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The zeroth-order general Randi\'{c} index of graphs with a given clique number

Xiaoling Sun|Jianwei Du|Yanling Shao
강원경기수학회 (The Kangweon-Kyungki Mathematical Society)|2020.09|한국수학논문집 (Korean Journal of Mathematics)|제28권 제3호|pp. 405-419

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키워드

chromatic numberclique numberzeroth-order general Randi'{c} index
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초록

The zeroth-order general Randi\'{c} index $^{0}R_{\alpha}(G)$ of the graph $G$ is defined as $\sum_{u\in V(G)}d(u)^{\alpha}$, where $d(u)$ is the degree of vertex $u$ and $\alpha$ is an arbitrary real number. In this paper, the maximum value of zeroth-order general Randi\'{c} index on the graphs of order $n$ with a given clique number is presented for any $\alpha\neq 0,1$ and $\alpha \notin (2,2n-1]$, where $n=|V(G)|$. The minimum value of zeroth-order general Randi\'{c} index on the graphs with a given clique number is also obtained for any $\alpha\neq 0,1$. Furthermore, the corresponding extremal graphs are characterized.

Abstract

The zeroth-order general Randi\'{c} index $^{0}R_{\alpha}(G)$ of the graph $G$ is defined as $\sum_{u\in V(G)}d(u)^{\alpha}$, where $d(u)$ is the degree of vertex $u$ and $\alpha$ is an arbitrary real number. In this paper, the maximum value of zeroth-order general Randi\'{c} index on the graphs of order $n$ with a given clique number is presented for any $\alpha\neq 0,1$ and $\alpha \notin (2,2n-1]$, where $n=|V(G)|$. The minimum value of zeroth-order general Randi\'{c} index on the graphs with a given clique number is also obtained for any $\alpha\neq 0,1$. Furthermore, the corresponding extremal graphs are characterized.

참고문헌 (31)

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