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In this paper, a novel and more general type of sequence of non-linear multivalued mappings as well as the corresponding contractions on a metric space equipped with a graph is initiated. Fixed point results of a single-valued mapping and the new sequence of multivalued mappings are examined under suitable conditions. A non-trivial comparative illustration is provided to support the assumptions of our main theorem. A few important results in ϵ-chainable metric space and cyclic contractions are deduced as some consequences of the concepts obtained herein. As a result of our findings, new criteria for solving a broader form of Fredholm integral equation are established. An open problem concerning discretized population balance model whose solution may be investigated using any of the ideas proposed in this note is highlighted as a future assignment.