$\cHC$ refers to the quantum coin, and $\cHG$ represents the graph on which the walker moves. The discrete time-evolution of the system is governed by the unitary operator $$U=SC\, ,$$ where $C$ is the coin operator which corresponds to flipping the quantum coin, and $S$ is the step or translation operator that moves the walker one step along some outgoing edge, depending on the coin state. Adopting a binary string representation of the vertices $V$ of the underlying graph $G=(V,E)$, the step operator $S$ (a permutation operator of the entire Hilbert space $\cH$) can be expressed as $$S = \sum_{d=0}^{n-1} \sum_{x\in V} {\left|d, x\oplus e_{dx}\right>} {\left<d,x\right|}\, .
\label{eq:S_gendef}$$ In (\[eq:S\_gendef\]) $x$ denotes the vertex index. Here, and in the rest of this paper we identify the vertices with their indices and understand $V$ as the set of vertex indices. The most remarkable fact about $S$ is that it contains all information about the topology of the graph. In particular, the actual binary string values of $e_{dx}$ are determined by the set of edges $E$. This is accomplished by the introduction of direction indices $d$, which run from $0$ to $n-1$ in case of the $n$ regular graphs which are used 