Perfect state transfer between non-antipodal vertices in integral circulant graphs. (Q2828846)

From the summary: In this paper we investigate the existence of perfect state transfer in integral circulant graphs between non-antipodal vertices. Perfect state transfer is considered on circulant quantum spin networks with nearest-neighbor couplings. The network is described by a circulant graph \(G\), which is characterized by its circulant adjacency matrix \(A\). Formally, we say that there exists perfect state transfer (PST) between vertices \(a,b\in V(G)\) if \(|F(\tau)_{ab}|=1\), for some positive real number \(\tau\), where \(F(t)=\exp (_{1}At)\). \textit{N. Saxena} et al. [Int. J. Quantum Inf. 5, No. 3, 417--430 (2007; Zbl 1119.81042)] proved that \(|F(\tau)_{aa}|=1\) for some \(a\in V(G)\) and \(\tau \in \mathbb{R}^{+}\) if and only if all the eigenvalues of \(G\) are integer. The integral circulant graph ICG\(_{n}(D)\) has the vertex set \(Z_{n}=\{0,1,\dots ,n-1\}\) and vertices \(a\) and \(b\) are adjacent if gcd\((a-b,n)\in D\), where \(D\subseteq \{d:d\mid n,1\leq d<n\}\). We characterize completely the class of integral circulant graphs having PST between non-antipodal vertices for \(|D|=2\). We have thus answered the question posed by Godsil on the existence of classes of graphs with PST between non-antipodal vertices. Moreover, for all values of \(n\) such that ICG\(_{n}(D)\) has PST \((n\in 4\mathbb{N})\), several clases of graphs ICG\(_{n}(D)\) are constructed such that PST exists between non-antipodal vertices.