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 2/d_i+d_j over all edges vivj of G, where d_i 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 Randićc index and the largest signless Laplacian eigenvalue conjectured by Hansen and Lucas [11]. [ABSTRACT FROM AUTHOR]