and M.-I. Lin. On a class of strongly regular designs and quasi-semisymmetric designs. In Recent developments in algebra and related areas, volume 8 of Adv. Lect. Math. (ALM), pages 129–153. Int. Press, Somerville, MA, 2009. [33] X. L. Hubaut. Strongly regular graphs. Discrete Math., 13(4):357–381, 1975. [34] Y. J. Ionin and H. Kharaghani. New families of strongly regular graphs. J. Combin. Des., 11(3):208–217, 2003. [35] Y. J. Ionin and H. Kharaghani. New families of strongly regular graphs. Preprint, Feb. 2016. http://www.cs.uleth.ca/~hadi/research/ IoninKharaghani.pdf. [36] Y. J. Ionin and M. S. Shrikhande. Combinatorics of symmetric designs, volume 5 of New Mathematical Monographs. Cambridge University Press, Cambridge, 2006. [37] A. A. Ivanov, M. H. Klin, and I. A. Faradjev. The primitive representations of the non-abelian simple groups of order less than 106, part II. Preprint (Russian), 1984. The Insitute for System Studies (VNIISI), Moscow. [38] Z. Janko and H. Kharaghani. A block negacyclic Bush-type Hadamard ma- trix and two strongly regular graphs. J. Combin. Theory Ser. A, 98(1):118– 126, 2002. [39] Z. Janko, H. Kharaghani, and V. D. Tonchev. The existence of a Bush-type Hadamard matrix of order 324 and two new inﬁnite classes of symmetric designs. Des. Codes Cryptogr., 24(2):225–232, 2001. [40] T. Jenrich and A. E. Brouwer. A 64-dimensional counterexample to Bor- suk’s conjecture. Electron. J. Combin., 21(4):Paper 4.29, 3, 2014. [41] L. K. Jørgensen and M. Klin. Switching of edges in strongly regular graphs. I. A family of partial diﬀerence sets on 100 vertices. Electron. J. Combin., 10:Research Paper 17, 31, 2003. 16

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text: Combin.,

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text: Hall,

text: Jr.

text: Combinatorial

text: theory.

text: Wiley-Interscience

text: Series

text: in

text: Discrete

text: Mathematics.

text: John

text: Wiley

text: Sons,

text: Inc.,

text: New

text: York,

text: second

text: edition,

text: A

text: Wiley-Interscience

text: Publication.

text: H.

text: D.

text: L.

text: Hollmann.

text: Association

text: schemes,

text: MSc

text: thesis,

text: Eindhoven

text: University

text: of

text: Technology.

text: T.

text: Huang,

text: L.

text: Huang,

text: and

text: M.-I.

text: Lin.

text: On

text: a

text: class

text: of

text: strongly

text: regular

text: designs

text: and

text: quasi-semisymmetric

text: designs.

text: In

text: Recent

text: developments

text: in

text: algebra

text: and

text: related

text: areas,

text: volume

text: of

text: Adv.

text: Lect.

text: Math.

text: (ALM),

text: pages

text: Int.

text: Press,

text: Somerville,

text: MA,

text: X.

text: L.

text: Hubaut.

text: Strongly

text: regular

text: graphs.

text: Discrete

text: Math.,

text: Y.

text: J.

text: Ionin

text: and

text: H.

text: Kharaghani.

text: New

text: families

text: of

text: strongly

text: regular

text: graphs.

text: J.

text: Combin.

text: Des.,

text: Y.

text: J.

text: Ionin

text: and

text: H.

text: Kharaghani.

text: New

text: families

text: of

text: strongly

text: regular

text: graphs.

text: Preprint,

text: Feb.

text: IoninKharaghani.pdf.

text: Y.

text: J.

text: Ionin

text: and

text: M.

text: S.

text: Shrikhande.

text: Combinatorics

text: of

text: symmetric

text: designs,

text: volume

text: of

text: New

text: Mathematical

text: Monographs.

text: Cambridge

text: University

text: Press,

text: Cambridge,

text: A.

text: A.

text: Ivanov,

text: M.

text: H.

text: Klin,

text: and

text: I.

text: A.

text: Faradjev.

text: The

text: primitive

text: representations

text: of

text: the

text: non-abelian

text: simple

text: groups

text: of

text: order

text: less

text: than

text: part

text: II.

