Unbalanced signed bipartite graphs containing no negative $C_4$ with maximum spectral radius

TL;DR

This paper characterizes unbalanced signed bipartite graphs with no negative 4-cycles that maximize spectral radius, extending spectral Turán type results.

Contribution

It determines the unique unbalanced signed bipartite graphs with maximum spectral radius avoiding negative 4-cycles, given fixed bipartite sizes and order.

Findings

Identifies the unique extremal graphs with maximum spectral radius

Establishes spectral Turán type results for signed bipartite graphs

Extends previous work on unbalanced signed bipartite graphs

Abstract

A signed graph $(G,\sigma)$ is a graph $G$ together with an assignment $\sigma$ of either a positive sign or a negative sign to each edge. A signed graph is unbalanced if it contains a cycle with odd number of negative edges. The spectral radius of a signed graph is the spectral radius of its adjacency matrix, in which for vertices $u,v$, the $(u,v)$-entry is $0$, $-1$, or $1$ depending on whether $uv$ represents no edge, a negative edge, or a positive edge, respectively. Recently, Conde, Dratman and Grippo [Discrete Math. 349 (2026) 114942] proved that there is only one unbalanced signed bipartite graph with maximum spectral radius, up to switching isomorphism. In this paper, we establish a spectral Tur\'an type results for signed bipartite graphs. More precisely, we determine the unique graphs containing no negative cycles of length four with maximum spectral radius, up to switching isomorphism, among unbalanced signed bipartite graphs with fixed bipartite sizes and order, respectively.