Some fixed point results concerning various contractions in extended <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg" display="inline" id="d1e48"><mml:mi>b</mml:mi></mml:math>- metric space endowed with a graph

Contraction type mappings are crucial for understanding fixed point theory under specific conditions. We propose generalized (Boyd–Wong) type A F and (S - N) rational type contractions in an enlarged b-metric space which are represented by a graphically. Also, we gave a contrast of generalized (Boyd–Wong) type A F — contraction in 2D and 3D. We use appropriate illustrations to demonstrate the validity and primacy of our outcomes. Additionally, we use our derived conclusions to solve the Fredholm integral problem.