University of IsfahanTransactions on Combinatorics2251-86576420171201On the average eccentricity, the harmonic index and the largest signless Laplacian eigenvalue of a graph43502147010.22108/toc.2017.21470ENHanyuanDengCollege of Mathematics and Computer Science, Hunan Normal University, Changsha, Hunan 410081, P. R. ChinaS.BalachandranDepartment of Mathematics, School of Humanities and Sciences,SASTRA University, Thanjavur, IndiaS. K.AyyaswamyDepartment of Mathematics, School of Humanities and Sciences,SASTRA University, Thanjavur, IndiaY. B.VenkatakrishnanDepartment of Mathematics, School of Humanities and Sciences,SASTRA University, Thanjavur, IndiaJournal Article20170308The eccentricity of a vertex is the maximum distance from it to another vertex and the average eccentricity $ecc\left(G\right)$ of a graph $G$ is the mean value of eccentricities of all vertices of $G$. The harmonic index $H\left(G\right)$ of a graph $G$ is defined as the sum of $\frac{2}{d_{i}+d_{j}}$ over all edges $v_{i}v_{j}$ of $G$, where $d_{i}$ denotes the degree of a vertex $v_{i}$ in $G$. In this paper, we determine the unique tree with minimum average eccentricity among the set of trees with given number of pendent vertices and determine the unique tree with maximum average eccentricity among the set of $n$-vertex trees with two adjacent vertices of maximum degree $\Delta$, where $n\geq 2\Delta$. Also, we give some relations between the average eccentricity, the harmonic index and the largest signless Laplacian eigenvalue, and strengthen a result on the Randi\'{c} index and the largest signless Laplacian eigenvalue conjectured by Hansen and Lucas \cite{hl}.https://toc.ui.ac.ir/article_21470_6107bccf810358fdefb9471c7d0ba0a8.pdf