A graph G possesses Hamiltonian s-properties when G is Hamilton-connected if s=1,Hamiltonian if s=0,and traceable if s=-1.Let S_A(G)=λ_n(G)-λ_1(G)and S_L(G)=μ_n(G)-μ_2(G)be the spread and the Laplacian spread of G...A graph G possesses Hamiltonian s-properties when G is Hamilton-connected if s=1,Hamiltonian if s=0,and traceable if s=-1.Let S_A(G)=λ_n(G)-λ_1(G)and S_L(G)=μ_n(G)-μ_2(G)be the spread and the Laplacian spread of G,respectively,whereλ_n(G)andλ_1(G)are the largest and smallest eigenvalues of G,andμ_n(G)andμ_2(G)are the largest and second smallest Laplacian eigenvalues of G,respectively.In this paper,we shall present two sufficient conditions involving S_A(G)and S_L(G)for a k-connected graph to possess Hamiltonian s-properties,respectively.We also derive a sufficient condition on the Laplacian eigenratio■for a k-connected graph to possess Hamiltonian s-properties.展开更多
Let D be a digraph. We call D primitive if there exists a positive integer ksuch that for all ordered pairs of venices u, v V(D) (not necessarily distinct), there isa directed walk of length k from u to v. In 1982, ...Let D be a digraph. We call D primitive if there exists a positive integer ksuch that for all ordered pairs of venices u, v V(D) (not necessarily distinct), there isa directed walk of length k from u to v. In 1982, J.A.Ross posed two problems: (1) If Dis a primitive digraph on n vertices with girth s>1 and (D) = n+s(n-2), does Dcontain an elementary circuit of length n? (2) Let D be a strong digraph on n verticeswhich contains a loop and suppose D is not isomorphic to Bi,n for i=1, 2, n-1(see Figure 1), if (D) =2n-2, does D contain an elementary circuit of length n?In this paper, we have solved both completely and obtained the following results: (1)Suppose that D is a primitive digraph on n vertices with girth s>1 and exponentn+s (n-2). Then D is Hamiltonian. (2) Suppose that D is a primitive digraph on nvertices which contains a loop, and (D)=2n-2. Then D is Hamiltonian if and only if max {d(u,v))=(u, v)= 2}=2} =n-2.展开更多
The author mainly uses the Galerkin approximation method and the iteration inequalities of the L-Maslov type index theory to study the properties of brake subharmonic solutions for the Hamiltonian systems z（t） = JVH...The author mainly uses the Galerkin approximation method and the iteration inequalities of the L-Maslov type index theory to study the properties of brake subharmonic solutions for the Hamiltonian systems z（t） = JVH（t, z（t））, where H（t, z） = 1/2（B（t）z, z） ＋ H（t, z）, B（t） is a semipositive symmetric continuous matrix and H is unbounded and not uniformly coercive. It is proved that when the positive integers j and k satisfy the certain conditions, there exists a jT-periodic nonconstant brake solution zj such that zj and zkj are distinct.展开更多
基金Supported by NSFC(Nos.12171089,12271235)NSF of Fujian Province(No.2021J02048)。
文摘A graph G possesses Hamiltonian s-properties when G is Hamilton-connected if s=1,Hamiltonian if s=0,and traceable if s=-1.Let S_A(G)=λ_n(G)-λ_1(G)and S_L(G)=μ_n(G)-μ_2(G)be the spread and the Laplacian spread of G,respectively,whereλ_n(G)andλ_1(G)are the largest and smallest eigenvalues of G,andμ_n(G)andμ_2(G)are the largest and second smallest Laplacian eigenvalues of G,respectively.In this paper,we shall present two sufficient conditions involving S_A(G)and S_L(G)for a k-connected graph to possess Hamiltonian s-properties,respectively.We also derive a sufficient condition on the Laplacian eigenratio■for a k-connected graph to possess Hamiltonian s-properties.
文摘Let D be a digraph. We call D primitive if there exists a positive integer ksuch that for all ordered pairs of venices u, v V(D) (not necessarily distinct), there isa directed walk of length k from u to v. In 1982, J.A.Ross posed two problems: (1) If Dis a primitive digraph on n vertices with girth s>1 and (D) = n+s(n-2), does Dcontain an elementary circuit of length n? (2) Let D be a strong digraph on n verticeswhich contains a loop and suppose D is not isomorphic to Bi,n for i=1, 2, n-1(see Figure 1), if (D) =2n-2, does D contain an elementary circuit of length n?In this paper, we have solved both completely and obtained the following results: (1)Suppose that D is a primitive digraph on n vertices with girth s>1 and exponentn+s (n-2). Then D is Hamiltonian. (2) Suppose that D is a primitive digraph on nvertices which contains a loop, and (D)=2n-2. Then D is Hamiltonian if and only if max {d(u,v))=(u, v)= 2}=2} =n-2.
基金supported by the National Natural Science Foundation of China(Nos.11501030,11226156)the Beijing Natural Science Foundation(No.1144012)
文摘The author mainly uses the Galerkin approximation method and the iteration inequalities of the L-Maslov type index theory to study the properties of brake subharmonic solutions for the Hamiltonian systems z（t） = JVH（t, z（t））, where H（t, z） = 1/2（B（t）z, z） ＋ H（t, z）, B（t） is a semipositive symmetric continuous matrix and H is unbounded and not uniformly coercive. It is proved that when the positive integers j and k satisfy the certain conditions, there exists a jT-periodic nonconstant brake solution zj such that zj and zkj are distinct.
