Edge-distance-regular graphs are distance-regular

Abstract

A graph is edge-distance-regular when it is distance-regular around each of its edges and it has the same intersection numbers for any edge taken as a root. In this paper we give some (combinatorial and algebraic) proofs of the fact that every edge-distance-regular graph Γ is distance-regular and homogeneous. More precisely, Γ is edge-distance-regular if and only if it is bipartite distance-regular or a generalized odd graph. Also, we obtain the relationships between some of their corresponding parameters, mainly, the distance polynomials and the intersection numbers.Postprint (published version

Completely regular clique graphs, II

“…By the result in [6], it follows from Theorem 1 that a graph Γ = (X, R) is a completely regular clique graph with parameters (1, 1) if and only if Γ is a distance-regular graph with a 1 = a 2 = · · · = a D−1 = 0. Hence bipartite distance-regular graphs and almost bipartite distance-regular graphs are the only distance-regular completely regular clique graphs if a 1 = 0.…”

A survey on the missing Moore graph

“…First, recall that a graph is edge-distance-regular when the distance partition induced by each edge is equitable, and with the same intersection numbers independently of the edge (see Cámara, Dalfó, Delorme, Fiol, and Suzuki [6]). A generalized odd graph is a distance-regular graph with diameter d and odd-girth (that is, the shortest cycle of odd length) 2d + 1.…”

“…A generalized odd graph is a distance-regular graph with diameter d and odd-girth (that is, the shortest cycle of odd length) 2d + 1. In [6] it is was proved that a graph is edge-distance-regular if and only if it is a bipartite distance-regular graph or a generalized odd graph. As a consequence, we show the first part of the next theorem, whereas the second part is due to Brouwer, Cohen, and Neumaier [3, Theorem 4.…”