2-Edge Distance-Balanced Graphs

TL;DR

This paper introduces the concept of 2-edge distance-balanced graphs, explores methods to recognize such graphs for k=2,3, and investigates their properties under graph products and subdivisions.

Contribution

It defines 2-edge distance-balanced graphs, provides recognition methods for k=2,3, and analyzes their behavior under Cartesian and lexicographic products.

Findings

Recognition methods for 2- and 3-edge distance-balanced graphs.

Conditions under which graph products preserve the 2-edge distance-balanced property.

Verification of the property in certain subdivision-related graphs.

Abstract

In a graph A, for each two arbitrary vertices g, h with d(g,h)=2,|MAg2h|=mAg2h is introduced the number of edges of A that are closer to g than to h. We say A is a 2-edge distance-balanced graph if we have mAg2h=mAh2g. In this article, we verify the concept of these graphs and present a method to recognize k-edge distance-balanced graphs for k = 2,3 using existence of either even or odd cycles. Moreover, we investigate situations under which the Cartesian and lexicographic products lead to 2-edge distance -balanced graphs. In some subdivision-related graphs 2-edge distance-balanced property is verified.