Distance-unbalancedness of graphs

TL;DR

This paper introduces a new graph invariant called distance-unbalancedness, which measures how balanced a graph is in terms of distances, providing formulas for various graph classes and exploring properties of trees.

Contribution

The paper defines the distance-unbalancedness invariant, derives explicit formulas for specific graphs, and investigates its properties in trees, opening new research directions.

Findings

Formulas for distance-unbalancedness in several graph classes

Analysis of distance-unbalancedness in trees

Open problems and conjectures proposed

Abstract

In this paper we propose and study a new structural invariant for graphs, called distance-unbalanced\-ness, as a measure of how much a graph is (un)balanced in terms of distances. Explicit formulas are presented for several classes of well-known graphs. Distance-unbalancedness of trees is also studied. A few conjectures are stated and some open problems are proposed.