A graph G is strongly distance-balanced if for every edge uv of G and every i ≥ 0 the number of vertices x with d(x, u) = d(x, v)−1 = i equals the number of vertices y with d(y, v) = d(y, u) − 1 = i. It is proved that the strong product of graphs is strongly distance-balanced if and only if both factors are strongly distancebalanced. It is also proved that connected components of the direct pro...

Throughout this paper, we present a new class of graphs so-called n-edgedistance-balanced graphs inspired by the concept of edge-distance-balanced property initially introduced by Tavakoli et al. [Tavakoli M., Yousefi-Azari H., Ashrafi A.R., Note on edge distance-balanced graphs, Trans. Combin. 1 (1) (2012), 1-6]. Moreover, we propose some characteristic results to recognize 2-edge-distance-bal...

An unweighted, connected graph is distance-balanced (also called self-median) if there exists a number d such that, for any vertex v, the sum of the distances from v to all other vertices is d. An unweighted connected graph is strongly distancebalanced (also called distance-degree regular) if there exist numbers d1, d2, d3, . . . such that, for any vertex v, there are precisely dk vertices at d...

in this paper, we introduce the notions of product vague graph, balanced product vague graph, irregularity and total irregularity of any irregular vague graphs and some results are presented. also, density and balanced irregular vague graphs are discussed and some of their properties are established. finally we give an application of vague digraphs.

A graph Γ is said to be distance-balanced if for any edge uv of Γ, the number vertices closer u than v equal u, and it called nicely in addition this independent chosen uv. strongly integer k, at distance k from k+1 v. In paper we solve an open problem posed by Kutnar Miklavič (2014) constructing several infinite families nonbipartite graphs which are not distance-balanced. We disprove a conjec...

In this paper, we first collect the earlier results about some graph operations and then we present applications of these results in working with chemical graphs.

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