Feasibility of distance-regular graphs

Abstract

The problem of deciding whether a distance-regular graph with a given intersection array exists is a widely studied topic in distance-regular graphs. In 1989 Brouwer, Cohen and Neumaier have compiled a list of intersection arrays that passed known feasibility conditions, but the existence of corresponding distance-regular graphs were unknown for many of those arrays. Since then the arrays from the list are studied and the existence and nonexistence of distance-regular graphs associated to many arrays from the list are proved but more than half are still unknown.

In this thesis, we study three intersection arrays from the list, {22, 16, 5; 1, 2, 20}, {27, 20, 10; 1, 2, 18}, and {36, 28, 4; 1, 2, 24}. We prove that distance-regular graphs with these intersection arrays do not exist. To prove these, we assume that such graphs exist and derive some combinatorial properties of their local graphs to get contradictions.

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