TITLE: A family of skew-symmetric matrices corresponding to cycles in graphs
QUESTION [8 upvotes]: When investigating loops in Markov chains I ran into the following observation.
A cycle in a graph $G$ with $n$ vertices may be represented by a matrix $\Gamma \in \mathbb R^{n \times n}$ having the following properties:
(i) $\Gamma = -\Gamma^T$.
(ii) $\Gamma \mathbb 1 = 0$, i.e. $\sum_{j = 1}^n \Gamma(i,j) = 0$ for all $i =1, \dots, n$.
(iii) $\Gamma(i,j) = 0$ whenever there is no edge between vertices $i$ and $j$.
(iv) $\Gamma \neq 0$.
The simplest example of such a matrix is $\Gamma = \begin{pmatrix} 0 & 1 & -1 \\ -1 & 0 & 1 \\ 1 & -1 & 0 \end{pmatrix}$.
The representation mentioned above is made more precise by the following proposition.
Proposition
Let $G$ be a graph over $n$ vertices and let $\Gamma$ satisfy the conditions (i)-(iv) above. Then there exists a cycle, i.e. a non-empty sub-graph $C = (V_C, E_C)$ with $V_C = \{ x_0, x_1, \dots, x_k\}$ and $E_C = \{x_0 x_1, x_1 x_2, \dots, x_k x_0 \}$, such that for every $x_i x_j \in E_C$, $\Gamma(x_i, x_j) > 0$.
Conversely, if there exists a non-empty cycle $C = (V_C, E_C)$, then there exists a matrix $\Gamma$ satisfying (i)-(iv), and $\Gamma(i,j) > 0$ for $(i,j)\in E_C$.
Proof:
"$\Rightarrow$": Since $\Gamma \neq 0$ and $\Gamma$ is skew-symmetric, there exists a pair $(i_0,i_1)$, $i_1 \neq i_0$, such that $\Gamma(i_0, i_1) > 0$. Since $\Gamma$ is skew-symmetric, $\Gamma(i_1, i_0) < 0$, and because rows sum to zero, there must be a positive element on row $i_1$. Suppose this is at position $(i_1, i_2)$. Again $i_2 \neq i_1$. We may repeat this procedure until we encounter a node $i_k$ that we already obtained (which will surely happen within $n-1$ steps). If this vertex is $i_0$, we are done. If this vertex is $i_l = i_k$ for some $0 < l < k - 1$ (note $l=k$ is impossible by skew-symmetry), we obtain a cycle $\{x_{i_l}, x_{i_{l+1}}, \dots, x_k\}$ with the required properties by removing vertices $i_0, \dots, i_{l-1}$.
"$\Leftarrow$": Let the entries of $\Gamma(i,j) = 1$ and $\Gamma(j,i) = -1$ whenever there is a edge between $i$ and $j$ in the directed cycle $x_0, x_1, \dots, x_k$, and $\Gamma(i,j) = 0$ otherwise. For any $i$, $\sum_{j=1}^n \Gamma(i,j) = \sharp \{\mbox{directed edges out of $i$}\} - \sharp \{\mbox{directed edges into $i$}\} = 0$, so that (ii) is satisfied. The other conditions (i), (iii), (iv) are clearly satisfied. $\square$
This result seems quite basic to me, but I have trouble finding references in the literature. What would a matrix satisfying (i)-(iv) be called? I would like to understand the structure of the set of matrices satisfying (i)-(iv) for particular adjacency structures of graphs. Also results on the spectra of such matrices might prove helpful. Basically anything related to matrices satisfying (i)-(iv) would be of interest to me.
Note: in the (applied) literature on Markov chains I have found one other reference to these matrices. This is Sun, Gomez, Schmidhuber, Improving the asymptotic performance of Markov chains by inserting vortices, 2010. They prove a couple of interesting results related to such matrices but do not provide a reference to any literature on this topic. The notion of skew-adjacency matrices (see e.g. Cavers et al, Linear Algebra and its Applications, 2012) seems related but is different. (In particular, a matrix satisfying (i)-(iv) and containing only 0 and +/- 1 entries is a skew adjacency matrix for the cycle it represents, but in general not for the original graph containing such a cycle.)

