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arXiv · 2607.06455

Generalized spectral closedness of $\mathcal{F}$-free graph classes

Wei Wang·Quanyu Tang
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Abstract

In this paper, we investigate the generalized spectral closedness of graph classes defined by a family $\mathcal{F}$ of forbidden induced subgraphs. To systematically study this property, we introduce a novel combinatorial concept of patterned closed walks (or $\beta$-closed walks), which naturally interlaces the edges of a graph with those of its complement. By establishing the induced-subgraph expansion of these $\beta$-closed walk counts, we obtain an algebraic sufficient condition for generalized spectral closedness based on the existence of a walk-realizable $\mathcal{F}$-supporter. Crucially, the search for such a walk-realizable supporter is reduced to a linear programming feasibility problem. As primary applications of this computational framework, we prove that the classes of threshold graphs and chain graphs are generalized spectrally closed.

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Wei Wang, Quanyu Tang. 2026-07-07. Generalized spectral closedness of $\mathcal{F}$-free graph classes. https://arxiv.org/abs/2607.06455

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