Energy of graphs and signed digraphs

Abstract

The energy of a graph is given by Pn q=1 j qj, where q0 s are the adjacency eigenvalues of the graph. A graph has real eigenvalues because its adjacency matrix is always symmetric. The energy of a sidigraph is defined by Pn q=1 jRe( q)j, where Re( q) represents the real part of eigenvalue q of the sidigraph. A sidigraph has complex eigenvalues because its adjacency matrix is not necessarily symmetric. A topological index is recognized as molecular descriptor that is a conversion of a molecular structure into some real number. In our disquisition, we first focused on the extremal energy of sidigraphs. We investigate the bicyclic sidigraphs having largest energy in the set of all bicyclic sidigraphs with fixed order. We construct some non-cospectral bicyclic sidigraphs having equal energy. We also investigate the energy ordering of signed digraphs in the class of all vertex-disjoint bicyclic sidigraphs. Our second focus is on the energy of graphs based on the inverse sum indeg matrix and generalized inverse sum indeg matrix. These matrices are defined by using definition of respective indices. We give inverse sum indeg energy formula of some graphs. Bounds on inverse sum indeg energy of graphs are obtained. Some non-cospectral equienergetic graphs with respect to inverse sum indeg energy are also obtained. In the end, we introduce generalized inverse sum indeg index and generalized inverse sum indeg energy of graphs. We study the generalized inverse sum indeg index and energy from an algebraic point of view. Extremal values of this index for some graph classes are determined. Some spectral properties of generalized inverse sum indeg matrix are studied. We also find Nordhaus-Gaddum-type results for generalized sum indeg energy and spectral radius of generalized inverse sum indeg matrix.