关于Jaco图的$f^\pm$指标的注记，$J_n(1), n \in \Bbb N$和Khazamula不规则性的引入
A note on $f^\pm$-Zagreb indices in respect of Jaco Graphs, $J_n(1), n \in \Bbb N$ and the introduction of Khazamula irregularity
摘要：
与\emph{first Zagreb index,} $M_1(G)$和\emph{second Zagreb index,} $M_2(G)$相关的拓扑指数$irr(G)$是研究的最古老的不规则性度量。Alberton $[3]$介绍了$G$的\emph{irregularity}作为$irr(G) = \sum\limits_{e \in E(G)}imb(e), imb(e) = |d(v) - d(u)|_{e=vu}$。在Fath-Tabar $[7]$的论文中，为了与化学图论的术语一致，将Alberton指数命名为\emph{third Zagreb indice}。最近Ado等人在$[1]$中引入了拓扑指数\emph{total irregularity}。后者可称为\emph{fourth Zagreb indice}。如果$d(v_i)$是不均匀的，则我们将顶点$v_i$的$\pm$ \emph{Fibonacci weight,} $f^\pm_i$定义为$-f_{d(v_i)},$，如果$d(v_i)$是均匀的，则定义为$f_{d(v_i)}$。根据上述内容，我们定义了$f^\pm$-Zagreb指数。本文介绍了Jaco图的无向底层图$J_n(1), n \leq 12$的一些初步结果。有关Jaco Graphs $J_n(1)$的更多信息，请参见$[9, 10]$。最后，我们介绍了\emph{Khazamula irregularity}作为一个新的拓扑变体。
英文摘要：
The topological indices $irr(G)$ related to the \emph{first Zagreb index,} $M_1(G)$ and the \emph{second Zagreb index,} $M_2(G)$ are the oldest irregularity measures researched. Alberton $[3]$ introduced the \emph{irregularity} of $G$ as $irr(G) = \sum\limits_{e \in E(G)}imb(e), imb(e) = |d(v) - d(u)|_{e=vu}$. In the paper of Fath-Tabar $[7]$, Alberton's indice was named the \emph{third Zagreb indice} to conform with the terminology of chemical graph theory. Recently Ado et.al. $[1]$ introduced the topological indice called \emph{total irregularity}. The latter could be called the \emph{fourth Zagreb indice}. we define the $\pm$\emph{Fibonacci weight,} $f^\pm_i$ of a vertex $v_i$ to be $-f_{d(v_i)},$ if $d(v_i)$ is uneven and $f_{d(v_i)}$, if $d(v_i)$ is even. From the aforesaid we define the $f^\pm$-Zagreb indices. This paper presents introductory results for the undirected underlying graphs of Jaco Graphs, $J_n(1), n \leq 12$. For more on Jaco Graphs $J_n(1)$ see $[9, 10]$. Finally we introduce the \emph{Khazamula irregularity} as a new topological variant.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
