On good pairs in edge-regular graphs

An undirected graph on $v$ vertices of valences equal to $k$, whose each edge belongs to exactly $\lambda$ triangles is called edge-regular with parameters $(v,k,\lambda)$. Let $b_1=k-\lambda-1$. We say that a pair of vertices $u$, $w$ is good if these vertices have exactly $k-2b_1+1$ common neighbours. We prove that if $k\ge3b_1-1$, then either for any vertex $u$ at most two vertices in $\Gamma$ form good pairs with $u$, or $k=3b_1-1$, $\Gamma$ is a polygon or the icosahedron graph, and any two vertices which are 2 distant from each other form good pairs. We give a new upper bound for the number of vertices in an edge-regular graph of diameter two with $k\ge3b_1-1$. We prove that an edge-regular graph with parameters of the triangular graph $T(n)$, $n=5,6$, the Clebsch graph, or the Schläfli graph coincides with the corresponding graph.
This research was supported by the Russian Foundation for Basic Research, grant 02–01–00772.