AI 中文总结
本文利用零阶广义兰迪奇指数，给出连通图、二分图和平衡二分图满足 \(l -\)亏缺性质的充分条件，且条件不可去掉，还借此加强和推广了前人相关结果。
AI 中文摘要
对于（分子）图 \(G\) 和任意非零实数 \(\alpha\)，零阶广义兰迪奇指数 \(^0R_\alpha\) 定义为 \(^0R_\alpha (G) =\sum_{v\in G}d_G (v) ^{\alpha}\)。图 \(G\) 的亏缺 \(def(G)\) 等于 \(G\) 中未被最大匹配覆盖的顶点数。若 \(def(G)\le l\)，则图 \(G\) 称为 \(l -\)亏缺的。本文利用该指数给出连通图、二分图和平衡二分图 \(G\) 满足 \(l -\)亏缺性质的充分条件，并表明这些条件均不可去掉。还将用这些结果加强和推广M. An和K. C. Das在2018年以及G. Su等人在2022年已得到的结果。
英文摘要
For a (molecular) graph $G$ and any real number $α\ne 0$ , the zero-order general Randić index , denote by $^0R_α$, is defined by the following equation: \begin{align*} {^0R_α} (G) =\sum_{v\in G}d_G (v) ^α (α\in \mathbb{R}-\left\{0\right\}) . \end{align*} The deficiency of $G$, denoted by $def(G)$, is equal to the cardinality of vertices which are not covered by a maximum matching in $G$. A graph G is called $l$-deficient if $def(G)\le l$. In this paper, we use this index to give sufficient conditions for a connected graph, bipartite graph and a balanced bipartite graph $G$ to satisfy the $l$-deficient property, and show that none of these conditions can be dropped. We will also use these results to enhance and generalise the results that already obtained by M. An and K. C. Das in 2018 and G. Su et al. in 2022.
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