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.