text: Preprint

text: (Russian),

text: The

text: Insitute

text: for

text: System

text: Studies

text: (VNIISI),

text: Moscow.

text: Z.

text: Janko

text: and

text: H.

text: Kharaghani.

text: A

text: block

text: negacyclic

text: Bush-type

text: Hadamard

text: ma-

text: trix

text: and

text: two

text: strongly

text: regular

text: graphs.

text: J.

text: Combin.

text: Theory

text: Ser.

text: A,

text: Z.

text: Janko,

text: H.

text: Kharaghani,

text: and

text: V.

text: D.

text: Tonchev.

text: The

text: existence

text: of

text: a

text: Bush-type

text: Hadamard

text: matrix

text: of

text: order

text: and

text: two

text: new

text: inﬁnite

text: classes

text: of

text: symmetric

text: designs.

text: Des.

text: Codes

text: Cryptogr.,

text: T.

text: Jenrich

text: and

text: A.

text: E.

text: Brouwer.

text: A

text: counterexample

text: to

text: Bor-

text: suk’s

text: conjecture.

text: Electron.

text: J.

text: Combin.,

text: L.

text: K.

text: Jørgensen

text: and

text: M.

text: Klin.

text: Switching

text: of

text: edges

text: in

text: strongly

text: regular

text: graphs.

text: I.

text: A

text: family

text: of

text: partial

text: diﬀerence

text: sets

text: on

text: vertices.

text: Electron.

text: J.

text: Combin.,

text: Paper


text: • RSHCD – graphs from (n, (cid:15))-regular symmetric Hadamard matrices M with constant diagonal, see Example 2.2 for the deﬁnition. Several con- structions from the literature (and one apparently new one, cf. Sect. ) for this class of Hadamard matrices were implemented in Sage and are available in its Hadamard matrices module. • Two-graph descendants. Each regular two-graph (a certain class of 3- uniform v-vertex hypergraphs having 2µ three-edges on each pair of points, cf. e.g. [8, Chap.10]) gives rise to a strongly regular graph with parameters − − (v 1, 2µ, 3µ v/2, µ) obtained by descendant construction, see e.g. [8, Sect.10.3]. • Switch OA Graphs – these strongly regular graphs are obtained from OA block graphs (see above). From such a graph G obtained from an OA(k, n), the procedure is to (1) add a new isolated vertex v; (2) perform Seidel { } switching on the union of v and several disjoint n-cocliques of G. Note that a n-coclique in G corresponds to a parallel class of the OA(k, n), and that those are easily obtained from an OA(k + 1, n) (i.e. a resolvable OA(k, n)). • Polhill Graphs – In [50], Polhill produced 5 new strongly regular graphs on 1024 vertices as Cayley graphs. His construction is able to produce ≥ larger strongly regular graphs of order 4096, though the current imple- mentation only covers the n = 1024 range. • Mathon’s pseudo-cyclic strongly regular graphs related to symmetric con- ference matrices, optionally parameterised by a strongly regular graph with parameters of a Paley graph, and a skew-symmetric Latin square [44, 51]. • Pseudo-Paley and Pasechnik graphs from skew-Hadamard matrices. These are constructions due to Goethals-Seidel [10] and Pasechnik [48], con- − structing graphs on (4m 1)2 vertices from skew-Hadamard matrices of order 4m. Sage builds the corresponding skew-Hadamard matrices from a small database featuring classical constructions of skew-Hadamard matri- ces from [30] and small examples from (anti)-circulant matrices [24, 56]. 5.3 Novel constructions Here we collect descriptions of constructions of graphs that are in our view suﬀciently novel and interesting to mention. Namely, Sect. describes another construction for a known graph, Sect. describes a working construction for a graph which was claimed to exist in the literature, although we were unable to verify a number of published constructions (see Sect. for details). Finally, Sect. discusses an unpublished construction by Ionin and Kharaghani. 5.3.1 Cossidente-Penttila hemisystems The construction of the hemisystem in [16] requires building GQ(q2, q), 10

label: list

text: RSHCD

text: graphs

text: from

text: (n,

text: symmetric

text: Hadamard

text: matrices

text: M

text: with

text: constant

text: diagonal,

text: see

text: Example

text: for

text: the

text: deﬁnition.

