Line Graphs and Circulants

The line graph of G, denoted L(G), is the graph with vertex set E(G), where vertices x and y are adjacent in L(G) i edges x and y share a common vertex in G. In this paper, we determine all graphs G for which L(G) is a circulant graph. We will prove that if L(G) is a circulant, then G must be one of three graphs: the complete graph K4, the cycle Cn, or the complete bipartite graph Ka;b, for some a and b with gcd(a; b) = 1