TITLE: Weighted Regular Graphs
QUESTION [8 upvotes]: The following graph theoretic notion appeared in an economics paper entitled: "Prize competition under limited comparability, by Michele Piccione and Ran Spiegler which studies models of economics were the firms are rational but the consumers are not. 
A graph is weighted-regular if vertices (nodes) can be assigned positive weights, such that each node has the same total weighted-expected-number of edges (links). Every regular graph is, of course, weighted regular (give all vertices weight 1). A star, tree with center vertex connected to k leaves is weighted-regular: Give the center weight 1 and every leaf weight 1/k.
(In the paper, every vertex is also contained in a loop. Therefore its weight also counts when we compute its weighted-expected-number of edges. The star is still weighted-regular: give weight 1 to the center and weight 0 to the leafs.)
Is there a characterization of weighted-regular graphs? Is this notion known? Does it have known applications or connections?

REPLY [2 votes]: Denoting by $A$ the $n\times n$ adjaceny matrix (with loops contributing $1$ 
on the diagonal) of a finite graph $\Gamma$ with $n$ vertices, a necessary condition for $A$ to be weighted-regular is the fact that the all $1$ vector 
$I_n=(1,1,\dots,1)$ of dimension $n$ is in the image of $A$. This is the case if and
only if $I_n$ is orthogonal to the kernel $\ker(A)$ of $A$.
If this necessary condition holds, the vector $I_n$ can be written uniquely
as
$$I_n=\sum_{\lambda\in{\mathrm Spec}(A)\setminus\{0\}}v_\lambda$$
where the sum is over all distinct non-zero eigenvalues of $A$ with 
$v_\lambda$ denoting the orthogonal projection of $I_n$
onto the eigenspace of eigenvalue $\lambda$.
The graph $\Gamma$ is now weighted-regular if and only if 
the affine subspace $\sum_{\lambda}\frac{1}{\lambda}v_\lambda+\ker(A)$
(corresponding to the set of preimages of $I_n$) 
intersects the open cone $({\mathbb R}_{>0})^n$
of vectors having only strictly positive coordinates.
This last condition can be checked by linear programming by 
optimizing an arbitrary linear functional (eg. the coordinate sum)
on $\ker(A)$ subject to the condition that all coefficients of 
the solution are strictly greater than the corresponding coefficients of 
$I_n-\sum_{\lambda}\frac{1}{\lambda}v_\lambda$.
In particular, a graph $\Gamma$ with invertible adjacency matrix $A$ is 
weighted-regular if and only if all coordinates of the vector 
$\sum_\lambda \frac{1}{\lambda}v_\lambda$ are strictly positive and these coordinates
give the unique set of weights (up to multiplication by a positive constant) turning 
$\Gamma$ into a weighted-regular graph.
This is of course nothing else than Kristall Cantwell's remark, fleshed out.