On the average eccentricity, the harmonic index and the largest signless Laplacian eigenvalue of a graph

The eccentricity of a vertex is the maximum distance from it to‎ ‎another vertex and the average eccentricity $eccleft(Gright)$ of a‎ ‎graph $G$ is the mean value of eccentricities of all vertices of‎ ‎$G$‎. ‎The harmonic index $Hleft(Gright)$ 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 $ngeq 2Delta$‎. ‎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}‎.