General Degree-Eccentricity Index of Trees

For a connected graph G and a , b ∈ R , the general degree-eccentricity index is defined as DEI a , b ( G ) = ∑ v ∈ 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.