On open neighborhood energy of graphs

Abstract

The open neighborhood of a vertex u denoted by N(u) in a simple connected graph G is the collection of all vertices other than u and adjacent to u. In this paper, we introduce a square matrix of order n, called the open neighborhood matrix, ONM(G) of a graph G whose (i, j)^th entry is |N(v_i) ∩ N(v_j)|/(d_i+d_j) whenever v_i~v_j,  i≠ j; and zero otherwise, where d_i and d_j, are the degrees of v_i and v_j respectively. We then establish the relationship between the connectedness of the graph G and the multiplicity of the eigenvalue zero of the matrix ONM(G), if it exists. Furthermore, we found the bounds for the largest open neighbourhood eigenvalue and open neighbourhood energy of graphs.