Parameters of Integral Circulant Graphs and Periodic Quantum Dynamics

The intention of the paper is to move a step towards a classification of network topologies that exhibit periodic quantum dynamics. We show that the evolution of a quantum system, whose hamiltonian is identical to the adjacency matrix of a circulant graph, is periodic if and only if all eigenvalues of the graph are integers (that is, the graph is integral). Motivated by this observation, we focus on relevant properties of integral circulant graphs. Specifically, we bound the number of vertices of integral circulant graphs in terms of their degree, characterize bipartiteness and give exact bounds for their diameter. Additionally, we prove that circulant graphs with odd order do not allow perfect state transfer.Comment: 12 page

Integral circulant graphs of prime power order with maximal energy

The energy of a graph is the sum of the moduli of the eigenvalues of its adjacency matrix. We study the energy of integral circulant graphs, also called gcd graphs, which can be characterized by their vertex count n and a set D of divisors of n in such a way that they have vertex set Zn and edge set {{a, b} : a, b in Zn; gcd(a - b, n) in D}. Using tools from convex optimization, we study the maximal energy among all integral circulant graphs of prime power order ps and varying divisor sets D. Our main result states that this maximal energy approximately lies between s(p - 1)p^(s-1) and twice this value. We construct suitable divisor sets for which the energy lies in this interval. We also characterize hyperenergetic integral circulant graphs of prime power order and exhibit an interesting topological property of their divisor sets.Comment: 25 page

Quantum state transfer on integral oriented circulant graphs

An oriented circulant graph is called integral if all eigenvalues of its Hermitian adjacency matrix are integers. The main purpose of this paper is to investigate the existence of perfect state transfer (\PST for short) and multiple state transfer (\MST for short) on integral oriented circulant graphs. Specifically, a characterization of \PST (or \MST) on integral oriented circulant graphs is provided. As an application, we also obtain a closed-form expression for the number of integral oriented circulant graphs with fixed order having \PST (or \MST).Comment: 15 page