Geometric distance-regular graphs without 4-claws

TL;DR

This paper classifies certain geometric distance-regular graphs without 4-claws, identifying conditions under which they exist and ruling out specific intersection arrays, thereby advancing understanding of their structure.

Contribution

It determines non-complete distance-regular graphs satisfying specific degree bounds and classifies geometric graphs with smallest eigenvalue -3, excluding certain intersection arrays.

Findings

Non-existence of 4-claws implies graphs are geometric with eigenvalue -3.

Classification of geometric ext{drg}s with smallest eigenvalue -3.

Seven feasible intersection arrays are ruled out.

Abstract

A non-complete \drg $\Gamma$ is called geometric if there exists a set $\mathcal{C}$ of Delsarte cliques such that each edge of $\Gamma$ lies in a unique clique in $\mathcal{C}$. In this paper, we determine the non-complete distance-regular graphs satisfying $\max \{3, 8/3}(a_1+1)\}<k<4a_1+10-6c_2$. To prove this result, we first show by considering non-existence of 4-claws that any non-complete distance-regular graph satisfying $\max \{3, \8/3}(a_1+1)\}<k<4a_1+10-6c_2$ is a geometric \drg with smallest eigenvalue -3. Moreover, we classify the geometric \drg s with smallest eigenvalue -3. As an application, 7 feasible intersection arrays in the list of \cite[Chapter 14]{bcn} are ruled out.