Korean J. Math.  Vol 28, No 3 (2020)  pp.405-419
DOI: https://doi.org/10.11568/kjm.2020.28.3.405

The zeroth-order general Randic index of graphs with a given clique number

Jianwei Du, Yanling Shao, Xiaoling Sun

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.


Keywords


zeroth-order general Randi\'{c} index, chromatic number, clique number

Subject classification

05C07, 92E10

Sponsor(s)

Shanxi Province Science Foundation for Youths [grant number 201901D211227]

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References


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