"Find the relation between the theory of Markov chains and the theory of graph spectra."
We shou that both theories are ruled by the PERRON-FROBEWIUS projeotion and three matrix representations of it, which yield a common generalization of the HOFFMAN theorems for graphs, directed graphs and point stable designs as uell as the ergodio theorem for stochastio matrices and the ranking procedure of WEI and KENDALL for tournaments.