Minimal spectral radius of graphs with given matching number
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arXiv
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| Hauptverfasser: | , , |
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| Format: | Preprint |
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2026
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| _version_ | 1866910000901783552 |
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| author | Liu, Jiaqi Lou, Zhenzhen Trevisan, Vilmar |
| author_facet | Liu, Jiaqi Lou, Zhenzhen Trevisan, Vilmar |
| contents | The Brualdi-Solheid problem asks which graph achieves the extremal (maximum or minimum) spectral radius for a given class of graphs. This paper addresses the Brualdi-Solheid problem for \( \mathcal{G}_{n,β} \), the family of graphs with order \( n \) and matching number \( β\), aiming to identify its spectrally minimal graphs i.e., those that minimize the spectral radius \(ρ(G)\).
We introduce the novel concept of ``quasi-adjacency'' relation, developing a unified structural classification framework for trees in \(\mathcal{G}_{n,β}\), which clarifies structural properties and provides a constructive method to generate trees with fixed \(β\). By showing that all spectrally minimal graphs in \( \mathcal{G}_{n,β} \) are trees, we further narrow the search for extremal graphs. Additionally, we apply this framework to the representative cases \(β=2,3,4\), obtaining the minimizers by explicit structural formulas involving parameters related to \(n\). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_18223 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Minimal spectral radius of graphs with given matching number Liu, Jiaqi Lou, Zhenzhen Trevisan, Vilmar Combinatorics The Brualdi-Solheid problem asks which graph achieves the extremal (maximum or minimum) spectral radius for a given class of graphs. This paper addresses the Brualdi-Solheid problem for \( \mathcal{G}_{n,β} \), the family of graphs with order \( n \) and matching number \( β\), aiming to identify its spectrally minimal graphs i.e., those that minimize the spectral radius \(ρ(G)\). We introduce the novel concept of ``quasi-adjacency'' relation, developing a unified structural classification framework for trees in \(\mathcal{G}_{n,β}\), which clarifies structural properties and provides a constructive method to generate trees with fixed \(β\). By showing that all spectrally minimal graphs in \( \mathcal{G}_{n,β} \) are trees, we further narrow the search for extremal graphs. Additionally, we apply this framework to the representative cases \(β=2,3,4\), obtaining the minimizers by explicit structural formulas involving parameters related to \(n\). |
| title | Minimal spectral radius of graphs with given matching number |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2601.18223 |