Total Irregularity and $f_t$-Irregularity of Linear Jaco Graphs$

TL;DR

This paper investigates the total irregularity and Fibonacci-based irregularity measures of linear Jaco graphs, introduces the concept of edge-joint, and explores their properties with an open problem for future research.

Contribution

It defines new irregularity measures based on Fibonacci weights and analyzes their properties specifically for linear Jaco graphs, including the introduction of edge-joint graphs.

Findings

Derived formulas for total irregularity of Jaco graphs.

Introduced and analyzed Fibonacci-based irregularity measure.

Proposed an open problem related to $firr_t^\pm(G)$.

Abstract

Total irregularity of a simple undirected graph $G$ is defined to be $irr_t(G) = \frac{1}{2}\sum\limits_{u, v \in V(G)}|d(u) - d(v)|$. See Abdo and Dimitrov [2]. We allocate the \emph{Fibonacci weight,} $f_i$ to a vertex $v_j$ of a simple connected graph, if and only if $d(v_j) = i$ and define the \emph{total fibonaccian irregularity} or $f_t-irregularity$ denoted $firr_t(G)$ for brevity, as: $firr_t(G) = \sum\limits_{i=1}^{n-1}\sum\limits_{j=i+1}^{n}|f_i - f_j|.$ The concept of an \emph{edge-joint} is also introduced to be the simple undirected graph obtained from two simple undirected graphs $G$ and $H$ by linking the edge $vu_{v \in V(G), u \in V(H)}$. This paper presents results for the undirected underlying graphs of Jaco Graphs, $J_n(x)$. Finally we pose an open problem with regards to $firr_t^\pm(G).$