A note on $f^{\pm }$-Zagreb indices in respect of Jaco Graphs, $J_n(1),n\in \mathbb{N}$ and the introduction of Khazamula irregularity

Abstract

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 $irr(G)=\sum _{e\in E(G)}imb(e),imb(e)=|d(v)-d(u){|}_{e=vu}$. 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,} $f_i^{\pm }$ 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, $J_n(1),n\le 12$. For more on Jaco Graphs $J_n(1)$ see $[9,10]$. Finally we introduce the \emph{Khazamula irregularity} as a new topological variant.

14 pages. On advice from arXiv Moderation this paper now incorporates similar ideas and variant results of another submission which has been removed