A new partial geometry pg(5,5,2)

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

“…(1, 4, 3)(2, 6, 9, 18, 7, 13) (5,20,10,21,11,12) The data in Table 2 distinguishes as nonisomorphic all but the three graphs Γ 5 , Γ 7 , Γ 8 , and the two graphs Γ 9 and Γ 10 . Let D i be the design on 81 points having as blocks the 6-cliques in Γ i .…”

“…One of the newly found graphs invariant under a group of order 972 gives rise to a new partial geometry pg(5, 5, 2) that is not isomorphic to the van Lint-Schrijver partial geometry. An isomorphic partial geometry was simultaneously and independently constructed by V. Krčadinac [12] by using a different method. The adjacency matrices of the two previously known graphs and the twelve newly found graphs are available online at http://www.math.uniri.hr/~asvob/SRGs81_pg552.txt…”

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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.

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Strongly regular configurations

“…The pg(q, q, 1) with q = 2, 3, 4, 5 are the classical generalized quadrangles W (q) and their duals, see [39]. Two non-isomorphic pg(5, 5, 2)'s are known [43,13,31], whereas the existence of a pg (6,6,4) is open. Six of the remaining 47 parameter sets are eliminated by Proposition 2.4.…”

Pseudo-Geometric Strongly Regular Graphs with a Regular Point

“…For example, Crnković, Švob, and Tonchev [11] found twelve new strongly regular graphs with parameters (81, 30, 9, 12) using automorphisms of the previously known graphs. In addition, this search led to the discovery of a new partial geometry pg (5,5,2), that was independently constructed by Krčadinac [28]. Feng and Xiang [15] and Feng, Momihara, and Xiang [14] gave a construction of strongly regular graphs using cyclotomic classes and skew Hadamard different sets, thereby extending the fundamental work of Van Lint and Schrijver [29].…”