Distance-balanced graphs are introduced as graphs in which every edge uv has the following property: the number of vertices closer to u than to v is equal to the number of vertices closer to v than to u. Basic properties of these graphs are obtained. The new concept is connected with symmetry conditions in graphs and local operations on graphs are studied with respect to it. Distance-balanced C...

It is shown that the graphs for which the Szeged index equals ‖G‖·|G| 2 4 are precisely connected, bipartite, distance-balanced graphs. This enables to disprove a conjecture proposed in [Some new results on distance-based graph invariants, European J. Combin. 30 (2009) 1149–1163]. Infinite families of counterexamples are based on the Handa graph, the Folkman graph, and the Cartesian product of ...

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...

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...

A connected graph $\G$ is called {\em nicely distance--balanced}, whenever there exists a positive integer $\gamma=\gamma(\G)$, such that for any two adjacent vertices $u,v$ of are exactly $\gamma$ which closer to $u$ than $v$, and $v$ $u$. Let $d$ denote the diameter $\G$. It known $d \le \gamma$, distance-balanced graphs with $\gamma = d$ precisely complete cycles length $2d$ or $2d+1$. In th...

A graph X is said to be distance–balanced if for any edge uv of X , the number of vertices closer to u than to v is equal to the number of vertices closer to v than to u. A graph X is said to be strongly distance–balanced if for any edge uv of X and any integer k, the number of vertices at distance k from u and at distance k+1 from v is equal to the number of vertices at distance k + 1 from u a...