Suppose I calculate the sum of all closed walks from vertex i in a subgraph of some graph (the subgraph is composed of a subset of vertices from the graph and all edges between those vertices from the graph).
Now I add one further vertex to that subset and find the new subgraph in the same way. I calculate the sum of all closed walks from vertex i in this new subgraph.
Are there results which can tell me anything about how much the sum of closed walks to i increases by when the new vertex is added?
(I feel like there must be an intuitive result here that breaks down walks in the subgraphs and says how they combine to give the increase in the number of walks.)