Cat and mouse are doing a random walk on nonbipartite connected graph G. Show that exp. time for C to catch M is bounded by $|E(G)|^2|V(G)|$

Question

I am trying to solve this exercise and am having trouble starting. The cat and mouse all move at the same time in discrete time steps. The cat catches the mouse when both are at the same location at the same time. I tried solving this by using a markov chain and trying to use some bounds on the expected time to cover the whole graph. But I am getting nowhere.

Any advice or help would be greatly appreciated