Generalized exponents of non-primitive graphs

The exponent of a primitive digraph is the smallest integer k such that for each ordered pair of (not necessarily distinct) vertices x and y there is a walk of length k from x to y. As a generalization of exponent, Brualdi and Liu (Linear Algebra Appl. 14 (1990) 483499) introduced three types of generalized exponents for primitive digraphs in 1990. In this Paper we extend their definitions of generalized exponents from primitive digraphs to general digraphs which are not necessarily primitive. We give necessary and sufficient conditions for the finiteness of these generalized exponents for graphs (undirected, corresponding to symmetric digraphs) and completely determine the largest finite values and the exponent sets of generalized exponents for the class of non-primitive graphs of Order n, the class of connected bipartite graphs of Order n and the class of trees of Order n.