图与矩阵的组合理论及其网络应用
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摘要
本文主要研究本原不可幂定号有向图基的界、两类特殊符号模式矩阵的惯量、有向图是有向Hamilton图的充分条件、有向图含有向Hamilton路的充分条件以及应用特殊的有向图—神经网络对学生成绩进行分类等内容。这些都是组合数学的重要内容，不仅在基础数学研究中具有极其重要的地位，在其它的学科邻域中也有重要的应用，如计算机科学、编码和密码学、物理、化学、生物等。
第一章阐述了符号模式矩阵、定号有向图基或局部基的界、符号模式矩阵惯量、双色(多色)有向图的本原指数、图论以及人工神经网络的发展与应用，详细给出了本论文的研究内容和安排．
第二章利用定号有向图基的定义和Frobenius数主要研究本原不可幂定号有向图基的界．首先，研究了一般仅有两个圈的本原不可幂定号有向图局部基的上界和特殊本原不可幂定号有向图局部基的上界和下界，得到某类特殊本原不可幂定号有向图的局部基．其次，研究了两类本原不可幂定号有向图S_(s，t)(仅有两个圈)和D_(s，t，q)(仅有三个圈)，得到了定号有向图D_(s，t)的基和定号有向图D_(s，t，q)基的上界，刻划了在s=t，q=s和q=t中仅有一个成立的情形下定号有向图D_(s，t，q)基的界．最后，研究了某类伴随图为(v_(11)，v_(k1)；v_(21)，v_(k2)；k_1，k_2；l)-lollipop或(v_(11)，v_(1k_1)；v_(21)，v_(2k_2)；…；v_(m1)，v_(mk_m)；k_1，k_2，…，k_m；l)-lollipop的本原不可幂零对称符号模式矩阵基的界，得到了(v_(11)，v_(k1)；v_(21)，v_(k2)；k_1，k_2；l)-lollipop或(v_(11)，v_(1k_1)；v_(21)，v_(2k_2)；…；v_(m1)，v_(mk_m)；k_1，k_2，…，k_m；l)-lollipop基的上界，并刻划了某些特殊(v_(11)，v_(k1)；v_(21)，v_(k2)；k_1，k_2；l)-lollipop或(v_(11)，v_(1k_1)；v_(21)，v_(2k_2)；…；v_(m1)，v_(mk_m)；k_1，k_2，…，k_m；l)-lollipop的基．
第三章利用代数学和矩阵分析的有关理论主要研究具有两个中心点的星符号模式矩阵和对称3-广义星符号模式矩阵的惯量．一方面，得到了二阶、三阶具有两个中心点的星符号模式矩阵的惯量和考虑前两行前两列的二阶子矩阵得到四阶及四阶以上的具有两个中心点的星符号模式矩阵的惯量．另一方面，考虑分块对角线上的矩阵的主对角线，得到对称3-广义星符号模式矩阵的惯量．
第四章利用有向图和代数学的有关理论主要研究有向图的Hamilton性质．一方面，论证了严格有向二部图含有向Hamilton路的一个充分条件和一个必要条件以及严格有向二部图为有向Hamilton图的一个充分条件．另一方面，论证了正则二部竞赛图删去某些弧所得的有向图的Hamilton性质．
第五章应用含一个隐含层的非线性的BP神经网络算法、径向基神经网络算法和感知器神经网络算法对学生学习成绩进行分类．通过运行程序可知，应用概率型神经网络算法当其分布密度∈[0．00121998 89，0．1186944 765]时分类的正确率达到99．06％，应用BP神经网络算法进行分类的正确率在98．51％与99．06％之间变化，但应用感知器神经网络算法进行分类的正确率很低，在20％与30％之间变化．因此通过考虑分类的正确率和整个分类时间，得到径向基神经网络算法是一种更适于对学生的成绩进行有效的分类算法．
第六章总结了本论文所研究的内容和需要进一步研究的内容．
In this thesis,we mainly study the bounds on the bases of primitive non-powerfulsigned digraphs,the inertias of two classes of sign pattern matrices and classificationof the students' scores based on some special digraphs--neural networks.All these arethe important contentsin combinational mathematics,which are not only in a extremelyimportant position in basic research in mathematics,but also have importantapplications in other disciplines such as computer science,coding and cryptography,physics,chemistry,biology.
