strongly regular graph

In this paper we classify distance-regular graphs‎, ‎including strongly regular graphs‎, ‎admitting a transitive action of the linear groups $L(3,2)$‎, ‎$L(3,3)$‎, ‎$L(3,4)$ and $L(3,5)$ for which the rank of the permutation representation is at most 15‎. ‎We give details about constructed graphs‎. ‎In addition‎, ‎we construct self-orthogonal codes from distance-regular graphs obtained in this paper‎.

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‎In this paper we construct transitive $t$-designs from the linear groups $L(2,q)‎, ‎q leq 23$‎.
‎Thereby we classify $t$-designs‎, ‎$t ge 2$‎, ‎admitting a transitive action of the linear groups $L(2,q)‎, ‎q leq 23$‎, ‎up to 35 points and obtained numerous transitive designs‎, ‎for $36leq vleq 55$‎. ‎In many cases we proved the existence of $t$-designs with certain parameter sets‎. ‎Among others we constructed $t$-designs with parameters $2$-$(55,10,4)$‎, ‎$3$-$(24,11,495)$‎, ‎$3$-$(24,12‎, ‎5m)‎, ‎m in {11‎, ‎12,22‎, ‎33‎, ‎44‎, ‎66‎, ‎132}$‎. ‎Furthermore‎, ‎we constructed strongly regular graphs admitting a transitive action of the linear groups $L(2,q)‎, ‎q leq 23$‎.

Let $G$ be a simple connected graph. In this paper, Szeged dimension and PI$_v$ dimension of graph $G$ are introduced. It is proved that if $G$ is a graph of Szeged dimension $1$ then line graph of $G$ is 2-connected. The dimensions of five composite graphs: sum, corona, composition, disjunction and symmetric difference with strongly regular components is computed. Also explicit formulas of Szeged and PI$_v$ indices for these composite graphs is obtained.