A switching for all strongly regular collinearity graphs from polar spaces

Abstract

We describe a general construction of strongly regular graphs from the collinearity graph of a finite classical polar spaces of rank at least 3 over a finite field of order q. We show that these graphs are non-isomorphic to the collinearity graphs and have the same parameters. To our knowledge for most of these parameters these graphs are new as the collinearity graphs were the only known examples.

New strongly regular graphs from finite geometries via switching

“…This note provides an algebraic explanation for the construction given in [15] by providing a specific family of orthogonal matrices Q. This gives rise to the following new switching which was recently discovered -in a slightly more general form -by Wang, Qiu, and Hu [25].…”

“…We apply Theorem 2 to several strongly regular graphs related to finite classical polar spaces, so that we obtain new, non-isomorphic strongly regular graphs with the same parameters. We refer the reader to [5, Section 9.4], [12], [15] and [24,Chapter 8] for more detailed descriptions of finite classical polar spaces. In the following we give a brief overview.…”

“…For instance this implies nowadays that the strongly regular collinearity graphs of symplectic polar spaces are not uniquely defined by their spectrum for sufficiently large q. In [15] the first author generalized all of the results obtained by switching to all finite classical polar spaces of rank at least 3 over F q by using purely geometrical arguments.…”

Strongly regular graphs satisfying the 4-vertex condition

“…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.…”

“…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. Similarly, we do not know when WQH-switching preserves the 4-vertex condition.…”