On General Degree-Eccentricity Index For Trees with Fixed Diameter and Number of Pendent Vertices

Document Type : Research Paper

Author

Department of Mathematics, Addis Ababa University, Addis Ababa, Ethiopia

Abstract

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.

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References

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Iranian Journal of Mathematical Chemistry
Volume 14, Issue 1
March 2023
Pages 19-32

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