On the eigenvalues of signed complete bipartite graphs

TL;DR

This paper determines the eigenvalues of signed complete bipartite graphs, analyzing how negative edges influence the spectrum and establishing relations with subgraph structures.

Contribution

It provides explicit eigenvalue characterizations for signed complete bipartite graphs with various negative edge configurations, extending spectral graph theory.

Findings

Eigenvalues depend on negative edge structure

Multiplicity of zero eigenvalue is bounded by subgraph parameters

Spectrum of graphs with negative edges forming specific subgraphs is determined

Abstract

Let $\Gamma=(G,\sigma)$ be a signed graph, where $\sigma$ is the sign function on the edges of $G$. The adjacency matrix of $\Gamma=(G, \sigma)$ is a square matrix $A(\Gamma)=A(G, \sigma)=\left(a_{i j}^{\sigma}\right)$, where $a_{i j}^{\sigma}=\sigma\left(v_{i} v_{j}\right) a_{i j}$. In this paper, we determine the eigenvalues of the signed complete bipartite graphs. Let $(K_{p, q},\sigma)$, $p\leq q$, be a signed complete bipartite graph with bipartition $(U_p, V_q)$, where $U_p=\{u_1,u_2,\ldots,u_p\}$ and $V_q=\{v_1,v_2,\ldots,v_q\}$. Let $(K_{p, q},\sigma)[U_r\cup V_s]$, $r\leq p$ and $s\leq q $, be an induced signed subgraph on minimum vertices $r+s$, which contains all negative edges of the signed graph $(K_{p, q},\sigma)$. We show that the multiplicity of eigenvalue $0$ in $(K_{p, q},\sigma)$ is at least $ p+q-2k-2$, where $k=min(r,s)$. We determine the spectrum of signed complete bipartite graph whose negative edges induce disjoint complete bipartite subgraphs and path. We obtain the spectrum of signed complete bipartite graph whose negative edges (positive edges) induce an $r-$ regular subgraph $H$. We find a relation between the eigenvalues of this signed complete bipartite graph and the non-negative eigenvalues of $H$.