The spectral excess theorem for graphs with few eigenvalues whose distance-2 or distance-1-or-2 graph is strongly regular

2016/08/01 by C. Dalfó, M.A. Fiol, Dalfó, C. +3
Mathematics · #Graph theory and applications #Finite Group Theory Research #Spectral Theory in Mathematical Physics

paper · doi:10.48550/arxiv.1608.00373

Abstract

We study regular graphs whose distance-2 graph or distance-1-or-2 graph is strongly regular. We provide a characterization of such graphs Γ (among regular graphs with few distinct eigenvalues) in terms of the spectrum and the mean number of vertices at maximal distance d from every vertex, where d+1 is the number of different eigenvalues of Γ. This can be seen as a another version of the so-called spectral excess theorem, which characterizes in a similar way those regular graphs that are distance-regular.

Related

• The spectral excess theorem for distance-regular graphs having distance-d graph with fewer distinct eigenvalues
• On some approaches to the spectral excess theorem for nonregular graphs
• On a version of the spectral excess theorem
• The distance-regular graphs such that all of its second largest local eigenvalues are at most one
• The spectral excess theorem for distance-biregular graphs