On regular graphs with four distinct eigenvalues

Let G(4,2)G(4,2) be the set of connected regular graphs with four distinct eigenvalues in which exactly two eigenvalues are simple, G(4,2,−1)G(4,2,−1) (resp. G(4,2,0)G(4,2,0)) the set of graphs belonging to G(4,2)G(4,2) with −1 (resp. 0) as an eigenvalue, and G(4,≥−1)G(4,≥−1) the set of connected regular graphs with four distinct eigenvalues and second least eigenvalue not less than −1. In this paper, we prove the non-existence of connected graphs having four distinct eigenvalues in which at least three eigenvalues are simple, and determine all the graphs in G(4,2,−1)G(4,2,−1). As a by-product of this work, we characterize all the graphs belonging to G(4,≥−1)G(4,≥−1) and G(4,2,0)G(4,2,0), respectively, and show that all these graphs are determined by their spectra.