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.

the wiener index $w(g)$ of a connected graph $g$‎ ‎is defined as $w(g)=sum_{u,vin v(g)}d_g(u,v)$‎ ‎where $d_g(u,v)$ is the distance between the vertices $u$ and $v$ of‎ ‎$g$‎. ‎for $ssubseteq v(g)$‎, ‎the {it steiner distance/} $d(s)$ of‎ ‎the vertices of $s$ is the minimum size of a connected subgraph of‎ ‎$g$ whose vertex set is $s$‎. ‎the {it $k$-th steiner wiener index/}‎ ‎$sw_k(g)$ of $g$ ...

For any graph G with fixed boundary there exists a layout in the plane, which minimizes the maximum Euclidean distance of any node to its neighbors. This layout balances the length of the graph edges and is therefore called a (length-) balanced layout of G. Furthermore the existence of a unique optimal balanced layout L with the following properties is proved: (i) L is the minimal element of an...

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...

the $k$-th semi total point graph $r^k(g)$ of a graph $g$, is a graph obtained from $g$ by adding $k$ vertices corresponding to each edge and connecting them to endpoint of edge considered. in this paper, we obtain formulae for the resistance distance and kirchhoff index of $r^k(g)$.

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...