Equitable Decompositions of Graphs
Wayne Barrett, Amanda Francis, Ben Webb

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 , it is possible to use to decompose any matrix 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 . 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…
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Taxonomy
TopicsNonlinear Dynamics and Pattern Formation · Complex Network Analysis Techniques · Graph theory and applications
