Spectral results on graphs with regularity constraints

Abstract

Graphs with ( k, τ)-regular sets and equitable partitions are examples of graphs with regularity constraints. A ( k, τ)-regular set of a graph G is a subset of vertices S ⊆ V( G) inducing a k-regular subgraph and such that each vertex not in S has τ neighbors in S. The existence of such structures in a graph provides some information about the eigenvalues and eigenvectors of its adjacency matrix. For example, if a graph G has a ( k 1, τ 1)-regular set S 1 and a ( k 2, τ 2)-regular set S 2 such that k 1 − τ 1 = k 2 − τ 2 = λ, then λ is an eigenvalue of G with a certain eigenvector. Additionally, considering primitive strongly regular graphs, a necessary and sufficient condition for a particular subset of vertices to be ( k, τ)-regular is introduced. Another example comes from the existence of an equitable partition in a graph. If a graph G, has an equitable partition π then its line graph, L( G), also has an equitable partition, π ¯ , induced by π, and the adjacency matrix of the quotient graph L ( G ) / π ¯ is obtained from the adjacency matrix of G/ π.