Abstract:
Let $G$ be a graph with vertex set $V(G)$, where the degree of a vertex $x\in V(G)$ is denoted by $d_x$. The total irregularity measure ($\mathrm{irr}_t$) of $G$ is defined as \begin{equation*} \mathrm{irr}_t(G)=\sum_{\{x,y\} \subseteq V(G)} |d_x - d_y|. \end{equation*} This note aims to establish the best possible upper and lower bounds on the total irregularity index of $n$-vertex trees with a fixed number of leaves (pendants), thereby resolving a problem posed in Yousaf et al. [“On total irregularity index of trees with given number of segments or branching vertices,” Chaos Soliton Fractals 157, 111925 (2022)]. Additionally, we extend our analysis to chemical trees, deriving corresponding bounds and exploring their structural implications within this class. Our results also yield similar findings for the total $\sigma$-irregularity index.
Keywords:total irregularity index, total $\sigma$-irregularity index, extremal tree, extremal chemical tree, leaf.
