On the energy of singular graphs

Abstract

The nullity, �(G), of a graph G is the algebraic multiplicity of the eigenvalue zero in the graph's spectrum. If �(G) > 0, then the graph G is said to be singular. The energy of a graph, E(G), was first defined by I. Gutman (1985) as the sum of the absolute values of the eigenvalues of the graph's adjacency matrix A(G). This paper considers the energy concept for singular graphs. In particular, it is proved that the change in energy upon the simple act of deleting a vertex is related to the type of vertices of the singular graph. Certain upper bounds are improved for the energy of the induced subgraph, G u, which is obtained by deleting vertex u, with the aid of a parameter known as the null spread of u, �u(G) = �(G) �(G u). Also, some new bounds are given for the energy of the minimal configuration graphs, an important class of singular graphs of nullity one that are related to the graph's core. Furthermore, certain graphs that increase their energy when an edge is deleted are considered, such as the complete multipartite graphs and the hypercubes of even dimensions.