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arXiv · 2512.07579

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

Dan Li·Mingsong Qin
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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 $n$.

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Dan Li, Mingsong Qin. 2025-12-08. The index of $t\mathcal{{C}}_{3}^{-}$-free signed graphs. https://arxiv.org/abs/2512.07579

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