Line Graphs and Circulants

Abstract

The line graph of \(G\), denoted \(L(G)\), is the graph with vertex set \(E(G)\), where vertices \(x\) and \(y\) are adjacent in \(L(G)\) if and only if 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 \(K_4\), the cycle \(C_n\), or the complete bipartite graph \(K_{a,b}\), for some \(a\) and \(b\) with \(\gcd(a,b) = 1\).