The Directed Graph Of Generalized Base Number And The Original Research Of Generalized Index Of Scrambling

The combinatorial properties of graph theory is a very important foundational question incombinatorial mathematics, its research and development prospect are very extensive. It hasimportant application prospect in many aspects such as information science, sociology,economic mathematics and computer science and so on. In this dissertation, we considered aspecial class of primitive digraphs which contained three cycles, two cycles of which is notintersect but equal length. We studied the local primitive exponents, scrambling index andgeneralized scrambling indices of such graphs. And studied the local bases of primitivenon-powerful signed digraph which basis on such graphs.In chapter1, we introduce some concepts of graph theory, the research progress andsome relevant basic knowledges of the primitive exponent, scrambling index of primitivedigraph and the base of primitive non-powerful signed digraph. At the same time, we get themain conclusion.In chapter2, we study a special class of primitive non-powerful signed digraphs whichcontained three cycles, two cycles of which is not intersect but equal length. Through theanalysis of whether there are a pair of SSSD walks in digraphs, by using some of thedefinition and nature about the primitive non-powerful signed digraph and Frobenius number,we get the local bases of such graphs.in chapter3, we study a special class of primitive digraphs which contained three cycles,two cycles of which is not intersect but equal length. Through analyses the vertex set of eachvertex can be reached by a walk of length t in digraph, by using some of the definitions andnature about the scrambling index and the generalized scrambling indices, we give thescrambling index and the upper and lower bounds of generalized scrambling indices of suchgraphs.In chapter4, we discuss the primitive digraph D and the primitive non-powerful signed digraph S, by using the graph theory method and the definitions about the primitive exponentand the scrambling index, give the relation of the primitive exponent and the scrambling indexabout D and D~k and the relation of the base about S andS k.