Partition and Colored Distances in Graphs Induced to Subsets of Vertices and Some of Its Applications

Mohammad Javad Nadjafi-Arani,Frank Emmert-Streib,Matthias Dehmer,Mahsa Mirzargar

doi:10.3390/sym12122027

Abstract

If G is a graph and P is a partition of V(G), then the partition distance of G is the sum of the distances between all pairs of vertices that lie in the same part of P. A colored distance is the dual concept of the partition distance. These notions are motivated by a problem in the facility location network and applied to several well-known distance-based graph invariants. In this paper, we apply an extended cut method to induce the partition and color distances to some subsets of vertices which are not necessary a partition of V(G). Then, we define a two-dimensional weighted graph and an operator to prove that the induced partition and colored distances of a graph can be obtained from the weighted Wiener index of a two-dimensional weighted quotient graph induced by the transitive closure of the Djoković–Winkler relation as well as by any partition that is coarser. Finally, we utilize our main results to find some upper bounds for the modified Wiener index and the number of orbits of partial cube graphs under the action of automorphism group of graphs.

Highlights

If G is a graph and P is a partition of V ( G ), the colored distance of G is the sum of the distances between all pairs of vertices that lie in the different parts of P
Dankelmann et al tackled a few applications of colored distance toward the facility location problem, median graphs, and the average distance of graphs
Studying partitions and colored distances has been crucial in metric graph theory, as the usefulness of those problems when defining/analyzing quantitative graph measures has been proved

Summary

Introduction

If G is a graph and P is a partition of V ( G ), the colored distance of G is the sum of the distances between all pairs of vertices that lie in the different parts of P This concept was defined by Dankelmann, Goddard, and Slater [1] and is based on a location problem [2]. Demonstrated that the dual concept has more practical value and addressed some applications in mathematical chemistry and network analysis [5,6] to obtain general bounds as well as to classify corresponding extremal graphs They expressed some basic graph invariants such as the diameter and the clique number by utilizing the partition distance. The main contributions of this paper involve applying the extended cut method and introducing new expressions and bounds for distance-based quantities (e.g., modified Wiener index).

Preliminaries

Extended Cut Methods on Some Subsets of Vertices

Some Bounds on the Modified Wiener Index and the Number of Orbits

Concluding Remarks