Characterization of Complete Bipartite Graphs via Resistance Spectra

Xiang-Yang Liu; Xiang-Feng Pan; Yong-Yi Jin; Li-Cheng Li

Paper

Characterization of Complete Bipartite Graphs via Resistance Spectra

Abstract

The notion of resistance distance, introduced by Klein and Randić, has become a fundamental concept in spectral graph theory and network analysis, as it captures both the structural and electrical properties of a graph. The associated resistance spectrum serves as a graph invariant and plays an important role in problems related to graph isomorphism. For an undirected graph

, the resistance distance

between two distinct vertices

and

is defined as the effective resistance between them when each edge of

is replaced by a

resistor. The multiset of all resistance distances over unordered pairs of distinct vertices is called the \emph{resistance spectrum} of

, denoted by

. A graph

is said to be \emph{determined by its resistance spectrum} if, for any graph

, the equality

implies that

is isomorphic to

. Complete bipartite graphs, denoted by

, are highly symmetric and constitute an important class of graphs in graph theory. In this paper, by exploiting properties of resistance distances, we prove that the complete bipartite graphs

, and

with

are uniquely determined by their resistance spectra.