We consider topological Markov chains (also called Markov shifts) on countable graphs. We show that a transient graph can be extended to a recurrent graph of equal entropy which is either positive recurrent of null recurrent, and we give an example of each type. We extend the notion of {\em local entropy} to topological Markov chains and prove that a transitive Markov chain admits a measure of maximal entropy (or {\em maximal measure}) whenever its local entropy is less than its (global) entropy.