图的 Rehan-Lanel 指数
Rehan-Lanel Indices of Graphs
AI总结：
本文提出四个 Rehan-Lanel 图指数，并结合多种度定义扩展出共 462 个新指数，在标准图及若干特殊图上完成计算，旨在服务于 QSPR/QSAR 研究。
摘要：
图 $G$ 由顶点集 $V(G)$ 和边集 $E(G)$ 组成。本文提出了四个新指数，分别定义并命名为第一 Rehan-Lanel 指数 $(RL_1)$、第二 Rehan-Lanel 指数 $(RL_2)$、第三 Rehan-Lanel 指数 $(RL_3)$ 和第四 Rehan-Lanel 指数 $(RL_4)$。顶点 $u, v \in V(G)$ 的度分别记为 $d_G(u)$ 和 $d_G(v)$。基于这些新指数以及 Revan 度、Domination 度、Banhatti 度、顶点温度、KV 指数的定义，我们随后又引入了额外的 448 个指数/指数型量，并针对每个后续定义的前四个新指数，对标准图（如 $r$-正则图、完全图、圈、路和完全二部图）计算了结果。此外，我们还对 Wheel 图、Sunflower 图和 French Windmill 图进行了计算。进一步地，利用顶点度的指数形式，即中心性概念，我们又引入了另外 8 个指数。此外，我们定义了一个称为图的顶点的 Chandana-Lanel 度（CL 度）的新度。利用这个度，定义了新的 6 个指数。同时，我们利用两个数的 Heronian 平均定义了称为 Heronian Rehan-Lanel 指数的指数。这些新颖的 462 个指数将在 QSPR/QSAR 研究中具有优势。
英文摘要：
A graph $G$ consists of vertices $V(G)$ and edges $E(G)$. In this paper, we propose four new indices defined and named as first Rehan-Lanel index of $G$ $(RL_1)$, second Rehan-Lanel index of $G$ $(RL_2)$, second Rehan-Lanel index of $G$, third Rehan-Lanel index of $G$, $(RL_3)$ and fourth Rehan-Lanel index of $G$ $(RL_4)$. The degrees of the vertices $u, v \in V(G)$ are denoted by $d_G(u)$ and $d_G(v)$. Based on these new indices and the definitions of Revan degree, Domination degree, Banhatti degree, Temperature of a vertex, KV indices, we subsequently introduced an additional 448 indices/exponentials and computed results for the first four new indices of each subsequent definition, for the standard graphs such as $r-$ regular graph, complete graph, cycle, path and compete bipartite graph. In addition, we performed calculations for the Wheel graph, Sunflower graph, and French Windmill graph. Furthermore, using the exponential of a degree of a vertex, the centrality concept, we introduced another 8 indices. Furthermore, we defined a new degree called Chandana-Lanel degree of a vertex of a graph(CL degree). Using this degree, new 6 indices were defined. Also, we defined the index called the Heronian Rehan-Lanel index using the Heronian mean of two numbers. These novel 462 indices would be advantageous in QSPR/QSAR studies.
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