On the Number of Synchronizing Colorings of Digraphs

We deal with $k$-out-regular directed multigraphs with loops (called simply \emph{digraphs}). The edges of such a digraph can be colored by elements of some fixed $k$-element set in such a way that outgoing edges of every vertex have different colors. Such a coloring corresponds naturally to an automaton. The road coloring theorem states that every primitive digraph has a synchronizing coloring. In the present paper we study how many synchronizing colorings can exist for a digraph with $n$ vertices. We performed an extensive experimental investigation of digraphs with small number of vertices. This was done by using our dedicated algorithm exhaustively enumerating all small digraphs. We also present a series of digraphs whose fraction of synchronizing colorings is equal to $1-1/k^d$, for every $d \ge 1$ and the number of vertices large enough. On the basis of our results we state several conjectures and open problems. In particular, we conjecture that $1-1/k$ is the smallest possible fraction of synchronizing colorings, except for a single exceptional example on 6 vertices for $k=2$.Comment: CIAA 2015. The final publication is available at http://link.springer.com/chapter/10.1007/978-3-319-22360-5_1

Bipartite Quantum Walks and the Hamiltonian

We study a discrete quantum walk model called bipartite walks via a spectral approach. A bipartite walk is determined by a unitary matrix U, i.e., the transition matrix of the walk. For every transition matrix U, there is a Hamiltonian H such that U = exp(iH). If there is a real skew-symmetric matrix S such that H = iS, we say there is a H-digraph associated to the walk and S is the skew-adjacency matrix of the H-digraph. The underlying unweighted non-oriented graph of the H-digraph is H-graph. Let G be a simple bipartite graph with no isolated vertices. The bipartite walk on G is the same as the continuous walk on the H-digraph over integer time. Two questions lie in the centre of this thesis are 1. Is there a connection between the H-(di)graph and the underlying graph G? If there is, what is the connection? 2. Is there a connection between the walk and the underlying graph G? If there is, what is the connection? Given a bipartite walk on G, we show that the underlying bipartite graph G determines the existence of the H-graph. If G is biregular, the spectrum of G determines the spectrum of U. We give complete characterizations of bipartite walks on paths and even cycles. Given a path or an even cycle, the transition matrix of the bipartite walk is a permutation matrix. The H-digraph is an oriented weighted complete graph. Using bipartite walks on even paths, we construct a in nite family of oriented weighted complete graphs such that continuous walks de- ned on them have universal perfect state transfer, which is an interesting but rare phenomenon. Grover's walk is one of the most studied discrete quantum walk model and it can be used to implement the famous Grover's algorithm. We show that Grover's walk is actually a special case of bipartite walks. Moreover, given a bipartite graph G, one step of the bipartite walk on G is the same as two steps of Grover's walk on the same graph. We also study periodic bipartite walks. Using results from algebraic number theory, we give a characterization of periodic walks on a biregular graph with a constraint on its spectrum. This characterization only depends on the spectrum of the underlying graph and the possible spectrum for a periodic walk is determined by the degrees of the underlying graph. We apply this characterization of periodic bipartite walk to Grover's walk to get a characterization of a certain class of periodic Grover's walk. Lastly, we look into bipartite walks on the incidence graphs of incidence structures, t-designs (t 2) and generalized quadrangles in particular. Given a bipartite walk on a t-design, we show that if the underlying design is a partial linear space, the H-graph is the distance-two graph of the line graph of the underlying incidence graph. Given a bipartite walk on the incidence graph of a generalized quadrangle, we show that there is a homogeneous coherent algebra raised from the bipartite walk. This homogeneous coherent algebra is useful in analyzing the behavior of the walk