A pythagorean fuzzy set is a generalized fuzzy structure that copes with imprecisions of real life accurately and with a larger domain. The aim of this manuscript is to develop the notion of graph for newly established concept of interval-valued Pythagorean fuzzy set. We introduced the concept of interval-valued Pythagorean fuzzy graph (IVPyFG) and discussed its related ideas with the help of examples. The significance of using interval-valued fuzzy concepts over non-interval-valued fuzzy concepts are demonstrated numerically. Further, the advantages of using the new approach over the pre-existing ideas are demonstrated with the help of numerical examples. The main goal of this article is to study three types of decision-making problems using the framework of IVPyFGs. These three problems include the problem of selection of the best university in a network of universities, a supply chain management problem and the shortest path problem. The comparison of the new approach toward these problems with existing approaches is also established.