Characterization of Complete Bipartite Graphs via Resistance Spectra

TL;DR

This paper proves that certain classes of complete bipartite graphs are uniquely identified by their resistance spectra, which encode electrical and structural properties, thus contributing to spectral graph theory and graph isomorphism problems.

Contribution

The paper establishes that specific complete bipartite graphs are uniquely determined by their resistance spectra, advancing understanding of graph invariants in spectral graph theory.

Findings

Complete bipartite graphs $K_{n,n}$, $K_{n,n+1}$, $K_{2,n}$, and $K_{m,n}$ with $m>3n+1$ are determined by their resistance spectra.

Resistance distances can be exploited to distinguish certain highly symmetric graphs.

The results contribute to the graph isomorphism problem by identifying classes of graphs uniquely characterized by spectral invariants.

Abstract

The notion of resistance distance, introduced by Klein and Randi\'c, 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 $G=(V,E)$, the resistance distance $R_G(u,v)$ between two distinct vertices $u$ and $v$ is defined as the effective resistance between them when each edge of $G$ is replaced by a $1\,\Omega$ resistor. The multiset of all resistance distances over unordered pairs of distinct vertices is called the \emph{resistance spectrum} of $G$, denoted by $\operatorname{RS}(G)$. A graph $G$ is said to be \emph{determined by its resistance spectrum} if, for any graph $H$, the equality $\operatorname{RS}(H)=\operatorname{RS}(G)$ implies that $H$ is isomorphic to $G$. Complete bipartite graphs, denoted by $K_{m,n}$, 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 $K_{n,n}$, $K_{n,n+1}$, $K_{2,n}$, and $K_{m,n}$ with $m>3n+1$ are uniquely determined by their resistance spectra.