University of Kashan Iranian Journal of Mathematical Chemistry 2228-6489 14 1 2023 03 01 On General Degree-Eccentricity Index For Trees with Fixed Diameter and Number of Pendent Vertices 19 32 113783 10.22052/ijmc.2023.248566.1675 EN Mesfin Masre Legese Department of Mathematics, Addis Ababa University, Addis Ababa, Ethiopia 0000-0002-2103-8847 Journal Article 2022 11 17 The general degree-eccentricity index of a graph $G$ is defined by, $DEI_{a,b} (G) = \sum_{v \in V(G)} d_{G}^{a}(v) ecc_{G}^{b}(v)$ for $a, b \in \mathbb{R}$, where $V(G)$ is the vertex set of $G$, $ecc_{G}(v)$ is the eccentricity of a vertex $v$ and $d_{G}(v)$ is the degree of $v$ in $G$. In this paper, we generalize results on the general eccentric connectivity index for trees. We present upper and lower bounds on the general degree-eccentricity index for trees of given order and diameter, and trees of given order and number of pendant vertices. The upper bounds hold for $a > 1$ and $b \in \mathbb{R}\setminus\{0\}$ and the lower bounds holds for $0 < a < 1$ and $b \in \mathbb{R}\setminus\{0\}$. We include the case $a = 1$ and $b \in \{-1, 1\}$ in those theorems for which the proof of that case is not complicated. We present all the extremal graphs, which means that our bounds are best possible.

https://ijmc.kashanu.ac.ir/article_113783_14dd4dde8e69844ae362265097053c7d.pdf