Flow trees for vertex-capacitated networks

Given a graph G=(V,E) with a cost function , we want to represent all possible min-cut values between pairs of vertices i and j. We consider also the special case with an additive cost c where there are vertex capacities c(v)?0∀v∈V, and for a subset S⊆V, c(S)=∑_v∈Sc(v). We consider two variants of cuts: in the first one, separation, {i} and {j} are feasible cuts that disconnect i and j. In the second variant, vertex-cut, a cut-set that disconnects i from j does not include i or j. We consider both variants for undirected and directed graphs. We prove that there is a flow-tree for separations in undirected graphs. We also show that a compact representation does not exist for vertex-cuts in undirected graphs, even with additive costs. For directed graphs, a compact representation of the cut-values does not exist even with additive costs, for neither the separation nor the vertex-cut cases.     

A random intersection digraph: Indegree and outdegree distributions

Let S(1),…,S(n),T(1),…,T(n) be random subsets of the set [m]={1,…,m}. We consider the random digraph D on the vertex set [n] defined as follows: the arc i→j is present in D whenever S(i)∩T(j)≠0?. Assuming that the pairs of sets (S(i),T(i)), 1≤i≤n, are independent and identically distributed, we study the in- and outdegree distributions of a typical vertex of D as n,m→∞.     

Characterization of the absence of a spectral gapfor a class of periodic Jacobi operators

Let q ∈ {2, 3} and let 0 = s₀ < s₁ < … < s_q = T be integers. For m, n ∈ Z, we put ¯m,n = {j ∈ Z| m? j ? n}. We set l_j = s_j − s_j−1 for j ∈ 1, q. Given (p₁,, p_q) ∈ R^q, let b: Z → R be a periodic function of period T such that b(·) = p_j on s_j−1 + 1, s_j for each j ∈ 1, q. We study the spectral gaps of the Jacobi operator (Ju)(n) = u(n + 1) + u(n − 1) + b(n)u(n) acting on l²(Z). By [λ_2j , λ_2j−1] we denote the jth band of the spectrum of J counted from above for j ∈ 1, T. Suppose that p_m ≠ p_n for m ≠ n. We prove that the statements (i) and (ii) below are equivalent for λ ∈ R and i ∈ 1, T − 1.     

A note on bideterminants for Schur superalgebras

Let S(m|n,r)_Z be a Z-form of a Schur superalgebra S(m|n,r) generated by elements ξ_i,j. We solve a problem of Muir and describe a Z-form of a simple S(m|n,r)-module D_λ,Q over the field Q of rational numbers, under the action of S(m|n,r)_Z. This Z-form is the Z-span of modified bideterminants [T_?:T_i] defined in this work. We also prove that each [T_?:T_i] is a Z-linear combination of modified bideterminants corresponding to (m|n)-semistandard tableaux T_i.     

Isomorphism of circulant graphs and digraphs

Let S? {1, …, n?1} satisfy ?S = S mod n. The circulant graph G(n, S) with vertex set {v₀, v₁,…, v_n?1} and edge set E satisfies v_iv_j?E if and only if j ? i ∈ S, where all arithmetic is done mod n. The circulant digraph G(n, S) is defined similarly without the restriction S = ? S. Ádám conjectured that $\text{G(n, S) ? G(n, S′)}$ if and only if S = uS′ for some unit u mod n. In this paper we prove the conjecture true if n = pq where p and q are distinct primes. We also show that it is not generally true when n = p², and determine exact conditions on S that it be true in this case. We then show as a simple consequence that the conjecture is false in most cases when n is divisible by p² where p is an odd prime, or n is divisible by 2⁴.     

