Spectral Properties of the Harary Signless Laplacian and Harary Incidence Energy

Let X be a partitioned matrix and let B its equitable quotient matrix. Consider a simple, undirected, connected graph G of order n. In this paper, we employ a technique based on quotient matrices derived from block-partitioned structures to establish new spectral results for the reciprocal distance signless Laplacian matrix. In particular, we identify a sequence of graphs whose eigenvalues are all integers. Furthermore, we introduce the concept of Harary incidence energy and extend known incidence energy results to the setting of the reciprocal distance signless Laplacian matrix. Finally, we characterize the Harary incidence energy of extremal graphs by examining vertex connectivity through the generalized graph join operation.