Hanyuan Deng - College of Mathematics and Computer Science, Hunan Normal University, Changsha, Hunan ۴۱۰۰۸۱, P. R. China
S. Balachandran - Department of Mathematics, School of Humanities and Sciences,SASTRA University, Thanjavur, India
‎S. K. Ayyaswamy - Department of Mathematics, School of Humanities and Sciences,SASTRA University, Thanjavur, India
Y. B. Venkatakrishnan - Department of Mathematics, School of Humanities and Sciences,SASTRA University, Thanjavur, India

The 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{۲}{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 ۲\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}‎.

‎Average eccentricity‎, ‎harmonic index‎, ‎signless‎ ‎Laplacian eigenvalue‎, ‎extremal value