Divisibility of determinants of power GCD matrices and power LCM matrices on finitely many quasi-coprime divisor chains

Let a, n ? 1 be integers and S = {x₁, … , x_n} be a set of n distinct positive integers. The matrix having the ath power (x_i, x_j)^a of the greatest common divisor of x_i and x_j as its i, j-entry is called ath power greatest common divisor (GCD) matrix defined on S, denoted by (S^a). Similarly we can define the ath power LCM matrix [S^a]. We say that the set S consists of finitely many quasi-coprime divisor chains if we can partition S as S = S₁ ∪ ? ∪ S_k, where k ? 1 is an integer and all S_i (1 ? i ? k) are divisor chains such that (max(S_i), max(S_j)) = gcd(S) for 1 ? i ≠ j ? k. In this paper, we first obtain formulae of determinants of power GCD matrices (S^a) and power LCM matrices [S^a] on the set S consisting of finitely many quasi-coprime divisor chains with gcd(S) ∈ S. Using these results, we then show that det(S^a)∣det(S^b), det[S^a]∣det[S^b] and det(S^a)∣det[S^b] if a∣b and S consists of finitely many quasi-coprime divisor chains with gcd(S) ∈ S. But such factorizations fail to be true if such divisor chains are not quasi-coprime.     

Effective Fast Algorithms for Polynomial Spectral Factorization

Let p(z) be a polynomial of degree n having zeros |ξ₁|≤???≤|ξ _m |<1<|ξ _m+1|≤???≤|ξ _n |. This paper is concerned with the problem of efficiently computing the coefficients of the factors u(z)=∏ _i=1 ^m (z?ξ _i ) and l(z)=∏ _i=m+1 ⁿ (z?ξ _i ) of p(z) such that a(z)=z ^?m p(z)=(z ^?m u(z))l(z) is the spectral factorization of a(z). To perform this task the following two-stage approach is considered: first we approximate the central coefficients x _?n+1,. . .x _n?1 of the Laurent series x(z)=∑ _i=?∞ ^+∞ x _i z ⁱ satisfying x(z)a(z)=1; then we determine the entries in the first column and in the first row of the inverse of the Toeplitz matrix T=(x _i?j ) _i,j=?n+1,n?1 which provide the sought coefficients of u(z) and l(z). Two different algorithms are analyzed for the reciprocation of Laurent polynomials. One algorithm makes use of Graeffe's iteration which is quadratically convergent. Differently, the second algorithm directly employs evaluation/interpolation techniques at the roots of 1 and it is linearly convergent only. Algorithmic issues and numerical experiments are discussed.     

A generalization of Sperner’s theorem and an application to graph orientations

A generalization of Sperner’s theorem is established: For a multifamily M={Y₁,…,Y_p} of subsets of {1,…,n} in which the repetition of subsets is allowed, a sharp lower bound for the number φ(M) of ordered pairs (i,j) satisfying i≠j and Y_i⊆Y_j is determined. As an application, the minimum average distance of orientations of complete bipartite graphs is determined.     

H-domination in graphs

Let H be some fixed graph of order p. For a given graph G and vertex set S⊆V(G), we say that S is H-decomposable if S can be partitioned as S=S₁∪S₂∪?∪S_j where, for each of the disjoint subsets S_i, with 1?i?j, we have |S_i|=p and H is a spanning subgraph of 〈S_i〉, the subgraph induced by S_i. We define the H-domination number of G, denoted as γ_H(G), to be the minimum cardinality of an H-decomposable dominating set S. If no such dominating set exists, we write γ_H(G)=∞. We show that the associated H-domination decision problem is NP-complete for every choice of H. Bounds are shown for γ_H(G). We show, in particular, that if δ(G)?2, then γ_P3(G)?3γ(G). Also, if γ_P3(G)=3γ(G), then every γ(G)-set is an efficient dominating set.     

Divisibility properties of power GCD matrices and power LCM matrices

Let a,b and n be positive integers and the set S={x₁,…,x_n} of n distinct positive integers be a divisor chain (i.e. there exists a permutation σ on {1,…,n} such that x_σ(1)|…|x_σ(n)). In this paper, we show that if a|b, then the ath power GCD matrix (S^a) having the ath power (x_i,x_j)^a of the greatest common divisor of x_i and x_j as its i,j-entry divides the bth power GCD matrix (S^b) in the ring M_n(Z) of n×n matrices over integers. We show also that if a?b and n?2, then the ath power GCD matrix (S^a) does not divide the bth power GCD matrix (S^b) in the ring M_n(Z). Similar results are also established for the power LCM matrices.     

