General Degree-Eccentricity Index of Trees

Abstract

For a connected graph G and a, b∈ R, the general degree-eccentricity index is defined as DEIa,b(G)=∑v∈V(G)dGa(v)eccGb(v), where V(G) is the vertex set of G, dG(v) is the degree of a vertex v and ecc G(v) is the eccentricity of v in G. We obtain sharp upper and lower bounds on the general degree-eccentricity index for trees of given order in combination with given matching number, independence number, domination number or bipartition. The bounds hold for 0 < a< 1 and b> 0 , or for a> 1 and b< 0. Many bounds hold also for a= 1. All the extremal graphs are presented.