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This work studies the relation between two graph parameters, rho and Lambda. For an undirected graph G, rho(G) is the growth rate of its universal covering tree, while Lambda(G) is a weighted geometric average of the vertex degree minus one, corresponding to the rate of entropy growth for the non-backtracking random walk (NBRW).
It is well known that rho(G) >= Lambda(G) for all graphs, and that graphs with rho=Lambda exhibit some special properties. In this work we derive an easy to check, necessary and sufficient condition for the equality to hold. Furthermore, we show that the variance of the number of random bits used by a length k NBRW is O(1) if rho = Lambda and Omega(k) if $rho > \Lambda. As a consequence we exhibit infinitely many non-trivial examples of graphs with rho = Lambda.

Joint work with Idan Eisner, Tel-Hai College