The normality and sum of normalities of trees

TL;DR

This paper introduces the concept of normality in trees, analyzing its properties, extremal structures, and differences from eccentricity, providing new insights into tree metrics and related extremal problems.

Contribution

It defines and studies the normality of vertices in trees, compares it with eccentricity, and explores extremal structures and open problems in this context.

Findings

Normality is minimized at peripheral vertices.

Extremal trees differ from classical path and star structures.

Differences between eccentricity and normality reveal new structural insights.

Abstract

The eccentricity of a vertex $v$ in a graph $G$ is the maximum distance from $v$ to any other vertex. The vertices whose eccentricity are equal to the diameter (the maximum eccentricity) of $G$ are called peripheral vertices. In trees the eccentricity at $v$ can always be achieved by the distance from $v$ to a peripheral vertex. From this observation we are motivated to introduce normality of a vertex $v$ as the minimum distance from $v$ to any peripheral vertex. We consider the properties of the normality as well as the middle part of a tree with respect to this concept. Various related observations are discussed and compared with those related to the eccentricity. Then, analogous to the sum of eccentricities we consider the sum of normalities. After briefly discussing the extremal problems in general graphs we focus on trees and trees under various constraints. As opposed to the path and star in numerous extremal problems, we present several interesting and unexpected extremal structures. Lastly we consider the difference between eccentricity and normality, and the sum of these differences. We also introduce some unsolved problems in the context.