The set of generalized exponents of primitive simple graphs Original Research Article

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. The exponent set (the set of those numbers attainable as exponents of primitive digraphs with n vertices) and bounds on the exponent have been extensively studied. As a generalization of exponent, R. A. Brualdi and B. Liu introduced three types of generalized exponents for primitive digraphs in 1990. We improve the bounds on these generalized exponents given by B. Liu for primitive simple graphs, and we express explicitly for this class of primitive graphs the exponent sets of all three types of generalized exponents.