On a Koolen -- Park inequality and Terwilliger graphs

TL;DR

This paper characterizes certain distance-regular graphs that meet a specific inequality bound, identifying them as well-known graphs like the icosahedron, Doro, or Conway-Smith graphs.

Contribution

It proves that graphs attaining equality in Koolen-Park's inequality with $c_2 \\ge 2$ are exactly the icosahedron, Doro, or Conway-Smith graphs.

Findings

Identifies the specific graphs meeting the equality condition

Extends Koolen-Park inequality to classify certain Terwilliger graphs

Provides a characterization of graphs with $c_2 \\ge 2$ in this context

Abstract

J.H. Koolen and J. Park have proved a lower bound for intersection number $c_2$ of a distance-regular graph $\Gamma$. Moreover, they showed that the graph $\Gamma$ which attains the equality in this bound is a Terwilliger graph. We prove that $\Gamma$ is the icosahedron, the Doro graph or the Conway-Smith graph, if equality is attained and $c_2\ge 2$.