In Chapter 1,we simply describes the developments and applications ofsign pattern matrices,the bounds on the bases or the local bases of signeddigraphs,the inertia sets of sign pattern matrices,the primitive exponents of multi-colordigraphs,graph theory and artificial neural networks.The detailed research contentsand arrangements about this thesis are given.
In Chapter 2,we mainly research the bounds on the bases of some primitive non-powerful signed digraphs by using the definition on the base of signed digraphs andthe Frobenius number.Firstly,the upper bounds on the local bases of the ordinaryprimitive non-powerful signed digraphs with two cycles and the upper and lowerbounds on the local bases of special primitive non-powerful signed digraphs withtwo cycles are studied.The local bases of some special primitive non-powerfulsigned digraphs with two cycles are obtained.Secondly,two classes of primitive non-powerful signed digraphs D_(s,t)where there only have two cycles and D_(s,t,q)wherethere have three cycles are researched.The equality cases on the bases of signeddigraphs D_(s,t)and the upper bounds on the bases signed digraphs D_(s,t,q)are obtained.The bounds on the basis of signed digraphs D_(s,t,q)are characterized when only one of equalities s=t,q=s and q=t exists.Finally,we study the bounds on the bases ofsome primitive non-powerful zero-symmetric sign patterns whose associated graphsare (v_(11),v_(k1);v_(21),v_(k1);k_1,k_2;l)-lollipop or (v_(11),v_(1k_1);v_(21),v_(2k_2);…;v_(m1),v_(mk_m);k_1,k_2,…,k_m;l).-lollipop The upper bounds on the bases of (v_(11),v_(k1);v_(21),v_(k2);k_1,k_2;l)-lollipopor (v_(11),v_(1k_1);v_(21),v_(2k_2);…;v_(m1),v_(mk_m);k_1,k_2,…,k_m;l)-lollipopare obtained and the bases of(v_(11),v_(k1);v_(21),v_(k2);k_1,k_2;l)-lollipop or (v_(11),v_(1k_1);v_(21),v_(2k_2);…;v_(m1),v_(mk_m);k_1,k_2,…,k_m;l)-lollipopare characterized.
In Chapter 3,the inertia sets of star sign patterns with two central vertices and thethe inertia set of a symmetric 3-generalized star sign patterns are researched by use ofthe theories of algebra and matrix alalysis.On the one hand,the inertia sets of star signpatterns with two central vertices of order 2 or order 3 are obtained.By the consideredsub-matrix of order 2 obtained from the first two rows and the first two columns,theinertia sets of star sign patterns with two central vertices of order n(n(?)4)are obtained.On the other hand,the inertia sets of some symmetric 3-generalized star sign patternsare obtained by the main diagonal entries of the block diagonal matrices.
In Chapter 4,the Hamiltonian properties of digraphs are researched by use of thethe theories of algebra and digraph.On the one hand,a sufficient condition and anecessary condition of a strict bipartite digraph with a directed Hamiltonian path and asufficient condition of a strict bipartite digraph to be Hamiltonian are characterized.Onthe other hand,the Hamiltonian properties of digraphs obtained from deleting somearcs from the regular bipartite tournament are studied.
In Chapter 5,the students' scores are classified by using the nonlinear BP neuralnetwork algorithm with a hidden layer,the probabilistic neural network algorithm andthe perceptron algorithm.The correct rate of the probabilistic neural network algorithmheads to 99.06% when net.spread∈[0.0012199889,0.1186944765].The correct rate of theBP neural network algorithm changes from 98.51% to 99.06%.But the correct rate of theperceptron neural network algorithm is too low and changes from 20% to 30%.Therefore by considering the correct rate and the whole time of classification,we obtain that theprobabilistic neural network algorithm is more suitable for solving the classification ofthe students' scores.
