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 \(\mathrm{irr}_t(G)=\sum_{\{x,y\} \subseteq V(G)} |d_x - d_y|.\) 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.
