Analysis of some graphs through resistance spectra

Abstract Let G = ( V ( G ), E ( G )) be a graph, where V ( G ) is the vertex set and E ( G ) is the edge set. The resistance distance R G ( u , v ) between vertices u and v in graph G is defined as the effective electrical resistance calculated between these two vertices in an electrical network, where each edge represents a unit resistor. The resistance spectrum of a graph is the multiset consisting of the resistance distances of all pairs of vertices in the graph. A graph is said to be uniquely determined by its resistance spectrum if there exists no other non-isomorphic graph having the same resistance spectrum. As reported in prior studies, after performing an in-depth computational analysis on simple graphs of order at most 11, it has been discovered that the overwhelming majority of such graphs can be uniquely characterized by their resistance spectra. Nevertheless, from a rigorous perspective, it is a challenging task to rigorously prove which specific graphs can be determined by their resistance spectra. In this work, by applying the principles of electrical networks and the rules of resistance distance, the authors demonstrate how the resistance spectra can be used to determine comb graphs, T -shape trees, ∞ -graphs, and dumbbell graphs.