Distance-balanced closure of some graphs

TL;DR

This paper proves that certain distance-balanced graphs with high maximum degree are regular and introduces the concept of distance-balanced closure, providing specific closures for trees with high maximum degree.

Contribution

It establishes regularity conditions for distance-balanced graphs with high maximum degree and defines the distance-balanced closure, including explicit closures for certain trees.

Findings

Distance-balanced graphs with maximum degree at least |V(G)|-3 are regular.

Introduces the notion of distance-balanced closure of a graph.

Finds distance-balanced closures for trees with maximum degree at least |V(T)|-3.

Abstract

In this paper we prove that any distance-balanced graph $G$ with $\Delta(G)\geq |V(G)|-3$ is regular. Also we define notion of distance-balanced closure of a graph and we find distance-balanced closures of trees $T$ with $\Delta(T)\geq |V(T)|-3$.