Characterizing tricyclic graphs with pendant vertices having largest $A_α$-spectral radius
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2026
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| _version_ | 1866912981549318144 |
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| author | Basunia, Mainak Panigrahi, Pratima |
| author_facet | Basunia, Mainak Panigrahi, Pratima |
| contents | For a graph $G$ with adjacency matrix $A(G)$ and degree diagonal matrix $D(G)$, the $A_α$-matrix of $G$ is defined as \begin{equation*}
A_α(G) = αD(G) + (1- α) A(G), \text{ for any } α\in [0,1]. \end{equation*} The $A_α$-spectral radius of $G$ is the largest eigenvalue of the matrix $A_α(G)$. A tricyclic graph of order $n$ is a simple connected graph with $n+2$ edges. In this paper, we characterize the unique graph having the largest $A_α$-spectral radius for $α\in [\frac{1}{2}, 1)$ among all tricyclic graphs of order $n$ with $k (\geq 1)$ pendant vertices. As an application, we derive a sufficient spectral condition (alternate to the edge condition) to guarantee the absence of the tricyclic structure in a graph with $k$ pendant vertices. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_23917 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Characterizing tricyclic graphs with pendant vertices having largest $A_α$-spectral radius Basunia, Mainak Panigrahi, Pratima Combinatorics 05C50, 05C35 For a graph $G$ with adjacency matrix $A(G)$ and degree diagonal matrix $D(G)$, the $A_α$-matrix of $G$ is defined as \begin{equation*} A_α(G) = αD(G) + (1- α) A(G), \text{ for any } α\in [0,1]. \end{equation*} The $A_α$-spectral radius of $G$ is the largest eigenvalue of the matrix $A_α(G)$. A tricyclic graph of order $n$ is a simple connected graph with $n+2$ edges. In this paper, we characterize the unique graph having the largest $A_α$-spectral radius for $α\in [\frac{1}{2}, 1)$ among all tricyclic graphs of order $n$ with $k (\geq 1)$ pendant vertices. As an application, we derive a sufficient spectral condition (alternate to the edge condition) to guarantee the absence of the tricyclic structure in a graph with $k$ pendant vertices. |
| title | Characterizing tricyclic graphs with pendant vertices having largest $A_α$-spectral radius |
| topic | Combinatorics 05C50, 05C35 |
| url | https://arxiv.org/abs/2603.23917 |