We introduce the notion of uniform number of a graph. The  uniform number of a connected graph $G$ is the least cardinality of a nonempty subset $M$ of the vertex set of $G$ for which the function $f_M: M^crightarrow mathcal{P}(X) - {emptyset}$ defined as $f_M(x) = {D(x, y): y in M}$ is a constant function, where $D(x, y)$ is the detour distance between $x$ and $y$ in $G$ and $mathcal{P}(X)$ ...

Graph theory has an important role in the area of applications of networks and clustering‎. ‎In the case of dealing with uncertain data‎, ‎we must utilize ambiguous data such as fuzzy value‎, ‎fuzzy interval value or values of fuzzy number‎. ‎In this study‎, ‎values of fuzzy number were used‎. ‎Initially‎, ‎we utilized the fuzzy number value fuzzy relation and then proposed fuzzy number-value f...

The concept of configuration of groups which is defined in terms of finite partitions and finite strings of elements of the group is presented by Rosenblatt and Willis. To each set of configurations, a finite system of equations known as configuration equations, is associated. Rosenblatt and Willis proved that a discrete group G is amenable if and only if every possible instance of its configur...

the crossing number of a graph  is the minimum number of edge crossings over all possible drawings of  in a plane. the crossing number is an important measure of the non-planarity of a graph, with applications in discrete and computational geometry and vlsi circuit design. in this paper we introduce vertex centered crossing number and study the same for maximal planar graph.

‎let $g=(v(g),e(g))$ be a graph‎, ‎$gamma_t(g)$. let $ooir(g)$ be the total domination and oo-irredundance number of $g$‎, ‎respectively‎. ‎a total dominating set $s$ of $g$ is called a $textit{total perfect code}$ if every vertex in $v(g)$ is adjacent to exactly one vertex of $s$‎. ‎in this paper‎, ‎we show that if $g$ has a total perfect code‎, ‎then $gamma_t(g)=ooir(g)$‎. ‎as a consequence, ...

a {em 2-rainbow dominating function} (2rdf) of a graph $g$ is a function $f$ from the vertex set $v(g)$ to the set of all subsets of the set ${1,2}$ such that for any vertex $vin v(g)$ with $f(v)=emptyset$ the condition $bigcup_{uin n(v)}f(u)={1,2}$ is fulfilled, where $n(v)$ is the open neighborhood of $v$. the {em weight} of a 2rdf $f$ is the value $omega(f)=sum_{vin v}|f (v)|$. the {em $2$-r...