The extension of Neutrosophic Set is the Quadripartitioned Neutrosophic Set. The division of indeterminacy component of the Neutrosophic Set into the contradiction and ignorance component is termed as the Quadripartitioned Neutrosophic Set. Fermatean Quadripartitioned Neutrosophic Fuzzy Graph is the integrating form of Fermatean and Quadripartitioned Neutrosophic Fuzzy Graph (FQNFG). In this article Regular, Totally Regular, Irregular, Totally Irregular, Neighbourly Irregular, Neighbourly Totally Irregular, Highly Irregular and Highly Totally Irregular of Fermatean Quadripartitioned Neutrosophic Fuzzy Graph are defined with examples. Also its properties were verified using theorems.