Sharp Bounds on General Sum-Connectivity Index Based on Lexicographic Product
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摘要: 对于图
$ G $ ，令$ E\left(G\right) $ ，$ {d}_{G}\left(v\right) $ 分别表示$ G $ 的边集和顶点$ v $ 的度。对于边$ e=uv $ ，定义广义和连通度指标$ {\chi }_{\alpha }\left(e\right)={({d}_{G}\left(u\right)+{d}_{G}(v\left)\right)}^{\alpha } $ ，其中$ \alpha $ 为任意实数。在对两个简单的连通图$ G $ 和H做乘积之前，先对其中一个图H进行$ S, R, Q, T $ 4种运算，运算后的图记为$ F\left(H\right) $ （其中$ F\in \{S, R, Q, T\} $ ），再对图$ G $ 和$ F\left(H\right) $ 做字典序乘积，给出了基于字典序乘积下图的广义和连通度的指标上下界，并且这些界都是最好的。Abstract: Give a graph$ G $ , let$ E\left(G\right) $ and$ {d}_{G}\left(v\right) $ represent the set of edges and the degree of the vertex$ v $ , respectively. For an edge$ e=uv $ , the general sum-connectivity index is$ {\chi }_{\alpha }\left(e\right)={({d}_{G}\left(u\right)+{d}_{G}(v\left)\right)}^{\alpha } $ , in which$ \alpha $ is any real number. Before taking the product of two simple connected graphs$ G $ and$ H $ , we first perform four operations of$ {S},{R},{Q},{T}$ on the graph$ H $ , denoted as$ F\left(H\right) $ , in which$ F\in \{S, R, Q, T\} $ , then take the lexicographical product of graphs$ G $ and$ F\left(H\right) $ . The sharp bounds on general sum-connectivity index of graphs for operations based on lexicographic product are given, and these bounds are sharp.-
Key words:
- general sum-connectivity index /
- lexicographic product /
- four operation on graphs /
- F-sum
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图 1
$F\in \{S, R, Q, T\} $ 时F(P6)的4种运算Figure 1. Four operations of F(P6)
${\rm{ at}} \;F\in \{S, R, Q, T\} $ -
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