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 ecc(g) of a‎ ‎graph g is the mean value of eccentricities of all vertices of‎ ‎g‎. ‎the harmonic index h(g) of a graph g is defined‎ ‎as the sum of 2di+dj over all edges vivj of‎ ‎g‎, ‎where di denotes the degree of a vertex vi 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 δ‎, ‎where n≥2δ‎. ‎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}‎.