text: Several

text: con-

text: structions

text: from

text: the

text: literature

text: (and

text: one

text: apparently

text: new

text: one,

text: cf.

text: Sect.

text: for

text: this

text: class

text: of

text: Hadamard

text: matrices

text: were

text: implemented

text: in

text: Sage

text: and

text: are

text: available

text: in

text: its

text: Hadamard

text: matrices

text: module.

text: Two-graph

text: descendants.

text: Each

text: regular

text: two-graph

text: (a

text: certain

text: class

text: of

text: uniform

text: v-vertex

text: hypergraphs

text: having

text: 2µ

text: three-edges

text: on

text: each

text: pair

text: of

text: points,

text: cf.

text: e.g.

text: gives

text: rise

text: to

text: a

text: strongly

text: regular

text: graph

text: with

text: parameters

text: (v

text: 3µ

text: µ)

text: obtained

text: by

text: descendant

text: construction,

text: see

text: e.g.

text: Switch

text: OA

text: Graphs

text: these

text: strongly

text: regular

text: graphs

text: are

text: obtained

text: from

text: OA

text: block

text: graphs

text: (see

text: above).

text: From

text: such

text: a

text: graph

text: G

text: obtained

text: from

text: an

text: OA(k,

text: n),

text: the

text: procedure

text: is

text: to

text: add

text: a

text: new

text: isolated

text: vertex

text: v;

text: perform

text: Seidel

text: switching

text: on

text: the

text: union

text: of

text: v

text: and

text: several

text: disjoint

text: n-cocliques

text: of

text: G.

text: Note

text: that

text: a

text: n-coclique

text: in

text: G

text: corresponds

text: to

text: a

text: parallel

text: class

text: of

text: the

text: OA(k,

text: n),

text: and

text: that

text: those

text: are

text: easily

text: obtained

text: from

text: an

text: OA(k

text: n)

text: (i.e.

text: a

text: resolvable

text: OA(k,

text: n)).

text: Polhill

text: Graphs

text: In

text: Polhill

text: produced

text: new

text: strongly

text: regular

text: graphs

text: on

text: vertices

text: as

text: Cayley

text: graphs.

text: His

text: construction

text: is

text: able

text: to

text: produce

text: larger

text: strongly

text: regular

text: graphs

text: of

text: order

text: though

text: the

text: current

text: imple-

text: mentation

text: only

text: covers

text: the

text: n

text: range.

text: Mathon’s

text: pseudo-cyclic

text: strongly

text: regular

text: graphs

text: related

text: to

text: symmetric

text: con-

text: ference

text: matrices,

text: optionally

text: parameterised

text: by

text: a

text: strongly

text: regular

text: graph

text: with

text: parameters

text: of

text: a

text: Paley

text: graph,

text: and

text: a

text: skew-symmetric

text: Latin

text: square

text: Pseudo-Paley

text: and

text: Pasechnik

text: graphs

text: from

text: skew-Hadamard

text: matrices.

