Enumeration of Cospectral Graphs

AMS classification: 05C50;graphs;eigenvalues;enumeration

Enumeration of Cospectral Graphs

AMS classification: 05C50

Cospectral Graphs and the Generalized Adjacency Matrix

AMS classifications: 05C50; 05E99;cospectral graphs;generalized spectrum;generalized adjacency matrix

Enumeration of cospectral and coinvariant graphs

We present enumeration results on the number of connected graphs up to 10 vertices for which there is at least one other graph with the same spectrum (a cospectral mate), or at least one other graph with the same Smith normal form (coinvariant mate) with respect to several matrices associated to a graph. The present data give some indication that possibly the Smith normal form of the distance Laplacian and the signless distance Laplacian matrices could be a finer invariant to distinguish graphs in cases where other algebraic invariants, such as those derived from the spectrum, fail. Finally, we show a new graph characterization using the Smith normal form of the signless distance Laplacian matrix

Developments on Spectral Characterizations of Graphs

In [E.R. van Dam and W.H. Haemers, Which graphs are determined by their spectrum?, Linear Algebra Appl. 373 (2003), 241-272] we gave a survey of answers to the question of which graphs are determined by the spectrum of some matrix associated to the graph. In particular, the usual adjacency matrix and the Laplacian matrix were addressed. Furthermore, we formulated some research questions on the topic. In the meantime some of these questions have been (partially) answered. In the present paper we give a survey of these and other developments.2000 Mathematics Subject Classification: 05C50Spectra of graphs;Cospectral graphs;Generalized adjacency matrices;Distance-regular graphs