Keywords: Fixed Point; directed graph; Geraghty contraction; metric space.

AMS(2010) Mathematics Subject Classification:  47H10, 54H25.

1. INTRODUCTION AND PRELIMINARIES

Banach contraction principle is one of the most fundamental results in fixed point theory; by extending the contractive condition and the ambi- ent space, there are several extensions and generalizations. Jachmski [17] extended the structure of orders is replaced by the structure of Graphs on metric spaces in extended fixed point theory. The intersection of theories of fixed point findings with single and multi valued mappings is known as fixed point theory and graph theory. Many researchers [2, 3, 6, 8, 9]studied fixed point results on various spaces endowed with graphs. Fixed point results ex- tended using Gerghty [15, 4]contractions with specific properties. Recently [7] proved the existence of fixed point theorems of auxiliary functions frac- tional differential equations with applications.

Note that metric fixed point and graph theory have common application environments. In the multivalued case, the authors in [3] proved a fixed point theorem for Mizoguchi–Takahashi-type contractions on a metric space endowed with a graph. For further results in this direction, we refer to [4–11]. Recently, in [12], the authors introduced a new concept of contrac- tions called F-Khan contractions and proved a related fixed point theorem. The investigation of iterative plans for different classes of contractive and nonexpansive mappings is a focal point in measurement fixed point hypothe- sis. It began with crafted by Banach who demonstrated an old style hypoth- esis, known as the Banach constriction guideline, for the presence of a one of a kind fixed point for a withdrawal. The significance of this outcome is that it likewise gives the intermingling of an iterative plan to the one of a kind fixed point. A few creators have likewise given results managing the pres- ence and estimate of fixed marks of specific classes of non expansive-type multi functions.   Suzuki laid out some strategies which broaden the no- table withdrawal techniques for mappings and multi functions. It is realized that consolidating a few branches is a regular movement in various areas of science particularly in math. Normally, it is prominent in fixed point hypothesis. Throughout recent many years, there have been a great deal of movement in fixed point hypothesis furthermore, one more branches in arithmetic such differential conditions, calculation and mathematical geog- raphy. In 2005, Echenique gave a short and useful evidence of an expansion of Tarski’s proper point hypothesis which is significant in the hypothesis of games. In 2006,  Espinola and Kirk provided useful results on combining fixed point theory and graph theory [6]. In 2008 and 2009, Jachymski con- tinued this idea by using different view (see [8] and [7]). Then,  Beg,  Butt and Radojevic obtained some results in 2010 (see [2]) in the same direction. In this paper, we present some iterative scheme results for G-contractive and G-nonexpansive maps on graphs.