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This article deals with a few outcomes ensuring the fixed points of a weak <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>G</mi><mo>,</mo><mi>ψ</mi><mo>)</mo></mrow></semantics></math></inline-formula>-contraction map of metric spaces comprised with a reflexive and transitive digraph <i>G</i>. To validate our findings, we furnish several examples. The findings we obtain enable us to seek out the unique solution of a nonlinear integral equation. The outcomes presented herewith sharpen, subsume, unify, improve, enrich, and compile a number of existing theorems.