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Highly regular graphs for which not all regularities are explainable bysymmetries are fascinating creatures. Some of them like, e.g., the line graphof W.~Kantor's non-classical $\mathrm{GQ}(5^2,5)$, are stumbling stones forexisting implementations of graph isomorphism tests. They appear to beextremely rare and even once constructed it is difficult to prove their highregularity. Yet some of them, like the McLaughlin graph on $275$ vertices andIvanov's graph on $256$ vertices are of profound beauty. This alone makes it anattractive goal to strive for their complete classification or, failing this,at least to get a deep understanding of them. Recently, one of the authorsdiscovered new methods for proving high regularity of graphs. Using thesetechniques, in this paper we study a classical family of strongly regulargraphs, originally discovered by A.E.~Brouwer, A.V.~Ivanov, and M.H.~Klin inthe late 80th. We analyze their symmetries and show that they are$(3,5)$-regular but not $2$-homogeneous. Thus we promote these graphs to thedistinguished club of highly regular graphs with few symmetries.

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