Fixed point theorems under w-distance on a metric space endowed with a graph

In the present article, we exend the concept of G -contraction by introducing a -contraction in which, we use w -distance in the contractive condition instead of a metric. Our work may be used to deduce many results in the area of fixed point theory particularly in partial ordered metric spaces where two comparable elements can be thought of to be connected by an edge of a graph. Applications and examples are produced to justify the usability of our results.