# Variations of the eccentricity and their properties in trees

## TL;DR

This paper introduces new distance-based functions in trees related to eccentricity and uniformity, explores extremal problems, compares these functions with classical measures, and analyzes their properties and bounds.

## Contribution

It proposes novel distance-based functions in trees, studies their extremal properties, and compares them with traditional eccentricity measures, revealing similarities and bounds.

## Key findings

- New distance-based functions are introduced and analyzed.
- Sharp bounds for these functions are established.
- The difference between eccentricity and uniformity behaves similarly to eccentricity.

## Abstract

Motivated from the study of eccentricity, center, and sum of eccentricities in graphs and trees, we introduce several new distance-based global and local functions based on the smallest distance from a vertex to some leaf (called the `uniformity' at that vertex). Some natural extremal problems on trees are considered. Then the middle parts of a tree is discussed and compared with the well-known center of a tree. The values of the global functions are also compared with the sum of eccentricities and some sharp bounds are established. Last but not the least, we show that the difference between the eccentricity and the uniformity, when considered as a local function, behaves in a very similar way as the eccentricity itself.