On regular graphs with four distinct eigenvalues

Let 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) (resp. g(4, 2, 0)) the set of graphs belonging to g(4, 2) with -1 (resp. 0) as an eigenvalue, and 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). As a by-product of this work, we characterize all the graphs belonging to g(4, >= -1) and g(4, 2,0), respectively, and show that all these graphs are determined by their spectra. (C) 2016 Published by Elsevier Inc.