The 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 $ 四种运算，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) $ denoted the set of edges and$ {d}_{G}\left(v\right) $ the degree of the vertex$ v $ , respectively. For an edge$ e=uv $ , the general sum-connectivity index$ {\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$ {\rm{S}},{\rm{R}},{\rm{Q}},{\rm{T}}$ operations 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) $ , we give the sharp bounds on general sum-connectivity index of graphs for operations based on lexicographic product, and these bounds are sharp.-
Key words:
- general sum-connectivity index /
- lexicographic product /
- operation on graphs /
- F-sum
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图 1
$ F\left({P}_{6}\right), F\in \{S, R, Q, T\} $ Figure 1. The four operations of graph
$ G $ -
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