Laplacian spectral characterization of disjoint union of paths and cycles

The Laplacian spectrum of a graph consists of the eigenvalues (together with multiplicities) of the Laplacian matrix. In this article we determine, among the graphs consisting of disjoint unions of paths and cycles, those ones which are determined by the Laplacian spectrum. For the graphs, which are not determined by the Laplacian spectrum, we give the corresponding cospectral non-isomorphic graphs.     

Uniform random colored complexes

We present here random distributions on (D + 1)‐edge‐colored, bipartite graphs with a fixed number of vertices 2p. These graphs encode D‐dimensional orientable colored complexes. We investigate the behavior of those graphs as p→∞. The techniques involved in this study also yield a Central Limit Theorem for the genus of a uniform map of order p, as p→∞.     

A new arithmetic criterion for graphs being determined by their generalized Q-spectrum

“Which graphs are determined by their spectrum (DS for short)?” is a fundamental question in spectral graph theory. It is generally very hard to show a given graph to be DS and few results about DS graphs are known in literature. In this paper, we consider the above problem in the context of the generalized $Q$-spectrum. A graph $G$ is said to be determined by the generalized $Q$-spectrum (DGQS for short) if, for any graph $H$, $H$ and $G$ have the same $Q$-spectrum and so do their complements imply that $H$ is isomorphic to $G$. We give a simple arithmetic condition for a graph being DGQS. More precisely, let $G$ be a graph with adjacency matrix $A$ and degree diagonal matrix $D$. Let $Q=A+D$ be the signless Laplacian matrix of $G$, and $W_Q\left(G\right)=[e,Qe,\dots ,Q^{n-1}e]$ ($e$ is the all-ones vector) be the $Q$-walk matrix. We show that if $\frac{det{W}_Q\left(G\right)}{2^{\lfloor \frac{3n-2}{2}\rfloor }}$ (which is always an integer) is odd and square-free, then $G$ is DGQS.