On General Degree-Eccentricity Index For Trees with Fixed Diameter and Number of Pendent Vertices

The general degree-eccentricity index of a graph G is defined by, DEIa,b(G) = ∑v∈V(G) daG(v)eccbG(v) for a, b ∈ R, where V (G) is the vertex set of G, eccG(v) is the eccentricity of a vertex v and dG(v) is the degree of v in G. In this paper, we generalize results on the general eccentric connectivity index for trees. We present upper and lower bounds on the general degree-eccentricity index for trees of given order and diameter and trees of given order and number of pendant vertices. The upper bounds hold for a 1 and b ∈ R \ {0} and the lower bounds hold for 0 a 1 and b ∈ R \ {0}. We include the case a = 1 and b ∈ {−1, 1} in those theorems for which the proof of that case is not complicated. We present all the extremal graphs, which means that our bounds are best possible.