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\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}‎.

Keywords:

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