The spectral excess theorem for distance-regular graphs having distance-d graph with fewer distinct eigenvalues

2014/09/17 by M.A. Fiol, M. A. Fiol, Fiol, M. A.
Mathematics · #05C50 #05E30 #Combinatorics (math.CO) #FOS: Mathematics #Finite Group Theory Research #Graph theory and applications #Spectral Theory in Mathematical Physics #math.CO #msc:05C50 #msc:05E30

paper · pdf · doi:10.48550/arxiv.1409.5146

arxiv created 2014/09/17 · openalex publication_date 2014/09/17 · arxiv updated 2014/09/19 · openalex created_date 2019/06/27 · openalex updated_date 2026/07/28

Abstract

Let Γ be a distance-regular graph with diameter d and Kneser graph K=Γ_d, the distance-d graph of Γ. We say that Γ is partially antipodal when K has fewer distinct eigenvalues than Γ. In particular, this is the case of antipodal distance-regular graphs (K with only two distinct eigenvalues), and the so-called half-antipodal distance-regular graphs (K with only one negative eigenvalue). We provide a characterization of partially antipodal distance-regular graphs (among regular graphs with d distinct eigenvalues) in terms of the spectrum and the mean number of vertices at maximal distance d from every vertex. This can be seen as a general version of the so-called spectral excess theorem, which allows us to characterize those distance-regular graphs which are half-antipodal, antipodal, bipartite, or with Kneser graph being strongly regular.

Related

• The spectral excess theorem for graphs with few eigenvalues whose distance-2 or distance-1-or-2 graph is strongly regular
• A spectral characterization of strongly distance-regular graphs with diameter four
• On some approaches to the spectral excess theorem for nonregular graphs
• The spectral excess theorem for distance-biregular graphs
• On a version of the spectral excess theorem