Using equality in the Krein conditions to prove nonexistence of certain distance-regular graphs

Abstract

We prove the nonexistence of a distance-regular graph with intersection array {74, 54, 15; 1, 9, 60} and of distance-regular graphs with intersection arrays {4r(3) + 8r(2) + 6r + 1, 2r(r + 1)(2r + 1), 2r(2) + 2r + 1; 1, 2r(r + 1), (2r + 1)(2r(2) + 2r + 1)} with r an integer and r >= 1. Both cases serve to illustrate a technique which can help in determining structural properties for distance-regular graphs and association schemes with a sufficient number of vanishing Krein parameters

Extremal 1-codes in distance-regular graphs of diameter 3

“…In this case c = ( p 2 − 4)/2 and by integrality of c also p = 2r for r ∈ N\{1}, which gives us precisely the intersection array (1). -p = c − 2, we obtain the feasible family (2).…”

“…For (1) and (3), the graphs with such intersection arrays have a nontrivial vanishing Krein parameter. Therefore, we can use the method of Coolsaet and Jurišić [2] to calculate some triple intersection numbers. This way we prove that such graphs indeed contain the desired codes (the family (3) actually generalizes to a two-parameter family with the same property).…”

Using Symbolic Computation to Prove Nonexistence of Distance-Regular Graphs

“…We call these numbers triple intersection numbers. They have first been studied in the case of strongly regular graphs [15], and later also for distance-regular graphs, see for example [18,36,37,38,58]. Unlike the intersection numbers, these numbers may depend on the particular choice of vertices u, v, w and not only on their pairwise distances.…”

“…However, feasibility is no guarantee for existence, so proofs of nonexistence of distance-regular graphs with feasible intersection arrays are also a contribution to the classification. In certain cases, single intersection arrays have been ruled out [40,43], while other proofs may show nonexistence for a whole infinite family of feasible intersection arrays [18,37,58]. In this paper we give proofs of nonexistence for distance-regular graphs belonging to a two-parameter infinite family, as well as for graphs with intersection arrays {135, 128, 16; 1, 16, 120}, {234, 165, 12; 1, 30, 198}, {55, 54, 50, 35, 10; 1, 5, 20, 45, 55}. We develop a package called sage-drg [60] for the Sage computer algebra system [54].…”

On Tight 4-Designs in Hamming Association Schemes

“…We then apply an analogue of the result in [25] to these subsets in the Hamming association schemes to construct another association scheme S, which, however, satisfies all known feasibility conditions. The association scheme S turns out to be Q-antipodal, and this property allows us to calculate the triple intersection numbers with respect to some triples of vertices of S. Triple intersection numbers can be thought of as a generalization of intersection numbers to triples of starting vertices instead of pairs, and, to our best knowledge, their investigation has been previously used to study strongly regular [8] and distance-regular graphs [9,13,16,17,18,26] only, but not strictly Q-polynomial association schemes. We hope that this approach will find more applications in the theory of association schemes.…”