Enumeration of cospectral and coinvariant graphs

TL;DR

This paper enumerates connected graphs up to 10 vertices that share spectra or Smith normal forms with other graphs, suggesting the Smith normal form of certain matrices may better distinguish non-isomorphic graphs.

Contribution

It provides the first enumeration data for cospectral and coinvariant graphs up to 10 vertices and proposes the Smith normal form as a finer graph invariant.

Findings

Smith normal form may distinguish graphs where spectra fail

Enumeration data for graphs up to 10 vertices

New characterization of graphs using Smith normal form

Abstract

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.