A short proof of the odd-girth theorem

TL;DR

This paper provides a more direct proof of the odd-girth theorem for graphs, avoiding the spectral excess theorem by using a characterization involving predistance polynomials.

Contribution

It introduces an alternative, more straightforward proof of the odd-girth theorem based on predistance polynomials, simplifying previous approaches.

Findings

The proof confirms the odd-girth theorem without spectral excess theorem.

It demonstrates the utility of predistance polynomial characterization.

The approach simplifies understanding of distance-regular graphs.

Abstract

Recently, it has been shown that a connected graph $\Gamma$ with $d+1$ distinct eigenvalues and odd-girth $2d+1$ is distance-regular. The proof of this result was based on the spectral excess theorem. In this note we present an alternative and more direct proof which does not rely on the spectral excess theorem, but on a known characterization of distance-regular graphs in terms of the predistance polynomial of degree $d$.