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. [ABSTRACT FROM AUTHOR]