General Degree-Eccentricity Index of Trees

For a connected graph G and a, b is an element of R, the general degree-eccentricity index is defined as DEIa,b(G) = Sigma(v is an element of V(G))d(G)(a)(v)ecc(G)(b)(v), where V(G) is the vertex set of G, d(G)(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.