University of KashanMathematics Interdisciplinary Research2538-36398120230301On Eccentric Adjacency Index of Graphs and Trees11711376110.22052/mir.2023.246384.1391ENReza SharafdiniPersian Gulf University0000-0002-3171-2209Mehdi AzadimotlaghDepartment of Computer Engineering of Jam, Persian Gulf University, Jam, IRAN0000-0003-0308-7132Vahid HashemiFaculty of intelligent systems engineering and data science,
Persian Gulf University, Bushehr 75169.0000-0002-3171-2209Fateme ParsanejadFaculty of intelligent systems engineering and data science,
Persian Gulf University, Bushehr 75169.0000-0002-3171-2209Journal Article20230108Let $G=(V(G),E(G))$ 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 $G$.<br />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 $x$.<br />The eccentric adjacency index (briefly EAI) of a connected graph $G$ is defined as<br />\[\xi^{ad} (G)=\sum_{u\in V(G)}\se_G(u)\varepsilon_G(u)^{-1},\]<br />\noindent<br />where $\se_G(u)=\displaystyle\sum_{\substack{v\in V(G)\\ d_G(u,v)=1}}\deg_{G}(v)$ and<br />$\varepsilon_G(u)=\max \{d_G(u,v)\mid v \in V(G)\}$.<br />In this article, we aim to obtain all extremal graphs based on the value of<br />EAI among all simple and connected graphs, all trees, and all trees with perfect matching.https://mir.kashanu.ac.ir/article_113761_25bbe911758dc04c41038b96ac151cd4.pdf