A Solution of the Open Problem on Total Irregularity of Trees with Specified Leaves

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.