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Title: Spectral results on graphs with regularity constraints
Author: Cardoso, D.M.
Rama, P.
Keywords: Equitable partitions
Graph spectra
Graph theory
Strongly regular graphs
Constrained optimization
Eigenvalues and eigenfunctions
Matrix algebra
Issue Date: 2007
Publisher: Elsevier
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 (k1, τ1)-regular set S1 and a (k2, τ2)-regular set S2 such that k1 - τ1 = k2 - τ2 = 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, over(π, ̄), induced by π, and the adjacency matrix of the quotient graph L (G) / over(π, ̄) is obtained from the adjacency matrix of G/π. © 2006 Elsevier Inc. All rights reserved.
Peer review: yes
ISSN: 0024-3795
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