In Chapter 6,we sum up the researched contents of this paper and give the furtherresearched contents.
第一章阐述了符号模式矩阵、定号有向图基或局部基的界、符号模式矩阵惯量、双色(多色)有向图的本原指数、图论以及人工神经网络的发展与应用，详细给出了本论文的研究内容和安排．
第二章利用定号有向图基的定义和Frobenius数主要研究本原不可幂定号有向图基的界．首先，研究了一般仅有两个圈的本原不可幂定号有向图局部基的上界和特殊本原不可幂定号有向图局部基的上界和下界，得到某类特殊本原不可幂定号有向图的局部基．其次，研究了两类本原不可幂定号有向图S_(s，t)(仅有两个圈)和D_(s，t，q)(仅有三个圈)，得到了定号有向图D_(s，t)的基和定号有向图D_(s，t，q)基的上界，刻划了在s=t，q=s和q=t中仅有一个成立的情形下定号有向图D_(s，t，q)基的界．最后，研究了某类伴随图为(v_(11)，v_(k1)；v_(21)，v_(k2)；k_1，k_2；l)-lollipop或(v_(11)，v_(1k_1)；v_(21)，v_(2k_2)；…；v_(m1)，v_(mk_m)；k_1，k_2，…，k_m；l)-lollipop的本原不可幂零对称符号模式矩阵基的界，得到了(v_(11)，v_(k1)；v_(21)，v_(k2)；k_1，k_2；l)-lollipop或(v_(11)，v_(1k_1)；v_(21)，v_(2k_2)；…；v_(m1)，v_(mk_m)；k_1，k_2，…，k_m；l)-lollipop基的上界，并刻划了某些特殊(v_(11)，v_(k1)；v_(21)，v_(k2)；k_1，k_2；l)-lollipop或(v_(11)，v_(1k_1)；v_(21)，v_(2k_2)；…；v_(m1)，v_(mk_m)；k_1，k_2，…，k_m；l)-lollipop的基．
第三章利用代数学和矩阵分析的有关理论主要研究具有两个中心点的星符号模式矩阵和对称3-广义星符号模式矩阵的惯量．一方面，得到了二阶、三阶具有两个中心点的星符号模式矩阵的惯量和考虑前两行前两列的二阶子矩阵得到四阶及四阶以上的具有两个中心点的星符号模式矩阵的惯量．另一方面，考虑分块对角线上的矩阵的主对角线，得到对称3-广义星符号模式矩阵的惯量．
第四章利用有向图和代数学的有关理论主要研究有向图的Hamilton性质．一方面，论证了严格有向二部图含有向Hamilton路的一个充分条件和一个必要条件以及严格有向二部图为有向Hamilton图的一个充分条件．另一方面，论证了正则二部竞赛图删去某些弧所得的有向图的Hamilton性质．
第五章应用含一个隐含层的非线性的BP神经网络算法、径向基神经网络算法和感知器神经网络算法对学生学习成绩进行分类．通过运行程序可知，应用概率型神经网络算法当其分布密度∈[0．00121998 89，0．1186944 765]时分类的正确率达到99．06％，应用BP神经网络算法进行分类的正确率在98．51％与99．06％之间变化，但应用感知器神经网络算法进行分类的正确率很低，在20％与30％之间变化．因此通过考虑分类的正确率和整个分类时间，得到径向基神经网络算法是一种更适于对学生的成绩进行有效的分类算法．
第六章总结了本论文所研究的内容和需要进一步研究的内容．
In this thesis,we mainly study the bounds on the bases of primitive non-powerfulsigned digraphs,the inertias of two classes of sign pattern matrices and classificationof the students' scores based on some special digraphs--neural networks.All these arethe important contentsin combinational mathematics,which are not only in a extremelyimportant position in basic research in mathematics,but also have importantapplications in other disciplines such as computer science,coding and cryptography,physics,chemistry,biology.
