Digraphs Of The Generalized Index

Generalized primitive exponents of digraphs are generalizations of traditional primitive exponents of primitive digraphs, with applications in memoryless communication systems. The determination of upper bounds and corresponding exponent sets for different digraph classes is an important problem in the study of generalized primitive exponents. In this paper, we are concerned with the generalized primitive exponents of the digraphs with given order and girth 2. The main results are:(1)We give an upper bound for kth generalized primitive exponents of primitive digraphs with order n and girth 2, and determine the exponent set of this class digraphs in section 2. (2)We give an upper bound for kth generalized primitive exponents of non-primitive digraphs with order n and girth 2, determine the exponent set of this class digraphs in section 3, and discuss the extreme cases. (3)We give an upper bound for Lewin index,which is related to generalized primitive exponents of primitive digraph of order n and girth 2, determine the index set of this class digraph and discuss the extreme cases.