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

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.

A New Technique to Uniquely Identify the Edges of a Graph

“…Various distance-related parameters have been studied, such as the partition distance of graphs were studied in [7], Alhevaz et al gave the sharp bounds for the generalized distance spectral radius of graphs [8], Wang studied distance bounds for generalized bicycle codes and Pryadko [9], Nadeem et al found the fault-tolerant partition dimension of oxide interconnection networks [10]. Concerning metric dimensions, that have been of more interest to the research community, one could remark of a few of them (although possibly not all of the most remarkable ones): partition dimension [11], strong metric dimension [12], k-metric dimension [13], identifying codes [14], k-metric anti-dimension [15], local metric dimension [16], edge metric dimension [17] and multiset dimension [18] (see also [19] for the outer-multiset dimension).…”