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1.
Let Qn,k(n≥3,1≤k≤n-1) be an n-dimensional enhanced hypercube which is an attractive variant of the hypercube and can be obtained by adding some complementary edges,fv and fe be the numbers of faulty vertices and faulty edges,respectively.In this paper,we give three main results.First,a fault-free path P [u,v] of length at least 2n-2fv-1(respectively,2n-2fv-2) can be embedded on Qn,k with fv+fe≤n-1 when d Qn,k(u,v) is odd(respectively,d Qn,k(u,v) is even).Secondly,an Qn,k is(n-2) edgefault-free hyper Hamiltonian-laceable when n(≥3) and k have the same parity.Lastly,a fault-free cycle of length at least 2n-2fv can be embedded on Qn,k with fe≤n-1 and fv+fe≤2n-4.  相似文献   

2.
The classical hypercube structure is a popular topological architecture in parallel computing environments and a large number of variations based on the hypercube were posed in the past three decades. Reliability evaluation of systems is important to the design and maintenance of multiprocessor systems. The h-extra edge-connectivity of graph G(V, E) is a kind of measure for the reliability of interconnection systems, which is defined as the minimum cardinality of a subset of edge set, if any, whose deletion disconnects G and such that every remaining component has at least h vertices. This paper shows that the h-extra edge-connectivity of the hypercube Qn is a constant 2~(n-1) for2~(n-1)/3 h ≤ 2~(n-1), and n ≥ 4, which extends the result of ["Bounding the size of the subgraph induced by m vertices and extra edge-connectivity of hypercubes, Discrete Applied Mathematics, 2013, 161(16): 2753-2757"].  相似文献   

3.
Let λ_k be the kth greatest eigenvalue of forest F (or tree T) on n vertices, then λ_k=-λ_(n-k 1). Hong Yuan proposed the following conjecture: Conjecture 1. Suppose T is a tree with n vertices and edge independence number q. For k≤q, λ_k(T)≥λ_k(S_(n-2k 2)~(2h-2) with equality iff T≌S_(n-2k 2)~(2k-2), where S_(n-2k 2)~(2k-2) is formed from a K_(1,n-2k 1) and a path P_(2k-2) by joining with an edge a vertex of degree one of P_(2k-2) to the vertex of degree n-2k 1 of K_(1,n-2k 1).  相似文献   

4.
In this paper, we study the enhanced hypercube, an attractive variant of the hypercube and obtained by adding some complementary edges from a hypercube, and focus on cycles embedding on the enhanced hypercube with faulty vertices. Let Fu be the set of faulty vertices in the n-dimensional enhanced hypercube Qn,k (n ≥ 3, 1 ≤ k 〈≤n - 1). When IFvl = 2, we showed that Qn,k - Fv contains a fault-free cycle of every even length from 4 to 2n - 4 where n (n ≥ 3) and k have the same parity; and contains a fault-free cycle of every even length from 4 to 2n - 4, simultaneously, contains a cycle of every odd length from n-k + 2 to 2^n-3 where n (≥ 3) and k have the different parity. Furthermore, when |Fv| = fv ≤ n - 2, we prove that there exists the longest fault-free cycle, which is of even length 2^n - 2fv whether n (n ≥ 3) and k have the same parity or not; and there exists the longest fault-free cycle, which is of odd length 2^n - 2fv + 1 in Qn,k - Fv where n (≥ 3) and k have the different parity.  相似文献   

5.
In this article, we present a Schwarz lemma at the boundary for pluriharmonic mappings from the unit polydisk to the unit ball, which generalizes classical Schwarz lemma for bounded harmonic functions to higher dimensions. It is proved that if the pluriharmonic mapping f ∈ P(D~n, B~N) is C~(1+α) at z0 ∈ E_rD~n with f(0) = 0 and f(z_0) = ω_0∈B~N for any n,N ≥ 1, then there exist a nonnegative vector λ_f =(λ_1,0,…,λ_r,0,…,0)~T∈R~(2 n)satisfying λ_i≥1/(2~(2 n-1)) for 1 ≤ i ≤ r such that where z'_0 and w'_0 are real versions of z_0 and w_0, respectively.  相似文献   

