Energy and Some Hamiltonian Properties of Graphs

We consider only finite undirected graphs without loops or multiple edges. Notation and terminology not defined here follow that in [2]. For a graph G = (V, E), n := |V |, e := |E|, and G := (V, E), where E := { xy : x ∈ V, y ∈ V, x = y, xy ∈ E }. For a bipartite graph GBPT = (X, Y ; E), GBPT := (X, Y ; E ), where E := { xy : x ∈ X, y ∈ Y, xy ∈ E }. The degree of vertex vi is denoted by di. The concept of closure of a general graph G was introduced by Bondy and Chvátal [1]. The k closure of a graph G, denoted clk(G), is a graph obtained from G by recursively joining two nonadjacent vertices such that their degree sum is at least k. The idea for the closure of a balanced bipartite graph can be found in [1] and [6]. The k closure of a balanced bipartite graph GBPT = (X, Y ; E), where |X | = |Y |, denoted clk(GBPT ), is a graph obtained from G by recursively joining two nonadjacent vertices x ∈ X and y ∈ Y such that their degree sum is at least k. We use C(n, r) to denote the number of r combinations of a set with n distinct elements.