Generalized Energy of F-Sum Graphs

Abstract

Graph spectrum has applications in computer science, chemistry, physics and mathematics itself. Graph energy is a significant parameter to characterize the structural attributes of graphs. Let [Formula: see text] be an [Formula: see text]-vertex connected graph. The generalized energy of [Formula: see text] is expressed as [Formula: see text] where [Formula: see text]. Graph operation serves as a powerful method to generate all manner of larger graphs from simpler factor graphs. Let [Formula: see text] [Formula: see text] represent four typical graph operations applied to [Formula: see text]. In this paper, we investigate the generalized energy of a class of composite graphs known as [Formula: see text]-sum graphs, constructed via the lexicographic product of a transformed graph [Formula: see text] and a graph [Formula: see text]. We derive the closed-form formulas for the generalized energy of these four classes of [Formula: see text]-sum graphs in terms of the generalized energy and structural parameters of factor graphs. Furthermore, we apply these results to derive the generalized energy and quasi-Laplacian energy of the [Formula: see text]-sum graphs generated by popular graphs, such as path, cycle, star, and complete graph. The deduction suggests an efficient approach to compute the energy of complex graphs without direct eigenvalue computations.