I don't understand how this property of discrete-time quantum walk relates to the property that the transition operator SC is unitary. Many papers mention this and continue without explaining.

• Suppose we have a directed graph, Let us give separate coin space for every vertex, if we have vertex A pointing to B and B not to A we can just not take the basis d corresponding to the direction of the edge AB in the coin space of B. What is wrong with my reasoning?

• Another reasoning I thought: Since we define an operator U which is unitary on the graph, this means U^t (conjugate transpose) is also an operator that is defined on the graph (sorry if this is very wrong). So if $U$ takes the walker from A to B. Then $U^t$ should take it from B to A. Which is not possible if the edge is directed.

I have spent a lot of time on this and can't seem to come to a good conclusion, please help.

Szegedy's walk is a discrete-time quantum walk(DTQW) but can be applied on a directed graph, it uses no coin space, so is the coin space the reason why DTQW is not possible on a directed graph?