On the extremal total reciprocal edge-eccentricity of trees

TL;DR

This paper introduces the total reciprocal edge-eccentricity as a new graph invariant and characterizes extremal trees with respect to this measure using novel transformations.

Contribution

It defines the reciprocal edge-eccentricity, introduces edge-grafting transformations, and characterizes extremal trees based on various parameters.

Findings

Sharp bounds on reciprocal edge-eccentricity of trees are established.

Extremal trees are characterized for various graph parameters.

Mathematical properties of the reciprocal edge-eccentricity are elucidated.

Abstract

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 $G=(V_G,E_G)$ is a simple connected graph, then the total reciprocal edge-eccentricity (REE) of $G$ is defined as $\xi^{ee}(G)=\sum_{uv\in E_G}(1/\varepsilon_G(u)+1/\varepsilon_G(v))$, where $\varepsilon_G(v)$ 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.