The minimum $\varepsilon$-spectral radius of $t$-clique trees with given diameter

Document Type : Research Paper

Authors

1 School of Computational Science and Electronics, Hunan Institute of Engineering, Xiangtan,411104, P. R. China.

2 College of Mathematics and Computer Science, Hunan Normal University, Changsha, Hunan 410081, P. R. China

3 School of Mathematics and Statistics, Hunan Normal University, Changsha, Hunan, China.

Abstract

The eccentricity matrix $\varepsilon(G)$ of a graph $G$ is defined as \begin{equation}
\varepsilon(G)_{uv}= \begin{cases}
d_{uv} & d_{uv}=min\{e(u),e(v)\},\\
0 & d_{uv} < min\{e(u),e(v)\}. \notag
\end{cases}
\end{equation} Let $T_t$ be a $t$-clique tree corresponding to the tree $T($underlying graph of $T_t)$ with order $n'=(n-1)t+1$ and diameter $d$. In this paper, we identify the extremal $t$-clique trees with given diameter having the minimum $\varepsilon$-spectral radius. Simultaneously, we calculate the lower bound of $\varepsilon$-spectral radius of $t$-clique trees when $n-d$ is odd.

Keywords

Main Subjects


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Volume 13, Issue 3 - Serial Number 3
September 2024
Pages 235-255
  • Receive Date: 16 July 2022
  • Revise Date: 11 July 2023
  • Accept Date: 22 July 2023
  • Published Online: 01 September 2024