Nonexistence of a Class of Distance-Regular Graphs

Yu-Pei Huang,Yeh-Jong Pan,Chih-Wen Weng

doi:10.37236/3356

Abstract

Let $\Gamma$ denote a distance-regular graph with diameter $D \geq 3$ and intersection numbers $a_1=0, a_2 \neq 0$, and $c_2=1$. We show a connection between the $d$-bounded property and the nonexistence of parallelograms of any length up to $d+1$. Assume further that $\Gamma$ is with classical parameters $(D, b, \alpha, \beta)$, Pan and Weng (2009) showed that $(b, \alpha, \beta)= (-2, -2, ((-2)^{D+1}-1)/3).$ Under the assumption $D \geq 4$, we exclude this class of graphs by an application of the above connection.

Highlights

Let Γ = (X, R) be a distance-regular graph with diameter D 3
We show a connection between the d-bounded property and the nonexistence of parallelograms of any length up to d + 1
Let Γ denote a distance-regular graph with diameter D the intersection numbers a1, a2, c2 satisfy one of the following

Summary

Introduction

∆ is distance-regular with intersection numbers ai(∆) = ai(Γ) and ci(∆) = ci(Γ) for 1 i d [14, Theorem 4.6]. Γ is said to be d-bounded whenever for all x, y ∈ X with ∂(x, y) d, there is a regular strongly closed subgraph of diameter ∂(x, y) which contains x and y. Other applications of D-bounded distance-regular graphs are given in [3, 12, 13, 15]. Let Γ denote a distance-regular graph with diameter D the intersection numbers a1, a2, c2 satisfy one of the following. Let Γ = (X, R) denote a distance-regular graph with diameter D 3, and intersection numbers a1 = 0, a2 = 0 and c2 = 1. Suppose Γ is a distance-regular graph with diameter D 3 and the intersection number a2 = 0.

Preliminaries

The Shape of Pentagons

D-bounded Property and Nonexistence of Parallelograms

Classical parameters