On a family of highly regular graphs by Brouwer, Ivanov, and Klin

Abstract

Highly regular graphs for which not all regularities are explainable by symmetries are fascinating creatures. Some of them like, e.g., the line graph of W. Kantor's non-classical GQ(5 2 , 5), are stumbling stones for existing implementations of graph isomorphism tests. They appear to be extremely rare and even once constructed it is difficult to prove their high regularity. Yet some of them, like the McLaughlin graph on 275 vertices and Ivanov's graph on 256 vertices are of profound beauty. This alone makes it an attractive goal to strive for their complete classification or, failing this, at least to get a deep understanding of them. Recently, one of the authors discovered new methods for proving high regularity of graphs. Using these techniques, in this paper we study a classical family of strongly regular graphs, originally discovered by A.E. Brouwer, A.V. Ivanov, and M.H. Klin in the late 80s. We analyze their symmetries and show that they are (3, 5)-regular but not 2-homogeneous. Thus we promote these graphs to the distinguished club of highly regular graphs with few symmetries.We are usually not so much interested into regularities for particular graph types but rather for whole classes.Definition 1.2. A graph is called • ( = m, = n)-regular, if it is T-regular for each graph type T of order (m, n), • ( = m, n)-regular, if it is ( = m, = l)-regular for all m ≤ l ≤ n, • (m, n)-regular, if it is ( = k, n)-regular for all k ≤ m.2010 Mathematics Subject Classification. 05E30(51A50).

On highly regular strongly regular graphs

“…Recently, in [48] a classical family of strongly regular graphs originally constructed by Brouwer, Ivanov, and Klin (see [7]) was analyzed for regularities. It was shown there that these graphs are (3, 5)-regular but not 2-homogeneous.…”

Strongly regular graphs satisfying the 4-vertex condition

“…The latter is proved in Appendix A. In [29] it is announced that Σ (m) is even (3, 5)-regular, but we are not aware of a proof in print.…”

“…In [29] it is shown that the graphs Γ (m) are triplewise 5-regular, a.k.a. (3,5)-regular, where (s, t)-regularity is the analog of the t-vertex condition where s instead of two vertices are distinguished.…”