Characterizing $A_α$-minimizer graphs: given order and independence number
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866911463548911616 |
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| author | Zhang, Jiaqi Li, Shuchao |
| author_facet | Zhang, Jiaqi Li, Shuchao |
| contents | For a given graph \( G \), let \( A(G) \), \( Q(G) \), and \( D(G) \) denote the adjacency matrix, signless Laplacian matrix, and diagonal degree matrix of \( G \), respectively. The \( A_α(G) \) matrix, proposed by Nikiforov, is defined as \( A_α(G)=αD(G)+(1 - α)A(G) \), where \( α\in[0,1] \). This matrix captures the gradual transition from \( A(G) \) to \( Q(G) \). Let \( \mathcal{G}_{n,γ} \) denote the family of all connected graphs with \( n \) vertices and independence number \( γ\). A graph in \( \mathcal{G}_{n,γ} \) is referred to as an \( A_α\)-minimizer graph if it achieves the minimum \( A_α\) spectral radius. In this paper, we first demonstrate that the \( A_α\)-minimizer graph in \( \mathcal{G}_{n,γ} \) must be a tree when \( γ\geq\left\lceil\frac{n}{2}\right\rceil \), and we provide several characterizations of such \( A_α\)-minimizer graphs. We then specifically characterize the \( A_α\)-minimizer graphs for the case \( γ= \left\lceil\frac{n}{2}\right\rceil + 1 \) when $n\geq 9$. Furthermore, we obtain a structural characterization for the \( A_α\)-minimizer graph when \( γ=n - c \), where \( c\geq4 \) is an integer. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_09770 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Characterizing $A_α$-minimizer graphs: given order and independence number Zhang, Jiaqi Li, Shuchao Combinatorics 05C75, 05C50 For a given graph \( G \), let \( A(G) \), \( Q(G) \), and \( D(G) \) denote the adjacency matrix, signless Laplacian matrix, and diagonal degree matrix of \( G \), respectively. The \( A_α(G) \) matrix, proposed by Nikiforov, is defined as \( A_α(G)=αD(G)+(1 - α)A(G) \), where \( α\in[0,1] \). This matrix captures the gradual transition from \( A(G) \) to \( Q(G) \). Let \( \mathcal{G}_{n,γ} \) denote the family of all connected graphs with \( n \) vertices and independence number \( γ\). A graph in \( \mathcal{G}_{n,γ} \) is referred to as an \( A_α\)-minimizer graph if it achieves the minimum \( A_α\) spectral radius. In this paper, we first demonstrate that the \( A_α\)-minimizer graph in \( \mathcal{G}_{n,γ} \) must be a tree when \( γ\geq\left\lceil\frac{n}{2}\right\rceil \), and we provide several characterizations of such \( A_α\)-minimizer graphs. We then specifically characterize the \( A_α\)-minimizer graphs for the case \( γ= \left\lceil\frac{n}{2}\right\rceil + 1 \) when $n\geq 9$. Furthermore, we obtain a structural characterization for the \( A_α\)-minimizer graph when \( γ=n - c \), where \( c\geq4 \) is an integer. |
| title | Characterizing $A_α$-minimizer graphs: given order and independence number |
| topic | Combinatorics 05C75, 05C50 |
| url | https://arxiv.org/abs/2508.09770 |