An example of using star complements in classifying strongly regular graphs

Abstract

In this paper we show how the star complement technique can be used to reprove the result of Wilbrink and Brouwer that the strongly regular graph with parameters (57, 14, 1, 4) does not exist.

There is no (75,32,10,16) strongly regular graph

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We show that there is no (75, 32,10,16) strongly regular graph. The result is obtained by a mix of algebraic and computational approaches.

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“…The state of affairs for all possible parameters on up to 1300 vertices is tracked by Brouwer on his web site [5]. It can be seen that on up to 100 vertices there are essentially 15 parameters whose classification is still open, the smallest three being (65,32,15,16), (69,20,7,5), and (75,32,10,16). We found the parameters of the last one the most intriguing since the existence of the SRG with (75, 32, 10, 16) is connected to the existence of certain so called two-graphs, and referred in [8, pp.…”

There is no (95, 40, 12, 20) strongly regular graph

“…Since all possible candidates for a star complement have compatibility graphs without cliques of order 75, the graph X cannot exist. See for a general overview of the star complement technique, for an application of this approach for classifying (57, 14, 1, 4) SRG's, and for an extended explanation and a proof of nonexistence of a (75, 32, 10, 16) SRG.…”

“…The theory of star complements then guarantees that X does not exist. See [9] for a general overview of the star complement technique and [14] for an application of this approach for classifying (57, 14, 1, 4) SRG's.…”

Graphs with least eigenvalue −2: Ten years on

“…On the other hand, star complements can be used to show that there is no strongly regular graph with parameters (57,14,1,4) [76,Theorem 4]. This last result requires (i) inspection of several possibilities for a star complement H for the eigenvalue 2, (ii) a computer search for maximal cliques in the associated compatibility graphs.…”