  We develop the theory of linear evolution equations associated with the
adjacency matrix of a graph, focusing in particular on infinite graphs of two
kinds: uniformly locally finite graphs as well as locally finite line graphs.
We discuss in detail qualitative properties of solutions to these problems by
quadratic form methods. We distinguish between backward and forward evolution
equations: the latter have typical features of diffusive processes, but cannot
be well-posed on graphs with unbounded degree. On the contrary, well-posedness
of backward equations is a typical feature of line graphs. We suggest how to
detect even cycles and/or couples of odd cycles on graphs by studying backward
equations for the adjacency matrix on their line graph.
