On General Degree-Eccentricity Index for Trees with Fixed Diameter and Number of Pendant Vertices

The general degree-eccentricity index of a graph G is defined by, DEIa,b(G) = & sigma;v & ISIN;V (G) daG(v)eccbG(v) for a, b E 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 con-nectivity 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 E R \ {0} and the lower bounds hold for 0 < a < 1 and b E R \ {0}. We include the case a = 1 and b E {-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.⠍c 2023 University of Kashan Press. All rights reserved