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...

Abstract:Minimumdisconnecting cuts of connected graphs provide fundamental information about the connectivity structure of the graph. Spectral methods are well-known as stable and efficient means of finding good solutions to the balanced minimum cut problem. In this paper we generalise the standard balanced bisection problem for static graphs to a new “dynamic balanced bisection problem”, in wh...