Inverse Problems in Graph Theory: Nets

Abstract

Let $\varGamma$ be a distance-regular graph of diameter 3 with strong regular graph $\varGamma _3$. The determination of the parameters $\varGamma _3$ over the intersection array of the graph $\varGamma$ is a direct problem. Finding an intersection array of the graph $\varGamma$ with respect to the parameters $\varGamma _3$ is an inverse problem. Previously, inverse problems were solved for $\varGamma _3$ by Makhnev and Nirova. In this paper, we study the intersection arrays of distance-regular graph $\varGamma$ of diameter 3, for which the graph ${\bar{\varGamma }}_3$ is a pseudo-geometric graph of the net $PG_{m}(n, m)$. New infinite series of admissible intersection arrays for these graphs are found. We also investigate the automorphisms of distance-regular graph with the intersection array $\{20,16,5; 1,1,16 \}$.