An odd characterization of the generalized odd graphs

TL;DR

This paper proves that connected regular graphs with specific spectral and girth properties are necessarily distance-regular, establishing a new characterization of generalized odd graphs based on eigenvalues and odd-girth.

Contribution

It provides a novel spectral characterization of generalized odd graphs, linking eigenvalues, girth, and distance-regularity in a unified framework.

Findings

Connected regular graphs with $d+1$ eigenvalues and odd-girth $2d+1$ are distance-regular.

Such graphs are characterized as generalized odd graphs.

The result bridges spectral properties and combinatorial structure in graph theory.

Abstract

We show that any connected regular graph with $d+1$ distinct eigenvalues and odd-girth $2d+1$ is distance-regular, and in particular that it is a generalized odd graph.