I would like to start with considering the time-dependent heat equation on a connected graph and consider its Laplacian matrix. Suppose we have a connected graph with unknown temperature on vertices. Let's for simplicity assume that we have few heat sources and we want to transfer this heat through some edges of the graph. Deciding which edges will contribute in transferring the heat would be of interest. We are given a laplacian matrix and we want to find those edges that will appear in the transferring (I think binary variables will be best choice).
Another interesting idea would be as following. Suppose we have some sensors that we want to install these in some vertices to measure the heat. The question is to find the best place for installing these sensors on the vertices.
My idea: To start, I will need to model it respect to time discretization. something like this: $$A T^{k+1} = B T^k + Cf^k$$ where the initial time is given and the time step is assumed to be constant. First, I have to determine the matrix $A$, $B$, and $C$.
Would you please let me know if you know some references or have some comments on this topic?
