On Generalized Harary Eigenvalues and Generalized Join of Graphs

In this work, we further study the generalized Harary matrix of a connected graph G, which we denote by RDα(G). This matrix is defined as the convex combination RDα(G) = αRT(G) + (1 − α)RD(G), where 0 ≤ α ≤ 1. In this formulation, RD(G) refers to the classical Harary matrix, and RT(G) represents a diagonal matrix of reciprocal transmission degrees of G. The construction of RDα(G) allows us to establish a unified framework that encompasses the spectral analysis of the Harary matrix and the reciprocal distance signless Laplacian matrix of G. Here, we determine the eigenvalues of RDα(G) for various graphs constructed via graph operations. Furthermore, we establish the generalized Harary spectrum for graphs formed by generalized join and lexicographic products, relating it to their adjacency spectrum. As an application, we determine the generalized Harary spectrum of power graphs of finite abelian groups. Furthermore, the generalized Harary auxiliary energy is investigated, leading to the construction of infinitely many pairwise equienergetic graphs.