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**Abstract:** In the present work, we introduce the concepts of $(G,\phi ,\psi )$-contraction and $(G,\phi ,\psi )$-graphic contraction defined on metric spaces endowed with a graph $G$ and we show that these two types of contractions generalize a large number of contractions. Subsequently, we investigate various results which assure the existence and uniqueness of fixed points for such mappings. According to the applications of our main results, we further obtain a fixed point theorem for cyclic operators and an existence theorem for the solution of a nonlinear integral equation. Moreover, some illustrative examples are provided to demonstrate our obtained results.

**Keywords:** Fixed point; connected graph; Picard operator; $(\phi ,\psi )$-type contraction.

**Classification (MSC2000):** 47H10; 05C40, 54H25

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*Electronic fulltext finalized on: 29 Dec 2016.
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