Pseudo-distance-regularised graphs are distance-regular or distance-biregular

TL;DR

This paper proves that graphs with pseudo-distance-regularity around each vertex are necessarily either distance-regular or distance-biregular, providing an algebraic alternative to a known combinatorial proof.

Contribution

It establishes a new algebraic proof that pseudo-distance-regular graphs are either distance-regular or distance-biregular, extending prior combinatorial results.

Findings

Pseudo-distance-regular graphs are either distance-regular or distance-biregular.

Provides an algebraic proof as an alternative to existing combinatorial proofs.

Extends understanding of graph regularity properties.

Abstract

The concept of pseudo-distance-regularity around a vertex of a graph is a natural generalization, for non-regular graphs, of the standard distance-regularity around a vertex. In this note, we prove that a pseudo-distance-regular graph around each of its vertices is either distance-regular or distance-biregular. By using a combinatorial approach, the same conclusion was reached by Godsil and Shawe-Taylor for a distance-regular graph around each of its vertices. Thus, our proof, which is of an algebraic nature, can also be seen as an alternative demonstration of Godsil and Shawe-Taylor's theorem.