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

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 $firr_t^\pm(G).$