Edge-distance-regular graphs are distance-regular

TL;DR

This paper proves that edge-distance-regular graphs are necessarily distance-regular and homogeneous, characterizing them as bipartite distance-regular graphs or generalized odd graphs, and explores their parameter relationships.

Contribution

It provides combinatorial and algebraic proofs that edge-distance-regular graphs are exactly bipartite distance-regular or generalized odd graphs, establishing their fundamental properties.

Findings

Edge-distance-regular graphs are distance-regular and homogeneous.

Such graphs are characterized as bipartite distance-regular or generalized odd graphs.

Relationships between parameters like distance polynomials and intersection numbers are established.

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 $\G$ is distance-regular and homogeneous. More precisely, $\G$ 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.