Mutually independent bipanconnected property of hypercube

A graph is denoted by G with the vertex set V(G) and the edge set E(G). A path P = 〈v₀, v₁, … , v_m〉 is a sequence of adjacent vertices. Two paths with equal length P₁ = 〈 u₁, u₂, … , u_m〉 and P₂ = 〈 v₁, v₂, … , v_m〉 from a to b are independent if u₁ = v₁ = a, u_m = v_m = b, and u_i ≠ v_i 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, d_G(u, v) ? l ? ∣V(G) − 1∣ with (l − d_G(u, v)) being even. We say that the pair of vertices u, v is (m, l)-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 d_Qn(u,v)?n-1, is (n − 1, l)-mutually independent bipanconnected for every with (l-d_Qn(u,v)) being even. As for d_Qn(u,v)?n-2, it is also (n − 1, l)-mutually independent bipanconnected if l?d_Qn(u,v)+2, and is only (l, l)-mutually independent bipanconnected if l=d_Qn(u,v).     

Eigenvalues and forbidden subgraphs I

Suppose a graph G have n vertices, m edges, and t triangles. Letting λ_n(G) be the largest eigenvalue of the Laplacian of G and μ_n(G) be the smallest eigenvalue of its adjacency matrix, we prove that

     

Locally compact abelian groups admitting non-trivial quasi-convex null sequences

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}∪{±x_n∣n∈N} is infinite and quasi-convex in G, and x_n?0;

(ii)

one of the subgroups {g∈G∣2g=0} or {g∈G∣3g=0} is open in G;

(iii)

G contains an open compact subgroup of the form or for some cardinal κ.

     

Laplacian energy of a graph

Let G be a graph with n vertices and m edges. Let λ₁, λ₂, … , λ_n be the eigenvalues of the adjacency matrix of G, and let μ₁, μ₂, … , μ_n 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.     

Lower bounds of Copson type for Hausdorff matrices

Let 1 ? p ? ∞, 0 < q ? p, and A = (a_n,k)_n,k?0 ? 0. Denote by L_p,q(A) the supremum of those L satisfying the following inequality:

     

Proof of conjecture involving the second largest signless Laplacian eigenvalue and the index of graphs

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]):

     

The number of reachable pairs in a digraph

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 u≠v and v is reachable from u (respectively, reachable from u by a path of length ?k). In this paper, we study the range S_n 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 S_n. These determine the set S_n for all n?208. We also determine for k?4 and show that whenever n?k+(k+1)^0.57+2, for arbitrary k.     

Partitions and sums with inverses in Abelian groups

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, g∈G and m<ω, one has

(g+FSI((xn)_n<ω))∩Am≠∅,     

A spectral condition for odd cycles in graphs

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).     

On the residue classes of integer sequences satisfying a linear three term recurrence formula

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.

     

Large matchings from eigenvalues

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

     

On the maximal order of numbers in the “factorisatio numerorum” problem

Let m(n) be the number of ordered factorizations of n?1 in factors larger than 1. We prove that for every ε>0

     

New results of general n-dimensional incompressible Navier-Stokes equations

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