Strongly regular graphs with parameters (81,30,9,12) and a new partial geometry pg(5,5,2)

Abstract

Twelve new strongly regular graphs with parameters (81, 30,9,12) are found as graphs invariant under certain subgroups of the automorphism groups of the two previously known graphs that arise from 2-weight codes. One of these new graphs is geometric and yields a partial geometry with parameters pg(5, 5, 2) that is not isomorphic to the partial geometry discovered by J. H. van Lint and A. Schrijver [13] in 1981.

Switching for Small Strongly Regular Graphs

“…The number of graphs after applying WQH switching up to i times: Note that our seed graph, the point graph of the vL-S partial geometry has an automorphism group of size 116,640. By comparing autormorphism group sizes, we see that our list cannot contain all of the 14 SRGs described in [8].…”

“…The point graph of this partial geometry is an SRG with parameters (81, 30, 9, 12). Recently, a second partial geometry of the same type was discovered by Krčadinac[19] and, almost at the same time, by Crnković, Švob and Tonchev[8]. More details on V NO − (4, 2) can be found in [4, §10.29].We can describe the SRG derived from the vL-S geometry as follows.…”