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Title

A CLASSIFICATION OF RAMANUJAN COMPLEMENTS OF UNITARY CAYLEY GRAPHS

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Abstract

 THE UNITARY CAYLEY GRAPH ON N VERTICES, XN, HAS VERTEX SET ZN, AND TWO VERTICES AAND B ARE CONNECTED BY AN EDGE IF AND ONLY IF THEY DIFFER BY A MULTIPLICATIVE UNIT MODULO N, I.E. GCD (AB, N) =1. A K-REGULAR GRAPH X IS RAMANUJAN IF AND ONLY IF (FORMULA) WHERE L(X) IS THE SECOND LARGEST ABSOLUTE VALUE OF THE EIGENVALUES OF THE ADJACENCY MATRIX OF X. WE OBTAIN A COMPLETE CHARACTERIZATION OF THE CASES IN WHICH THE COMPLEMENTS OF UNITARY CAYLEY GRAPH XN IS A RAMANUJAN GRAPH.

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