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A graph is denoted by G with the vertex set V(G) and the edge set E(G). A path P = 〈v0v1, … , vm〉 is a sequence of adjacent vertices. Two paths with equal length P1 = 〈 u1u2, … , um〉 and P2 = 〈 v1v2, … , vm〉 from a to b are independent if u1 = v1 = a, um = vm = b, and ui ≠ vi for 2 ? i ? m − 1. Paths with equal length from a to b are mutually independent if they are pairwisely independent. Let u and v be two distinct vertices of a bipartite graph G, and let l be a positive integer length, dG(uv) ? l ? ∣V(G) − 1∣ with (l − dG(uv)) being even. We say that the pair of vertices u, v is (ml)-mutually independent bipanconnected if there exist m mutually independent paths with length l from u to v. In this paper, we explore yet another strong property of the hypercubes. We prove that every pair of vertices u and v in the n-dimensional hypercube, with dQn(u,v)?n-1, is (n − 1, l)-mutually independent bipanconnected for every with (l-dQn(u,v)) being even. As for dQn(u,v)?n-2, it is also (n − 1, l)-mutually independent bipanconnected if l?dQn(u,v)+2, and is only (ll)-mutually independent bipanconnected if l=dQn(u,v).  相似文献   

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Suppose a graph G have n vertices, m edges, and t triangles. Letting λn(G) be the largest eigenvalue of the Laplacian of G and μn(G) be the smallest eigenvalue of its adjacency matrix, we prove that
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In this paper, we show that, for every locally compact abelian group G, the following statements are equivalent:
(i)
G contains no sequence such that {0}∪{±xnnN} is infinite and quasi-convex in G, and xn?0;
(ii)
one of the subgroups {gG∣2g=0} or {gG∣3g=0} is open in G;
(iii)
G contains an open compact subgroup of the form or for some cardinal κ.
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Let G be a graph with n vertices and m edges. Let λ1λ2, … , λn be the eigenvalues of the adjacency matrix of G, and let μ1μ2, … , μn be the eigenvalues of the Laplacian matrix of G. An earlier much studied quantity is the energy of the graph G. We now define and investigate the Laplacian energy as . There is a great deal of analogy between the properties of E(G) and LE(G), but also some significant differences.  相似文献   

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Let 1 ? p ? ∞, 0 < q ? p, and A = (an,k)n,k?0 ? 0. Denote by Lp,q(A) the supremum of those L satisfying the following inequality:
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Let G = (V, E) be a simple graph. Denote by D(G) the diagonal matrix of its vertex degrees and by A(G) its adjacency matrix. Then the signless Laplacian matrix of G is Q(G) = D(G) + A(G). In [5], Cvetkovi? et al. have given the following conjecture involving the second largest signless Laplacian eigenvalue (q2) and the index (λ1) of graph G (see also Aouchiche and Hansen [1]):
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Let G be an Abelian group and let be infinite. We construct a partition of A such that whenever (xn)n<ω is a one-to-one sequence in A, gG and m<ω, one has
(g+FSI((xn)n<ω))∩Am≠∅,  相似文献   

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A.R. Rao 《Discrete Mathematics》2006,306(14):1595-1600
For a digraph G, let R(G) (respectively, R(k)(G)) be the number of ordered pairs (u,v) of vertices of G such that uv and v is reachable from u (respectively, reachable from u by a path of length ?k). In this paper, we study the range Sn of R(G) and the range of R(k)(G) as G varies over all possible digraphs on n vertices. We give a sufficient condition and a necessary condition for an integer to belong to Sn. These determine the set Sn for all n?208. We also determine for k?4 and show that whenever n?k+(k+1)0.57+2, for arbitrary k.  相似文献   

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Let G be a graph of sufficiently large order n, and let the largest eigenvalue μ(G) of its adjacency matrix satisfies . Then G contains a cycle of length t for every t?n/320This condition is sharp: the complete bipartite graph T2(n) with parts of size n/2 and n/2 contains no odd cycles and its largest eigenvalue is equal to .This condition is stable: if μ(G) is close to and G fails to contain a cycle of length t for some t?n/321, then G resembles T2(n).  相似文献   

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In this paper we investigate linear three-term recurrence formulae with sequences of integers (T(n))n?0 and (U(n))n?0, which are ultimately periodic modulo m, e.g.
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We find lower bounds on the difference between the spectral radius λ1 and the average degree of an irregular graph G of order n and size e. In particular, we show that, if n ? 4, then
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Let m(n) be the number of ordered factorizations of n?1 in factors larger than 1. We prove that for every ε>0
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Let u=u(x,t,u0) represent the global strong/weak solutions of the Cauchy problems for the general n-dimensional incompressible Navier-Stokes equations
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