The largest eigenvalue of $\mathcal{C}_4^{-}$-free signed graphs

TL;DR

This paper determines the maximum spectral radius of unbalanced signed graphs that do not contain a negative 4-cycle, extending classical spectral graph results to signed graphs and characterizing extremal structures.

Contribution

It provides the first complete characterization of the maximum eigenvalue for $ ext{C}_4^-$-free unbalanced signed graphs, generalizing known results from unsigned to signed graphs.

Findings

Identified the maximum spectral radius among $ ext{C}_4^-$-free unbalanced signed graphs.

Characterized the extremal signed graph achieving this maximum.

Extended classical spectral bounds to the context of signed graphs.

Abstract

Let $\mathcal{C}_{k}^{-}$ be the set of all negative $C_k$. For odd cycle, Wang, Hou and Li [29] gave a spectral condition for the existence of negative $C_3$ in unbalanced signed graphs. For even cycle, we determine the maximum index among all $\mathcal{C}_4^{-}$-free unbalanced signed graphs and completely characterize the extremal signed graph in this paper. This could be regarded as a signed graph version of the results by Nikiforov [23] and Zhai and Wang [37].