On Eccentric Adjacency Index of Graphs and Trees

Document Type : Original Scientific Paper

Authors

1 Persian Gulf University

2 Department of Computer Engineering of Jam, Persian Gulf University, Jam, IRAN

3 Faculty of intelligent systems engineering and data science, Persian Gulf University, Bushehr 75169.

Abstract

Let $G=(V(G),E(G))$ be a simple and connected graph. The distance between any two vertices $x$ and $y$, denoted by $d_G(x,y)$, is defined as the length of a shortest path connecting $x$ and $y$ in $G$.
The degree of a vertex $x$ in $G$, denoted by $\deg_G(x)$, is defined as the number of vertices in $G$ of distance one from $x$.
The eccentric adjacency index (briefly EAI) of a connected graph $G$ is defined as
\[\xi^{ad} (G)=\sum_{u\in V(G)}\se_G(u)\varepsilon_G(u)^{-1},\]
\noindent
where $\se_G(u)=\displaystyle\sum_{\substack{v\in V(G)\\ d_G(u,v)=1}}\deg_{G}(v)$ and
$\varepsilon_G(u)=\max \{d_G(u,v)\mid v \in V(G)\}$.
In this article, we aim to obtain all extremal graphs based on the value of
EAI among all simple and connected graphs, all trees, and all trees with perfect matching.

Keywords

Main Subjects

References

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Mathematics Interdisciplinary Research
Volume 8, Issue 1
Special Issue: Proceedings of the 27th Iranian Algebra Seminar (IAS27) --- Editors: Reza Sharafdini and Mojtaba Sedaghatjoo
March 2023
Pages 1-17

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