A Pythagorean fuzzy soft set, an extension of intuitionistic fuzzy soft set, plays an essential role to handle the vagueness in many real-life problems. We apply this concept to graph theory, and present certain new notions including, perfectly regular Pythagorean fuzzy soft graphs (PFSGs), perfectly edge-regular PFSGs and explore some of their properties. We formulate the notion of perfectly irregular PFSGs, perfectly edge-irregular PFSGs and open neighborhood degree sum and closed neighborhood degree sum of PFSGs. Finally, we discuss some decision-making problems of PFSGs.
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