math.CO — Combinatorics
Strongly regular graphs on 306, 400, 576, 784 and 1296 vertices from multi-orbit coset fusion and Bush-type Hadamard matrices
We construct strongly regular graphs with parameters (306,55,4,11), (400,171,74,72), (576,253,112,110), (784,351,158,156) and (1296,595,274,272), and hence also their complements. All five parameter sets were marked "?" (existence unknown) in Brouwer's online table on 10 and 11 October 2026. Three of the graphs were found by a computer search that extends the transitive coset fusion of our earlier papers to groups acting with two orbits: the search fuses the orbitals of a group G on G/H_1⊔ G/H_2, screens with tables of marks, solves the exact quadratic equations on the structure constants with a SAT-based solver, and rebuilds every solution from scratch. The groups are 2× M_12, 5^(1+2)_+:4A_5 and Sp(6,2). Also deposited on Zenodo with the full data archive (the citable record): https://doi.org/10.5281/zenodo.23296158 The graphs on 400, 576, 784 and 1296 vertices come from a short step beyond known constructions. A symmetric Bush-type Hadamard matrix of order n^2 whose n× n blocks have unit diagonal, for instance one of Kharaghani's matrices, defines a strongly regular graph Γ_0 with the parameters of the Latin square graph LS_(n/2+1)(n). In Γ_0 the block rows and the block columns form two orthogonal spreads of Delsarte cliques. Deleting one spread gives the known graphs with λ=μ of Bonato, Holzmann and Kharaghani. By an elementary spread lemma, deleting both gives a strongly regular graph with the parameters of LS_(n/2-1)(n). Kharaghani's matrices exist for every Hadamard order n, so such a graph exists for every n=4m≥8, for example srg(1600,741,344,342) and srg(1936,903,422,420). A second source, found by analysing the search hits, is the rule (a,b)∼(c,d) iff a≠ c, b≠ d and H_abH_bcH_cdH_da=1 for a symmetric Hadamard matrix H. Together the two sources give at least 5, 7, 5 and 6 pairwise non-isomorphic graphs on 400, 576, 784 and 1296 vertices. None of the graphs we tested on these four vertex numbers is a Latin square graph, so they say nothing about the orthogonal arrays OA(n,n/2-1) named in the table. The graph on 306 vertices has a short description on the 300 non-square points of the parabolic quadric Q(4,5) together with the six points of a totally singular line; this line is a Delsarte clique. The graph has the parameters of the collinearity graph of a generalized quadrangle GQ(5,10) but is not geometric, and the same rule over 𝔽_3 gives the collinearity graph of Q(4,3). Strong regularity of every graph is also checked by an exact, standalone, standard-library Python program.