Equitable Decompositions of Graphs

TL;DR

This paper explores how symmetries in graphs allow for equitable decompositions of associated matrices, enabling spectral analysis and bounds on eigenvalues, which are useful for understanding network dynamics.

Contribution

It introduces equitable decomposition techniques for graphs with symmetries, linking spectral properties to automorphisms and providing bounds on eigenvalues and spectral gaps.

Findings

Decomposition preserves eigenvalues across smaller matrices.

Bounds on spectral radius and spectral gap are derived.

Results include sharp bounds on the number of simple eigenvalues.

Abstract

We investigate connections between the symmetries (automorphisms) of a graph and its spectral properties. Whenever a graph has a symmetry, i.e. a nontrivial automorphism $\phi$, it is possible to use $\phi$ to decompose any matrix $M\in\mathbb{C}^{n \times n}$ appropriately associated with the graph. The result of this decomposition is a number of strictly smaller matrices whose collective eigenvalues are the same as the eigenvalues of the original matrix $M$. Some of the matrices that can be decomposed are the graph's adjaceny matrix, Laplacian matrix, etc. Because this decomposition has connections to the theory of equitable partitions it is referred to as an equitable decomposition. Since the graph structure of many real-world networks is quite large and has a high degree of symmetry, we discuss how equitable decompositions can be used to effectively bound both the network's spectral radius and spectral gap, which are associated with dynamic processes on the network. Moreover, we show that the techniques used to equitably decompose a graph can be used to bound the number of simple eigenvalues of undirected graphs, where we obtain sharp results of Petersdorf-Sachs type.