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 toý ýanother vertex and the average eccentricity ecc(G) of aý ýgraph G is the mean value of eccentricities of all vertices ofý ýGý. ýThe harmonic index H(G) of a graph G is definedý ýas the sum of 2di over all edges vivj ofý ýGý, ýwhere di denotes the degree of a vertex vi in Gý. ýIný ýthis paperý, ýwe determine the unique tree with minimum averageý ýeccentricity among the set of trees with given number of pendentý ývertices and determine the unique tree with maximum averageý ýeccentricity among the set of n-vertex trees with two adjacentý ývertices of maximum degree Δý, ýwhere n≥2Δý. ýAlsoý, ýweý ýgive some relations between the average eccentricityý, ýthe harmonicý ýindex and the largest signless Laplacian eigenvalueý, ýand strengthený ýa result on the Randi'{c} index and the largest signless Laplacianý ýeigenvalue conjectured by Hansen and Lucas \cite{hl}ý.