Nonexistence of a Subfamily of a Family of Edge-Regular Graphs

A simple, non-edgeless, regular graph is said to be edge-regular if the cardinality of the intersection of the neighborhoods of every pair of adjacent vertices is a fixed integer \(\lambda .\) Edge-regular graphs necessarily have a constant number p of common non-neighbors for every adjacent vertex pair. Much work has been done to determine the extremal edge-regular graphs for the inequality \(n\le 3(\lambda + p)\) , as well as the edge-regular graphs satisfying \(n=3(\lambda + p) -2\) with added structural conditions, where n is the number of vertices and \(\lambda > 0\) . We consider the case where \( n = 3(\lambda + p) -4\) , with the added condition that the common neighbor set of every pair of adjacent vertices induces \(\frac{\lambda }{2}K_2\) , a disjoint union of edges.