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

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 $β$-closed walks), which naturally interlaces the edges of a graph with those of its complement. By establishing the induced-subgraph expansion of these $β$-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.

Disclosure

“ion of this work, the authors used AI systems as auxiliary tools. Specif- ically, GPT-5.5 Pro was used to assist with the early formulation and presentation of the walk- realizable supporter construction for threshold graphs. Additionally, Gemini 3.1 Pro was used to facilitate literature searching, assist with algorithmic programming, and support linguistic refinement. All AI-generated outputs and suggestions were checked, critically evaluated, and re- structured by the authors. The author”

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2026-08-24T00:34:12.480Z

Structural counts