Quotient-polynomial graphs

TL;DR

This paper introduces quotient-polynomial graphs, a generalization of orbit-polynomial and distance-regular graphs, exploring their properties, characterizations, and their connection to association schemes.

Contribution

It defines quotient-polynomial graphs, studies their properties, and shows they generate symmetric association schemes, extending the understanding of graph regularity and algebraic structure.

Findings

All quotient-polynomial graphs are walk-regular.

They are distance-polynomial.

They generate symmetric association schemes.

Abstract

As a generalization of orbit-polynomial and distance-regular graphs, we introduce the concept of a quotient-polynomial graph. In these graphs every vertex $u$ induces the same regular partition around $u$, where all vertices of each cell are equidistant from $u$. Some properties and characterizations of such graphs are studied. For instance, all quotient-polynomial graphs are walk-regular and distance-polynomial. Also, we show that every quotient-polynomial graph generates a (symmetric) association scheme.