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

Abstract

The final publication is available at Springer via http://dx.doi.org/10.1007/s10801-015-0654-6Let Gamma be a distance-regular graph with diameter d and Kneser graph K = Gamma(d), the distance-d graph of Gamma. We say that Gamma is partially antipodal when K has fewer distinct eigenvalues than Gamma. 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 + 1 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 more 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.Peer ReviewedPostprint (author's final draft

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

“…Similarly, the distance-i-or-j graph is the graph Γ i,j with the same vertices as Γ and in which two vertices are adjacent if and only if they are at distance i or j in Γ. In the recent works of Brouwer and Fiol [4,15], it was studied the situation in which the distance-d graph Γ d of Γ (or the Kneser graph K of Γ) with adjacency matrix A d (= p d (A)), where p d is the distance-d polynomial, has fewer distinct eigenvalues than Γ. Examples are the so-called half antipodal (K with only one negative eigenvalue, up to multiplicity), and antipodal distance-regular graphs (where K consists of disjoint copies of a complete graph).…”

“…Notice that, in this case, Γ 1,2 is strongly regular if and only if its complement Γ 3 is. As commented in the Introduction, the last case was studied for general diameter by Brouwer and Fiol [4] and Fiol [15]. (i) The distance-2 graph Γ 2 is strongly regular if and only if a 2 − c 3 is an eigenvalue of Γ.…”