Graph Energy Change Due to Vertex Deletion

Abstract

Let G be a graph with adjacency matrix A. The energy of G is denoted by E(G) and defined as the sum of the absolute values of the eigenvalues of A. In this paper we study the problem of variation of the energy when a vertex is deleted. Concretely, we show that if G is a graph and G (j) is the graph obtained from G by deleting the vertex vj of G, thenwhere dj is the degree of vj. Moreover, equality occurs if and only if the connected component of G containing vj is isomorphic to a star tree and vj is its center. Afterwards, we introduce a new approach to the local energy of a vertex and initiate the study of its basic properties.