The index of $t\mathcal{{C}}_{3}^{-}$-free signed graphs

TL;DR

This paper extends spectral Turán theory to signed graphs, characterizing extremal unbalanced signed graphs free of multiple triangles and identifying those with maximum and second maximum spectral index.

Contribution

It provides a characterization of extremal graphs avoiding multiple unbalanced triangles, advancing spectral Turán theory for signed graphs.

Findings

Characterized extremal unbalanced signed graphs avoiding multiple triangles.

Identified graphs with the maximum spectral index for t≥2.

Determined second maximum spectral index graphs for t≥3.

Abstract

The classical spectral Tur\'{a}n problem is to determine the maximum spectral radius of an $F$-free graph of order $n$. This paper extends this framework to signed graphs. Let $\mathcal{C}_r^-$ be the set of all unbalanced signed graphs with underlying graphs $C_r$. Wang, Hou and Li [Linear Algebra Appl, 681 (2024) 47-65] previously determined the spectral Tur\'{a}n number of $\mathcal{C}_{3}^{-}$. In the present work, we characterize the extremal graphs that achieve the maximum index among all unbalanced signed graphs of order $n$ that are $t\mathcal{C}{3}^{-}$-free for $t\geq 2$. Furthermore, for $t\geq 3$, we identify the graphs with the second maximum index among all $t\mathcal{C}{3}^{-}$-free unbalanced signed graphs of fixed order $n$.