text: These

text: are

text: constructions

text: due

text: to

text: Goethals-Seidel

text: and

text: Pasechnik

text: con-

text: structing

text: graphs

text: on

text: vertices

text: from

text: skew-Hadamard

text: matrices

text: of

text: order

text: Sage

text: builds

text: the

text: corresponding

text: skew-Hadamard

text: matrices

text: from

text: a

text: small

text: database

text: featuring

text: classical

text: constructions

text: of

text: skew-Hadamard

text: matri-

text: ces

text: from

text: and

text: small

text: examples

text: from

text: (anti)-circulant

text: matrices

text: Novel

text: constructions

text: Here

text: we

text: collect

text: descriptions

text: of

text: constructions

text: of

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text: COHANA MONET-M MONET-S PG-M PG-S 106 106 106 106 111111000000012345 11111110000000-1012345 111111000000012345 1111111100000000--21012345 10-1 10-2 10-1 10-3 1 2 4 8 16 32 64 1 2 4 8 16 32 64 1 2 4 8 16 32 64 1 2 4 8 16 32 64 scale scale scale scale (a) Q1 (b) Q2 (c) Q3 (d) Q4 Figure 11: Performance comparison among diﬀerent evaluation schemes 13, 15]. Techniques such as data compression [16, 18], query [4] D. Abadi, S. Madden, and M. Ferreira. Integrating processing on compressed data [4, 6, 12], array based aggre- compression and execution in column-oriented gation [5, 17], and materialized view based approaches [14] database systems. In SIGMOD, pages 671–682, 2006. are proposed for speeding up OLAP queries. Albeit target- [5] S. Agarwal, R. Agrawal, P. Deshpande, A. Gupta, ing OLAP queries which are deﬁned on relational operators J. F. Naughton, R. Ramakrishnan, and S. Sarawagi. that are generally not applicable to cohort queries, the above On the computation of multidimensional aggregates. techniques can also be used to accelerate the processing of In VLDB, pages 506–521, 1996. cohort queries as we have shown in Section . [6] S. Amer-Yahia and T. Johnson. Optimizing queries on Cohort analysis originates from social science [9]. How- compressed bitmaps. In VLDB, pages 329–338, 2000. ever, the cohort analysis approach presented in social science [7] P. A. Boncz, M. Zukowski, and N. Nes. literatures has two limitations: 1) lack of a way for specify- Monetdb/x100: Hyper-pipelining query execution. In ing a subset of users or activity tuples for analysis; 2) only CIDR, pages 225–237, 2005. use time attribute to identify cohorts. These two limitations [8] Q. Cai and K.-T. Lo. A multi-faced measurement are recognized in modern analytical software package [3, 1, study on a large private bittorrent community. 2]. These software somehow try to solve these limitations Peer-to-Peer Networking and Applications, 8(1):32–48, in their respective application domains. For example, Mix- 2015. Panel allows data analysts to select user segment for cohort [9] N. D. Glenn. Cohort Analysis. Sage Publications, Inc., analysis. But none of the solutions we investigated so far is London, 2005. general. For example, none of the software support Birth() [10] J. Gray, A. Bosworth, A. Layman, and H. Pirahesh. ﬁltering we present in this paper. An implicit cohort analysis Data cube: A relational aggregation operator is conducted in [8] to study the behavior of users in a private generalizing group-by, cross-tab, and sub-total. In BitTorrent community using the SQL approach. Compared ICDE, pages 152–159, 1996. to the above works, our eﬀort not only generalizes the co- [11] A. Hall, O. Bachmann, R. Bu¨ssow, S. Ganceanu, and hort analysis for broader spectrum of applications, but also M. Nunkesser. Processing a trillion cells per mouse is the ﬁrst attempt to extend database systems to support click. PVLDB, 5(11):1436–1446, 2012. the generalized cohort analysis. [12] Y. Li and J. M. Patel. Bitweaving: Fast scans for 7. CONCLUSIONS main memory data processing. In SIGMOD, pages 289–300, 2013. Cohort analysis is a powerful tool for ﬁnding unusual [13] S. Manegold, P. A. Boncz, and M. L. Kersten. user behavioral trends in large activity tables. This pa- Optimizing database architecture for the new per has conducted the ﬁrst investigation of database sup- bottleneck: Memory access. The VLDB Journal, port for cohort analysis. Consequently, we have introduced 9(3):231–246, 2000. an extended relation for modeling activity data and ex- [14] M. Staudt and M. Jarke. Incremental maintenance of tended SQL with three new operators for composing cohort externally materialized views. In VLDB, pages 75–86, queries. We have developed a columnar based query engine, 1996. COHAHA, for eﬃcient cohort query processing. Our ex- [15] M. Stonebraker, D. J. Abadi, A. Batkin, X. Chen, perimental results showed that COHANA can achieve two M. Cherniack, M. Ferreira, E. Lau, A. Lin, orders faster query performance than simply running SQL S. Madden, E. O’Neil, P. O’Neil, A. Rasin, N. Tran, queries over conventional database systems, demonstrating and S. Zdonik. C-store: A column-oriented dbms. In the possible beneﬁt of extending a database system for co- VLDB, pages 553–564, 2005. hort queries over implementing cohort queries on top of it. [16] T. Westmann, D. Kossmann, S. Helmer, and 8. REFERENCES G. Moerkotte. The implementation and performance of compressed databases. SIGMOD Record, [1] Retention. https://mixpanel.com/retention/. 29(3):55–67, 2000. [2] Rjmetrics. https://rjmetrics.com/. [17] Y. Zhao, P. M. Deshpande, and J. F. Naughton. 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text: REFERENCES

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