On the extension of connected irregular (λ, μ)-graphs

Abstract

Let λ and μ be any nonnegative integers. A graph G is called a (λ, μ)-graph if every pair of adjacent vertices have λ common neighbors, and every pair of non-adjacent vertices have μ common neighbors. Thus, (λ, μ)-graphs are generalization of strongly regular graphs obtained by lessening the regularity property on vertex degrees. In this paper, we will show that a (λ, 1)-graph, with n vertices, is of type K1 ∨ (s2K2 ∪ s3K3 ∪ … ∪ st Kt) such that si ≥ 0, for all i = 1, … , t and 1 + 2s2 + … + tst = n. Note that λ can take several values in the same graph.