Extremal Values on the General Degree–Eccentricity Index of Unicyclic Graphs of Fixed Diameter

For a connected graph G and two real numbers a,b, the general degree–eccentricity index of G is given by DEIa,bG=∑v∈VGdGavecGbv, where VG represent the vertex set of graph G, dGv denotes the degree of vertex v, and ecGv is the eccentricity of v in G, that is, the maximum distance from v to another vertex of G. This index generalizes several well-known “topological indices†of graphs such as the eccentric connectivity index. We characterize the unique unicyclic graph with the maximum general degree–eccentricity index for the case a≥1,b≤0 and the minimum general degree–eccentricity index for the case 0≤a≤1,b≥0, among all n-vertex unicyclic graphs with fixed order and diameter.