Perfectly Regular and Perfectly Edge-regular Fuzzy Graphs

Abstract

A perfectly regular fuzzy graph is a fuzzy graph that is both regular and totally regular. A perfectly edge-regular fuzzy graph is a fuzzy graph that is both edge-regular and totally edge-regular. In this paper, we introduce and classify these types of fuzzy graphs and study several of their properties, including how these two classes of fuzzy graphs structurally relate to one another and several of their spectral properties such as isospectral fuzzy graphs and when the energy of fuzzy graphs is proportional to the energy of their underlying crisp graphs. These properties are studied in particular due to their having at least one constant function between and .