On some distance-regular graphs with many vertices

We construct distance-regular graphs, including strongly regular graphs, admitting a transitive action of the Chevalley groups $G_2(4)$ and $G_2(5)$, the orthogonal group $O(7,3)$ and the Tits group $T=$$^2F_4(2)'$. Most of the constructed graphs have more than 1000 vertices, and the number of vertices goes up to 28431. Some of the obtained graphs are new.Comment: 18 page

On the PSU(4, 2)-invariant vertex-transitive strongly regular (216, 40, 4, 8) graph

In 2018 the first, Rukavina and the third author constructed with the aid of a computer the first example of a strongly regular graph $\Gamma$ with parameters (216, 40, 4, 8) and proved that it is the unique PSU(4,2)-invariant vertex-transitive graph on 216 vertices. In this paper, using the geometry of the Hermitian surface of PG(3, 4), we provide a computer-free proof of the existence of the graph $\Gamma$. The maximal cliques of $\Gamma$ are also determined

Distance-regular graphs obtained from the Mathieu groups

In this paper we construct distance-regular graphs admitting a transitive action of the five sporadic simple groups discovered by E. Mathieu, the Mathieu groups $M_{11}$, $M_{12}$, $M_{22}$, $M_{23}$ and $M_{24}$. From the code spanned by the adjacency matrix of the strongly regular graph with parameters (176,70,18,34) we obtain block designs having the full automorphism groups isomorphic to the Higman-Sims finite simple group. Further, we discuss a possibility of permutation decoding of the codes spanned by the adjacency matrices of the graphs constructed and find small PD-sets for some of the codes.Comment: 22 pages. arXiv admin note: text overlap with arXiv:1809.1019

Graphs cospectral with NU(n+1,q2)(n + 1, q^2), n≠3n \ne 3

Let $H(n, q^2)$ be a non-degenerate Hermitian variety of $PG(n, q^2)$, $n \ge 2$. Let NU$(n+1, q^2)$ be the graph whose vertices are the points of $PG(n, q^2) \setminus H(n, q^2)$ and two vertices $P_1$, $P_2$ are adjacent if the line joining $P_1$ and $P_2$ is tangent to $H(n, q^2)$. Then NU$(n + 1, q^2)$ is a strongly regular graph. In this paper we show that NU$(n + 1, q^2)$, $n \ne 3$, is not determined by its spectrum