Статьи, опубликованные в английской версии журнала
A Solution of the Open Problem on Total Irregularity of Trees with Specified Leaves
S. Ahmada,
R. Farooqa,
K. C. Dasb a National University of Sciences and Technology, Islamabad, Pakistan
b SungKyunKwan University, Suwon, Republic
of Korea
Аннотация:
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.
Ключевые слова:
total irregularity index, total
$\sigma$-irregularity index, extremal tree, extremal chemical tree, leaf.
Поступило: 18.02.2025
Исправленный вариант: 09.07.2025
Язык публикации: английский
