Parameters of Integral Circulant Graphs and Periodic Quantum Dynamics

Abstract

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.

Eigenspaces of Hamming graphs and unitary Cayley graphs

“…In [16] he characterizes the integral circulant graphs as those circulants on n vertices whose indices (less than n) of ones in the first row of the associated adjacency matrix can be partitioned into complete sets of numbers that have same greatest common divisor with n. This condition is trivially fulfilled for unitary Cayley graphs. It is interesting to note that integral circulants play a role in quantum physics [15].…”

Graphs for small multiprocessor interconnection networks

“…This fact could be relevant in managing load balancing in multiprocessor systems. It is noteworthy that integral graphs have already several applications: in quantum computing (perfect state transfer [22], in quantum spin networks with periodic dynamics [4]) and in theoretical chemistry (hyperenergetic and equienergetic graphs [21]). We expect that some further applications can be found in load balancing (see Section 4).…”

Further results on the perfect state transfer in integral circulant graphs

“…Some properties of quantum dynamics on circulant graphs were studied in [4]. Saxena et al [5] considered circulant graphs as the potential candidates for modeling quantum spin networks having the PST. They show that a circulant graph is periodic if and only if all eigenvalues of the graph are integers (i.e.…”