Circulant graphs are an extremely well-studied subclass of regular graphs, partially because they model many practical computer network topologies. It has long been known that the number of spanning trees in n-node circulant graphs with constant jumps satisfies a recurrence relation in n. For the non-constant-jump case, i.e., where some jump sizes can be functions of the graph size, only a few ...

In this paper we present a time-polynomial recognition algorithm for certain classes of circulant graphs. Our approach uses coherent configurations and Schur rings generated by circulant graphs for elucidating their symmetry properties and eventually finding a cyclic automorphism.

For a given graph G, denote by A its adjacency matrix and F (t) = exp(iAt). We say that there exists a perfect state transfer (PST) in G if |F (τ)ab| = 1, for some vertices a, b and a positive real number τ . Such a property is very important for the modeling of quantum spin networks with nearest-neighbor couplings. We consider the existence of the perfect state transfer in integral circulant g...

Circulant graphs are an important class of interconnection networks in parallel and distributed computing. Integral circulant graphs play an important role in modeling quantum spin networks supporting the perfect state transfer as well. The integral circulant graph ICGn(D) has the vertex set Zn = {0, 1, 2, . . . , n− 1} and vertices a and b are adjacent if gcd(a− b, n) ∈ D, where D ⊆ {d : d | n...

All graphs considered in the paper are directed. Let % be a graph on n vertices which we identify with the elements of the additive cyclic group Zn 1⁄4 f0; 1; . . . ; n 1g. The graph % is called circulant if it has a cyclic symmetry, that is, if the permutation ð0; 1; 2; . . . ; n 1Þ is an automorphism of the graph. Each circulant graph is completely determined by its connection set S which is ...

Let D be a digraph, V (D) and A(D) will denote the sets of vertices and arcs of D, respectively. A digraph D is 3-transitive if the existence of the directed path (u, v, w, x) of length 3 in D implies the existence of the arc (u, x) ∈ A(D). In this article strong 3-transitive digraphs are characterized and the structure of non-strong 3-transitive digraphs is described. The results are used, e.g...

The random walk formalism is used across a wide range of applications, from modelling share prices to predicting population genetics. Likewise, quantum walks have shown much potential as a framework for developing new quantum algorithms. Here we present explicit efficient quantum circuits for implementing continuous-time quantum walks on the circulant class of graphs. These circuits allow us to...

We estimate the norms of standard Gaussian random Toeplitz and circulant matrices and their inverses, mostly by means of combining some basic techniques of linear algebra. In the case of circulant matrices we obtain sharp probabilistic estimates, which show that these matrices are expected to be very well conditioned. Our probabilistic estimates for the norms of standard Gaussian random Toeplit...

A first kind Frobenius graph is a Cayley graph Cay(K,S) on the Frobenius kernel of a Frobenius group K o H such that S = a for some a ∈ K with 〈a〉 = K, where H is of even order or a is an involution. It is known that such graphs admit ‘perfect’ routing and gossiping schemes. A circulant graph is a Cayley graph on a cyclic group of order at least three. Since circulant graphs are widely used as ...