6.
关于布尔矩阵行空间基数的若干存在区间   总被引:1,自引:0,他引:1  
Let B_n be the set of all n×n Boolean Matrices;R(A) denote the row space of A∈B_n,|R(A)| denote the cardinality of R(A),m,n,k,l,t,i,γ_i be positive integers,S_i,λ_i be non negative integers.In this paper,we prove the following two results: (1)Let n≥13,n-3≥k > S_l,S_(i+1)> S_i,i = 1,2,…,l-1.if k+l≤n,then for any m=2~k+2~(S_(l)) + 2~(S_(l-1))+…+ 2~(S_(1)),there exists A∈B_n,such that |R(A)|= m. (2)Let n≥13,n-3≥k>S_(n-k-1)> S_(n-k-2)>…>S_1>λ_t>λ_(t-1)>…>λ_1,2≤t≤n-k.If existγ_i(k+1≤γ_i≤n-1,i=1,2,…,t-1)γ_i<γ_...  相似文献   

7.
Let π be a self-dual irreducible cuspidal automorphic representation of GL_2(AQ) with trivial central character. Its Hecke eigenvalue λ_π(n) is a real multiplicative function in n. We show that λ_π(n) 0 for some n ? Q_π~(2/5), where Qπ denotes(a special value of) the analytic conductor. The value2/5 is the first explicit exponent for Hecke-Maass newforms.  相似文献   

8.
An invariant σ2(G) of a graph is defined as follows: σ2(G) := min{d(u) + d(v)|u, v ∈V(G),uv ∈ E(G),u ≠ v} is the minimum degree sum of nonadjacent vertices (when G is a complete graph, we define σ2(G) = ∞). Let k, s be integers with k ≥ 2 and s ≥ 4, G be a graph of order n sufficiently large compared with s and k. We show that if σ2(G) ≥ n + k- 1, then for any set of k independent vertices v1,..., vk, G has k vertex-disjoint cycles C1,..., Ck such that |Ci| ≤ s and vi ∈ V(Ci) for all 1 ≤ i ≤ k.
The condition of degree sum σs(G) ≥ n + k - 1 is sharp.  相似文献   

9.
Let f be a holomorphic Hecke eigenform of weight k for the modular groupΓ = SL2(Z) and let λf(n) be the n-th normalized Fourier coefficient. In this paper, by a new estimate of the second integral moment of the symmetric square L-function related to f, the estimate 1λf(n21) x2 k2(log(x + k))6n≤x is established, which improves the previous result.  相似文献   

10.
A vertex x in a graph G strongly resolves a pair of vertices v, w if there exists a shortest x-w path containing v or a shortest x-v path containing w in G. A set of vertices S■V(G) is a strong resolving set of G if every pair of distinct vertices of G is strongly resolved by some vertex in S. The strong metric dimension of G, denoted by sdim(G), is the minimum cardinality over all strong resolving sets of G. For a connected graph G of order n≥2, we characterize G such that sdim(G) equals 1, n-1, or n-2, respectively. We give a Nordhaus-Gaddum-type result for the strong metric dimension of a graph and its complement: for a graph G and its complement G, each of order n≥4 and connected, we show that 2≤sdim(G)+sdim(G)≤2( n-2). It is readily seen that sdim(G)+sdim(G)=2 if and only if n=4; we show that, when G is a tree or a unicyclic graph, sdim(G)+sdim(G)=2(n 2) if and only if n=5 and G ~=G ~=C5, the cycle on five vertices. For connected graphs G and G of order n≥5, we show that 3≤sdim(G)+sdim(G)≤2(n-3) if G is a tree; we also show that 4≤sdim(G)+sdim(G)≤2(n-3) if G is a unicyclic graph of order n≥6. Furthermore, we characterize graphs G satisfying sdim(G)+sdim(G)=2(n-3) when G is a tree or a unicyclic graph.  相似文献   

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