Spectral Tur\'{a}n problem for $\mathcal{K}_{2,t}^-$-free unbalanced signed graphs

TL;DR

This paper characterizes the maximum spectral index and the corresponding extremal unbalanced signed graphs that are free of a specific subgraph, extending spectral Turán theory to signed graphs.

Contribution

It determines the extremal unbalanced signed graphs with maximum and second maximum spectral index avoiding a certain subgraph, for fixed graph order and parameters.

Findings

Identified the maximum spectral index among $\\mathcal{K}_{2,t}^-$-free unbalanced signed graphs.

Characterized the graphs achieving the maximum and second maximum spectral index.

Extended spectral Turán problem results to unbalanced signed graphs.

Abstract

We determine the maximum index and the signed graphs with the maximum index among all $\mathcal{K}_{2,t}^-$-free unbalanced signed graphs with fixed order for $t\geq 3$, as well as the second maximum index and the signed graphs with the second maximum index among all $\mathcal{K}_{2,t}^-$-free unbalanced signed graphs with fixed order for $t\geq 4$.