A note on the Nordhaus-Gaddum type inequality to the second largest eigenvalue of a graph

Let G = (V,E) be a simple undirected graph with n vertices and m edges and let G be its complement. Given a vertex v ∈ V, the set of neighbors of v is N(v) = {w ∈ V |{v, w} ∈ E} and d(v) = |N(v)| is the degree of v. The girth of G is the length of the shortest cycle in G. Write A(G) = [aij ], where aij = 1 if {i, j} ∈ E(G) and aij = 0, if {i, j} / ∈ E(G), for the adjacency matrix of G. The ∗Corresponding author. André E. Brondani 2010 Mathematics Subject Classification. 05C50.