Two Laplacians for the resistance distance matrix of a graph

Abstract

In this paper, we study the resistance Laplacian and the resistance signless Laplacian matrix associated with a connected graph. We provide the general form of these matrices for various types of graphs, including complete graphs, complete bipartite graphs, and cycles. We investigate their spectral characteristics, determining the eigenvalues and eigenvectors for complete graphs and complete bipartite graphs, and obtaining bounds on the eigenvalues in the resistance transmission regular case. Moreover, we introduce the resistance Laplacian energy of a graph and discuss some bounds for this energy.