For an ordered set W = {w1, w2, . . . , wk} of vertices and a vertex v in a connected graph G, the ordered k-vector r(v|W ) := (d(v,w1), d(v,w2), . . . , d(v,wk)) is called the (metric) representation of v with respect to W , where d(x, y) is the distance between the vertices x and y. The set W is called a resolving set for G if distinct vertices of G have distinct representations with respect ...

In [5] Witzgall proved that any weak metric defined on a real vector space, which is convex in each of the arguments, is determined by a weak gauge. In this paper we extend this result to any continuous weak metric defined on the positive cone in a totally ordered vector space, which is convex in each of the arguments.

the sequential $p$-convergence in a fuzzy metric space, in the sense of george and veeramani, was introduced by d. mihet as a weaker concept than convergence. here we introduce a stronger concept called $s$-convergence, and we characterize those fuzzy metric spaces in which convergent sequences are $s$-convergent. in such a case $m$ is called an $s$-fuzzy metric. if $(n_m,ast)$ is a fuzzy metri...

‎In this paper‎, ‎first we introduce the notion of $frac{1}{2}$-modular metric spaces and weak $(alpha,Theta)$-$omega$-contractions in this spaces and we establish some results of best proximity points‎. ‎Finally‎, ‎as consequences of these theorems‎, ‎we derive best proximity point theorems in modular metric spaces endowed with a graph and in partially ordered metric spaces‎. ‎We present an ex...

For an ordered set W = {w1, w2, · · · , wk} of vertices and a vertex v in a connected graph G, the representation of v with respect to W is the ordered k-tuple r(v|W ) = (d(v, w1), d(v, w2), · · · , d(v, wk)) where d(x, y) represents the distance between the vertices x and y. The set W is called a resolving set for G if every vertex of G has a distinct representation. A resolving set containing...