Spectral Tur\'{a}n problem for $\mathcal{K}_5^{-}$-free signed graphs

TL;DR

This paper investigates the maximum spectral radius of unbalanced signed graphs that do not contain certain unbalanced complete subgraphs, providing specific results and characterizations for the case when k=5.

Contribution

It advances the spectral Turán problem by focusing on unbalanced signed graphs and offers a complete characterization for the extremal case when k=5.

Findings

Determined the maximum spectral radius for $ ext{K}_5^-$-free unbalanced signed graphs.

Provided a complete characterization of extremal graphs for the case k=5.

Abstract

The classical spectral Tur\'{a}n problem is to determine the maximum spectral radius of an $\mathcal{F}$-free graph of order $n.$ Let $\mathcal{K}_{k}^{-}$ be the set of all unbalanced $K_k.$ In this paper, we focus on the spectral Tur\'{a}n problem of $\mathcal{K}_{k}^{-}$-free unbalanced signed graph for $k\geq5$. Moreover, we give an answer for $k=5$ and completely characterize the corresponding extremal signed graph.