Characterization of Complete Bipartite Graphs via Resistance Spectra

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.