On the extension of connected irregular (λ, μ)-graphs. (3rd April 2017) - Hailat, Mohammad
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 K 1 ∨ ( s 2 K 2 ∪ s 3 K 3 ∪ … ∪ st Kt ) such that si ≥ 0, for all i = 1, … , t and 1 + 2 s 2 + … + ts t = n . Note that λ can take several values in the same graph.