The Smith Groups And Critical Groups Of Some Regular Signed Graphs

The smith group and critical group of a connected graph G are subtle invariants of graph G,which are closely related to the adjacency matrix and laplacian matrix of graph G,respectively.The definition of the smith groups of the unsigned connected graph G are same as the smith groups of the signed connected graph Γ.Both of them regard the adjacency matrix A(G)and A(Γ)as the mapping of Zn→Zn,and their cokernels cokerA(G)=Zn/A(G)Zn and cokerA(Γ)=Zn/A(Γ)Zn are the Smith groups of the unsigned connected graph G and the signed connected graph Γ,respectively.That is to say,the invariant factorizations of the Smith groups of a unsigned connected graph G and a signed connected graph Γ can be given by the Smith canonical form of their adjacency matrices.Assume that unsigned connected graph G has n vertices,then the lapla-cian matrix of G is L(G)=D(G)-A(G),where D(G)=diag(d1,d2,…,dn)is the degree matrix of G,and A(G)is the adjacency matrix of G.Similarly,we can give the definition of the laplacian matrix L(Γ)of signed connected graphΓ.Regarding the laplacian matrix L(G)of an unsigned connected graph G as a mapping of Zn→Zn,then its cokernel can be written as cokerlL(G)=Zn/L(G)Zn≌Z(?)K(G),where K(G)representing the critical group of graph G,is a finite Abelian group,and furthermore the order of K(G)is equal to the number of spanning trees of an unsigned connected graph G.For signed connected graph Γ,its laplacian matrix L(Γ)is regarded as the mapping of Zn→Zn,and its cokerl L(Γ)=Zn/L(Γ)Zn is the critical group of graph Γ.In this paper,we mainly study the smith groups and critical groups of signed lattice graph S Rn,maximal smith signed graph S14,S16,and T2n,and the smith groups and critical groups of C2n(1,n-1),which is the underlying graph of T2n.In Chapter 1,we introduce the background of our research,the basic concepts of smith group and critical group of unsigned connected graph G and signed connected graph Γ,and some useful lemmas.In Chapter 2,we study the smith groups and critical groups of signed lattice graph SRn and maximal smith signed graph S14,S16,T2n via elementary integer determinant transformation and elementary factor method.In Chapter 3,we mainly explore the smith group and critical group of C2n(1,n-1),which is the underlying graph of T2n.By transforming the the adjacency matrix of C2n(1,n-1)to obtain its smith standard form,the direct sum decomposition of smith group of C2n(1,n-1)can be derived.Then the number of spanning trees of C2n(1,n-1)can be obtained with the help of expansion of the Matrix-tree theorem,i.e.,the order of the critical group of C2n(1,n-1)can be obtained.Finally,the direct sum decomposition of critical group of C2n(1,n-1)can be successfully given by transforming the relation matrix of generators of critical group to obtain its smith standard form.In Chapter 4,we give the conclusion of this paper,and furthermore,some future research directions are proposed to emphasize the perspectiveness and evolvability of this paper.