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

• Published in: arXiv.org 2014-09
• Main Authors: Kok, Johan, Mukungunugwa, Vivian
• Format: Article
• Language: English
• Subjects: Graph theory; Graphs; Irregularities; Organic chemistry; Topology; Weight
• ISSN: 2331-8422

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.