Combinational Theory Of Graph And Matrix As Well As Network's Applications

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,twhere there only have two cycles and D_s,t,qwherethere have three cycles are researched.The equality cases on the bases of signeddigraphs D_s,tand the upper bounds on the bases signed digraphs D_s,t,qare obtained.The bounds on the basis of signed digraphs D_s,t,qare 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 (v₁₁,v_k1;v₂₁,v_k1;k₁,k₂;l)-lollipop or (v₁₁,v_1k1;v₂₁,v_2k2;…;v_m1,v_mkm;k₁,k₂,…,k_m;l).-lollipop The upper bounds on the bases of (v₁₁,v_k1;v₂₁,v_k2;k₁,k₂;l)-lollipopor (v₁₁,v_1k1;v₂₁,v_2k2;…;v_m1,v_mkm;k₁,k₂,…,k_m;l)-lollipopare obtained and the bases of(v₁₁,v_k1;v₂₁,v_k2;k₁,k₂;l)-lollipop or (v₁₁,v_1k1;v₂₁,v_2k2;…;v_m1,v_mkm;k₁,k₂,…,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.