On the extremal total reciprocal edge-eccentricity of trees

The total reciprocal edge-eccentricity is a novel graph invariant with vast potential in structure activity/property relationships. This graph invariant displays high discriminating power with respect to both biological activity and physical properties. If G=(VG,EG)G=(VG,EG) is a simple connected graph, then the total reciprocal edge-eccentricity (REE) of G   is defined as ξee(G)=∑uv∈EG(1/εG(u)+1/εG(v))ξee(G)=∑uv∈EG(1/εG(u)+1/εG(v)), where εG(v)εG(v) is the eccentricity of the vertex v. In this paper we first introduced four edge-grafting transformations to study the mathematical properties of the reciprocal edge-eccentricity of G. Using these elegant mathematical properties, we characterize the extremal graphs among n-vertex trees with given graphic parameters, such as pendants, matching number, domination number, diameter, vertex bipartition, et al. Some sharp bounds on the reciprocal edge-eccentricity of trees are determined.