Relationships between net regularity, strong regularity and walk regularity of signed graphs

In this paper, we consider relationships between net-regular, strongly regular and walk-regular signed graphs. There is no inclusion between any two of these classes, and we investigate conditions under which a signed graph is in the intersection of two or all three classes. In this context, we deduce which strongly regular signed graphs are net-regular, and we provide several sufficient conditions for a walk-regular signed graph to be net-regular or strongly regular, or both. It occurs that, in many situations, signed graphs belonging to at least two classes have a comparatively small number of distinct eigenvalues. Also, we investigate strongly regular signed graphs that induce, or are induced by, symmetric 2-class or 3-class association schemes. Many necessary and sufficient conditions are established, and in all results a limited number of distinct eigenvalues figures as one of them. Some problems that arose during the research are formulated.