The Nonexistence of a Distance-Regular Graph with Intersection Array {22,16,5;1,2,20}

Supalak Sumalroj,Chalermpong Worawannotai

doi:10.37236/5486

Abstract

We prove that a distance-regular graph with intersection array {22,16,5;1,2,20} does not exist. To prove this, we assume that such a graph exists and derive some combinatorial properties of its local graph. Then we construct a partial linear space from the local graph to display the contradiction.

Highlights

We prove that a distance-regular graph with intersection array {22, 16, 5; 1, 2, 20} does not exist
One of the main problems in distance-regular graphs is to decide whether a distanceregular graph with a given intersection array exists
Cohen and Neumaier [3] have compiled a list of intersection arrays that passed known feasibility conditions, but the existence of the corresponding distance-regular graphs was unknown for many of those arrays

Summary

Introduction

One of the main problems in distance-regular graphs is to decide whether a distanceregular graph with a given intersection array exists. By Lemma 7 and since ∆ is a regular graph of degree 5, it follows that each vertex of S is adjacent to at least 3 vertices of R, that is, each line in S is incident with at least 3 points in R.

Results

Conclusion