Combinational Properties Of Graph Theory And Sign Pattern Matrix As Well As The Applications In Intelligent System

The Combinatorial Matrix Theory is an organic combination of the combinatorialmathematics, graph theory and linear algebra. Over the recent20years, as a newly arising andfast developing branch of the mathematics, it uses the theory of matrices and linear algebra toapprove the combinatorial theorem, analyze and describe the combinatorial properties ofsome qualitative combinations; and it also applies the combinatorial methods to analyze andapprove some algebraic problems (such as the classic Theorem—Cayley-Hamilton). Themodels and systems developed in many fields of the science is consistent with the mostqualitative properties in the mathematical model, thus the combinatorial matrix theory is notonly closely connected with many mathematical fields, but also it frequently appears and hasparticular and practical application backgrounds in many other research fields includinginformation science, sociology, economics, biology, chemistry, theoretical computer scienceand control theory et al., therefore, Combinatorial Matrix Theory is a very active andimportant research field. Since the emergence of Combination Matrix Theory developed byBrualdi and Ryser, it has largely driven the study and application of the combinatorial matrixtheory.As important components of the combinatorial matrix theory–the study of graph theoryand sign pattern matrix, it is currently one very young and active research subject among theinternational combinatorial matrix theories. Graph theory mainly studies Topologicalproperties of graph, provides the model for any system which includes some kind of binaryrelation to analyse and describe. The sign pattern matrix mainly studies the qualitativecomposite structures, namely not considering the value of its elements. As is well-known, themode of information exchanging and sharing is determined by the communication topology ofan agent system. Thus the status of communication links is crucial for system stability andtask achievement. As a result, great effort has been devoted to investigation of the following two problems. One is under what communication link conditions can the networked systemobtain the desired status, such as achieving a steady state. The other is how to design theconnection pattern between each element for agents to meet these conditions required by thespecific practical tasks.We use incidence matrix and some properities of sign pattern matrix astheoretical basis to state the close relationship between the proposed condition and loopcircuit in directed graph and be used to study some other network problems, such as Edgeconsistency of directed network. And as the important roles in communication system are thegeneralized scrambling index and the generalized μ-scrambling index of the special primitiveno-powerful signed digraphs, in this paper we study them.In the meanwhile studying underwhat conditions can a digraph be balanced; the combinatorial properties of Sign patternmatrix and the application on the intelligent system.The main research contents are asfollowing:(1) The basic concepts of graph theory and sign pattern matrix were introduced. Theresearch progress of sign pattern matrix, generalized scrambling index and generalizedμ-scrambling index, the application of the sign pattern matrix in intelligent system weredescribed.(2) We studied the application of graph theory in mechanical devices. Compensation ofVariable Pitch Roller Chains for the Polygon Effect was researched by use of the graph.The roller chain with variable pitches to abate and reduce polygon effect was designed.(3) We linked the relationship between each component of intelligent system to the degreeof influence between them by edges,the sign graph was composed of edges between positiveelement which were identified by “1” or “-1” and edges between negative element whichwere also identified by “1” or “-1”.It made the system achieved balance by converting digraphinto equilibrium diagram.We judged the element arranged in the following order aboutwhether can achieve the stability of the intelligent system based on the influence degreebetween each element showing the relationships between function and its correlation andsystem uptime.(4) The generalized μ-scrambling index of a class of primitive signed digraph was studied, some exact th lower and upper bounds of the μ-scrambling indices of this class of primitivedigraph were given respectively. We provided the generalized scrambling indices and thegeneralized μ-scrambling indices for the two special primitive digraphs at the same time.(5) By use of weighted graph and other knowledge, an effective Dijkstra algorithm ofsingle-source shortest paths was presented to implement the Dijkstra algorithm for the singlesource shortest path problem with few distinct positive lengths. The method to calculate theshortest path considering mixed weights based on the Dijkstra algorithm was presented in thispaper, which was based on Dijkstra algorithm mixing with shortest path of weighted graph tosearch mathematical model dynamically.