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 $$\{20,16,5; 1,1,16 \}$$ .