It would help a lot if any of you could provide a reference for this kind of theory. Many thanks in advance.

REPLY [2 votes]: The object you're interested in is an element of the kernel of the combinatorial divergence operator whose definition you give in (ii).  But there's quite a nice theory of these kind of things that may be in the spirit of what you (were, I see this is from 2014) after.  In what follows, rather than have your matrix be all zeroes and $\pm1$, I'm going to generalize and allow it to have any real numbers.  
Let $X$ be a finite set of vertices and consider the following three spaces for $k \in \{0,1,2\}$:
$$
\Omega^k(X) = \big\{\phi : X^{k+1} \to \mathbb{R} \; | \; \phi(x_{\sigma(0)}, \ldots, x_{\sigma(k)}) = \textrm{sign}(\sigma) \phi(x_0,\ldots, x_k)\big \}$$
where $\sigma$ is a permutation of $\{0, \ldots, k\}$. The space $\Omega^0$ is the space of real valued functions from your vertex set and so isomorphic to $\mathbb{R}^{|X|}$.  The space $\Omega^1$ is the set of edge flows on the complete graph on $|X|$ vertices (where if we simply don't want an edge, we give it a zero flow weight). The permutation condition implies these are skew symmetric $|X| \times |X|$ matrices.  The space $\Omega^2$ is the space of all triangular flows on the graph (an element of it is an alternating 3-tensor, or, a skew symmetric $|X| \times |X| \times |X|$ hypermatrix).
We may also define some natural linear maps between these spaces that are natural combinatorial analogues of the classic vector calculus operations. The gradient is a map:
$$
\textrm{grad}: \Omega^0 \to \Omega^1
$$  
via $\textrm{grad}(a)_{ij} = a_j - a_i$.  Similarly, the curl is a map:
$$
\textrm{curl}: \Omega^1 \to \Omega^2
$$
via $\textrm{curl}(A)_{ijk} = A_{ij} + A_{jk} + A_{ki}$.
Note that $\Omega^0$ inherits an inner product from $\mathbb{R} ^{|X|}$. 
 Define a nice simple inner product structure on $\Omega^1$ via:
$$
\langle A, B \rangle_1 = \sum_{\{i,j\} \in E} A_{ij} B_{ij}
$$
where $E$ is just the edge set of your graph.  Similarly for $\Omega^2$:
$$
\langle\mathcal{A}, \mathcal{B}\rangle_2 = \sum_{\{ijk\} \in T} \mathcal{A}_{ijk} \mathcal{B}_{ijk}.
$$
where $T$ is the set of triangles.  Now using these we can construct the adjoints of $\textrm{grad}$ and $\textrm{curl}$:
$$
\textrm{grad}^* = -\textrm{div} = -\sum_{j \in X} A_{ij}
$$
$$
\textrm{curl}^* = \sum_{k \in X} \mathcal{A}_{ijk}.
$$
Then we have the two sequences:
$$
\Omega^0(X) \xrightarrow{\text{grad}} \Omega^1(X) \xrightarrow{\text{curl}} \Omega^2(X)
$$
and
$$
\Omega^0(X) \xleftarrow{\text{  grad}^*} \Omega^1(X) \xleftarrow{\text{  curl}^*}  \Omega^2(X).
$$

Theorem (Combinatorial Hodge/Helmholtz Decomposition): $\Omega^1$ admits an orthogonal decomposition as:
  $$
\Omega^1 = \textrm{im}(\textrm{grad}) \oplus \underbrace{\textrm{ker}(\textrm{curl}) \cap \textrm{ker}(\textrm{div}) \oplus \textrm{im}(\textrm{curl}^*)}_{= \textrm{ker}(div)}.
$$

Elements in the first component are precisely those flows that admit a potential function; flows in $\textrm{ker}(\textrm{div})$ are precisely those in your example: they are flows with everywhere zero divergence, and hence are sums of cycles and occupy a linear subspace orthogonal to those flows that admit a potential function.