A $Q$-polynomial structure associated with the projective geometry $L_N(q)$

TL;DR

This paper explores a generalized $Q$-polynomial property in graphs related to projective geometry $L_N(q)$, extending the concept beyond distance-regular graphs and providing detailed examples.

Contribution

It introduces a generalized $Q$-polynomial structure for graphs not necessarily distance-regular, linked to the projective geometry $L_N(q)$, with detailed examples.

Findings

Identification of a generalized $Q$-polynomial property

Detailed example involving projective geometry $L_N(q)$

Extension of $Q$-polynomial concepts beyond distance-regular graphs

Abstract

There is a type of distance-regular graph, said to be $Q$-polynomial. In this paper we investigate a generalized $Q$-polynomial property involving a graph that is not necessarily distance-regular. We give a detailed description of an example associated with the projective geometry $L_N(q)$.