On Eccentric Adjacency Index of Graphs and Trees

Document Type : Original Scientific Paper


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.


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
\[\xi^{ad} (G)=\sum_{u\in V(G)}\se_G(u)\varepsilon_G(u)^{-1},\]
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.


Main Subjects

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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