On the extremal values of the eccentric distance sum of trees

Abstract

Let G be a simple connected graph. The eccentric distance sum (EDS) of G is defined as ξd(G)=∑v∈VεG(v)DG(v), where εG(v) is the eccentricity of the vertex v and DG(v)=∑u∈VdG(u,v) is the sum of all distances from the vertex v. In this paper, the trees having the maximal EDS among n-vertex trees with maximum degree Δ and among those with domination number 3 are characterized. The trees having the maximal or minimal EDS among n-vertex trees with independence number α and the trees having the maximal EDS among n-vertex trees with matching number m are also determined.