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_{uin v(g)}se_g(u)varepsilon_g(u)^{-1},]noindentwhere $se_g(u)=displaystylesum_{substack{vin v(g) d_g(u,v)=1}}deg_{g}(v)$ 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.