On Eccentric Adjacency Index of Graphs and Trees

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\[\xi^{ad} (G)=\sum_{u\in V(G)}\se_G(u)\varepsilon_G(u)^{-1},\]\noindentwhere $\se_G(u)=\displaystyle\sum_{\substack{v\in V(G)\\ d_G(u,v)=1}}\deg_{G}(v)$ 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 ofEAI among all simple and connected graphs, all trees, and all trees with perfect matching.