In Chapter 1,we simply describes the developments and applications ofsign pattern matrices,the bounds on the bases or the local bases of signeddigraphs,the inertia sets of sign pattern matrices,the primitive exponents of multi-colordigraphs,graph theory and artificial neural networks.The detailed research contentsand arrangements about this thesis are given.
In Chapter 2,we mainly research the bounds on the bases of some primitive non-powerful signed digraphs by using the definition on the base of signed digraphs andthe Frobenius number.Firstly,the upper bounds on the local bases of the ordinaryprimitive non-powerful signed digraphs with two cycles and the upper and lowerbounds on the local bases of special primitive non-powerful signed digraphs withtwo cycles are studied.The local bases of some special primitive non-powerfulsigned digraphs with two cycles are obtained.Secondly,two classes of primitive non-powerful signed digraphs D_(s,t)where there only have two cycles and D_(s,t,q)wherethere have three cycles are researched.The equality cases on the bases of signeddigraphs D_(s,t)and the upper bounds on the bases signed digraphs D_(s,t,q)are obtained.The bounds on the basis of signed digraphs D_(s,t,q)are characterized when only one of equalities s=t,q=s and q=t exists.Finally,we study the bounds on the bases ofsome primitive non-powerful zero-symmetric sign patterns whose associated graphsare (v_(11),v_(k1);v_(21),v_(k1);k_1,k_2;l)-lollipop or (v_(11),v_(1k_1);v_(21),v_(2k_2);…;v_(m1),v_(mk_m);k_1,k_2,…,k_m;l).-lollipop The upper bounds on the bases of (v_(11),v_(k1);v_(21),v_(k2);k_1,k_2;l)-lollipopor (v_(11),v_(1k_1);v_(21),v_(2k_2);…;v_(m1),v_(mk_m);k_1,k_2,…,k_m;l)-lollipopare obtained and the bases of(v_(11),v_(k1);v_(21),v_(k2);k_1,k_2;l)-lollipop or (v_(11),v_(1k_1);v_(21),v_(2k_2);…;v_(m1),v_(mk_m);k_1,k_2,…,k_m;l)-lollipopare characterized.
In Chapter 3,the inertia sets of star sign patterns with two central vertices and thethe inertia set of a symmetric 3-generalized star sign patterns are researched by use ofthe theories of algebra and matrix alalysis.On the one hand,the inertia sets of star signpatterns with two central vertices of order 2 or order 3 are obtained.By the consideredsub-matrix of order 2 obtained from the first two rows and the first two columns,theinertia sets of star sign patterns with two central vertices of order n(n(?)4)are obtained.On the other hand,the inertia sets of some symmetric 3-generalized star sign patternsare obtained by the main diagonal entries of the block diagonal matrices.
In Chapter 4,the Hamiltonian properties of digraphs are researched by use of thethe theories of algebra and digraph.On the one hand,a sufficient condition and anecessary condition of a strict bipartite digraph with a directed Hamiltonian path and asufficient condition of a strict bipartite digraph to be Hamiltonian are characterized.Onthe other hand,the Hamiltonian properties of digraphs obtained from deleting somearcs from the regular bipartite tournament are studied.
In Chapter 5,the students' scores are classified by using the nonlinear BP neuralnetwork algorithm with a hidden layer,the probabilistic neural network algorithm andthe perceptron algorithm.The correct rate of the probabilistic neural network algorithmheads to 99.06% when net.spread∈[0.0012199889,0.1186944765].The correct rate of theBP neural network algorithm changes from 98.51% to 99.06%.But the correct rate of theperceptron neural network algorithm is too low and changes from 20% to 30%.Therefore by considering the correct rate and the whole time of classification,we obtain that theprobabilistic neural network algorithm is more suitable for solving the classification ofthe students' scores.
In Chapter 6,we sum up the researched contents of this paper and give the furtherresearched contents.
引文
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