New strongly regular graphs from finite geometries via switching

Abstract

We show that the strongly regular graph on non-isotropic points of one type of the polar spaces of type U (n, 2), O(n, 3), O(n, 5), O + (n, 3), and O − (n, 3) are not determined by its parameters for n ≥ 6. We prove this by using a variation of Godsil-McKay switching recently described by Wang, Qiu, and Hu. This also results in a new, shorter proof of a previous result of the first author which showed that the collinearity graph of a polar space is not determined by its spectrum. The same switching gives a linear algebra explanation for the construction of a large number of non-isomorphic designs.

Switching for Small Strongly Regular Graphs

“…The aim of this paper is to provide a large collection of SRGs which can be generated by WQH switching with a partition of type 2 2 , n − 4 or 3 2 , n − 6. Note that WQH switching with a partition of type 2 2 , n − 4 corresponds to Godsil-McKay switching as remarked in [13]. It was shown by Abiad and Haemers [1] (for q = 2), and the author [12,13] (for all q) that this technique works for certain families of SRGs.…”

Strongly regular graphs satisfying the 4-vertex condition

“…This is an example of WQH-switching (Wang, Qiu & Hu [33], cf. [19]) and yields a graph cospectral with Γ 0 . One can repeat this interchange of hyperplanes and get arbitrary permutations of all hyperplanes.…”

“…Further graphs with the same parameters satisfy the 4-vertex condition. Additional examples can be obtained by repeated WQH-switching, see §7.4 and [19], and there are more examples among the graphs constructed in [18]. We have not tried (much) to determine precisely which graphs in [18] do satisfy the 4-vertex condition.…”

A general construction of strictly Neumaier graphs and a related switching

“…The following switching, which produces cospectral graphs, was discovered in [18] and applied in [14] to obtain new stongly regular graphs.…”

“…After that, we discuss small strictly Neumaier graphs obtained from the general construction and give a geometric description for some of them. Finally, we apply a variation of the Godsil-McKay switching (see [14] and [18]) to the graphs from the general construction of strictly Neumaier graphs.…”