In application domains such as routing, network analysis, scheduling, and planning, directed graphs are widely used as both formal models and core data structures for the development of efficient algorithmic solutions. In these areas, graphs are often evolving in time: for example, connection links may fail due to temporary technical issues, meaning that edges of the graph cannot be traversed for some time interval and alternative paths have to be followed. In classical computation graphs have been implemented both explicitly through adjacency matrices/lists and symbolically as ordered binary decision diagrams. Moreover, ad-hoc visit procedures have been developed to deal with dynamically evolving graphs. Quantum computation, exploiting interference and entanglement, has provided an exponential speed-up for specific problems, e.g., database search and integer factorization. In the quantum framework everything must be represented and manipulated using reversible operators. This poses a challenge when one has to deal with traversals of dynamically evolving directed graphs. Graph traversals are not intrinsically reversible because of converging paths. In the case of dynamically evolving graphs also the creation/destruction of paths comes into play against reversibility. In this paper we propose a novel high level graph representation in quantum computation supporting dynamic connectivity typical of real-world network applications. Our procedure allows to encode any multigraph into a unitary matrix. We devise algorithms for computing the encoding that are optimal in terms of time and space and we show the effectiveness of the proposal with some examples. We describe how to react to edge/node failures in constant time. Furthermore, we present two methods to perform quantum random walks taking advantage of this encoding: with and without projectors. We implement and test our encoding obtaining that the theoretical bounds for the running time are confirmed by the empirical results and providing more details on the behavior of the algorithms over graphs of different densities.