Nonsingularity of matrices associated with classes of arithmetical functions on lcm-closed sets

Let S = {x₁, … , x_n} be a set of n distinct positive integers and f be an arithmetical function. Let [f(x_i, x_j)] denote the n × n matrix having f evaluated at the greatest common divisor (x_i, x_j) of x_i and x_j as its i, j-entry and (f[x_i, x_j]) denote the n × n matrix having f evaluated at the least common multiple [x_i, x_j] of x_i and x_j as its i, j-entry. The set S is said to be lcm-closed if [x_i, x_j] ∈ S for all 1 ? i, j ? n. For an integer x > 1, let ω(x) denote the number of distinct prime factors of x. Define ω(1) = 0. In this paper, we show that if S = {x₁, … , x_n} is an lcm-closed set satisfying , and if f is a strictly increasing (resp. decreasing) completely multiplicative function, or if f is a strictly decreasing (resp. increasing) completely multiplicative function satisfying (resp. f(p) ? p) for any prime p, then the matrix [f(x_i, x_j)] (resp. (f[x_i, x_j])) defined on S is nonsingular. By using the concept of least-type multiple introduced in [S. Hong, J. Algebra 281 (2004) 1-14], we also obtain reduced formulas for det(f(x_i, x_j)) and det(f[x_i, x_j]) when f is completely multiplicative and S is lcm-closed. We also establish several results about the nonsingularity of LCM matrices and reciprocal GCD matrices.     

Two equivalent measures on weighted hypergraphs

Let H=(N,E,w) be a hypergraph with a node set N={0,1,…,n-1}, a hyperedge set E⊆2^N, and real edge-weights w(e) for e∈E. Given a convex n-gon P in the plane with vertices x₀,x₁,…,x_n-1 which are arranged in this order clockwisely, let each node i∈N correspond to the vertex x_i and define the area A_P(H) of H on P by the sum of the weighted areas of convex hulls for all hyperedges in H. For 0?i<j<k?n-1, a convex three-cut C(i,j,k) of N is {{i,…,j-1}, {j,…,k-1}, {k,…,n-1,0,…,i-1}} and its size c_H(i,j,k) in H is defined as the sum of weights of edges e∈E such that e contains at least one node from each of {i,…,j-1}, {j,…,k-1} and {k,…,n-1,0,…,i-1}. We show that the following two conditions are equivalent:

•

A_P(H)?A_P(H^′) for all convex n-gons P.

•

c_H(i,j,k)?c_H^′(i,j,k) for all convex three-cuts C(i,j,k).

From this property, a polynomial time algorithm for determining whether or not given weighted hypergraphs H and H^′ satisfy “A_P(H)?A_P(H^′) for all convex n-gons P” is immediately obtained.     

Integer version of the multipath flow network synthesis problem

We consider the following integer multipath flow network synthesis problem. We are given two positive integers q, n, (1<q<n), and a non-negative, integer, symmetric, n×n matrix R, each non-diagonal element r_ij of which represents the minimum requirement of q-path flow value between nodes i and j in an undirected network on the node set N={1,2,…,n}. We want to construct a simple, undirected network G=[N,E] with integer edge capacities {u_e:e∈E} such that each of these flow requirements can be realized (one at a time) and the sum of all the edge capacities is minimum. We present an O(n³) combinatorial algorithm for the problem and we show that the problem has integer rounding property.     

On the h-triangles of sequentially (S r ) simplicial complexes via algebraic shifting

Recently, Haghighi, Terai, Yassemi, and Zaare-Nahandi introduced the notion of a sequentially (S _r ) simplicial complex. This notion gives a generalization of two properties for simplicial complexes: being sequentially Cohen–Macaulay and satisfying Serre’s condition (S _r ). Let Δ be a (d?1)-dimensional simplicial complex with Γ(Δ) as its algebraic shifting. Also let (h _i,j (Δ))_{0≤j≤i≤d} be the h-triangle of Δ and (h _i,j (Γ(Δ)))_{0≤j≤i≤d} be the h-triangle of Γ(Δ). In this paper, it is shown that for a Δ being sequentially (S _r ) and for every i and j with 0≤j≤i≤r?1, the equality h _i,j (Δ)=h _i,j (Γ(Δ)) holds true.     

The six classes of trees with the largest algebraic connectivity

In this paper, we study the algebraic connectivity α(T) of a tree T. We introduce six Classes (C1)-(C6) of trees of order n, and prove that if T is a tree of order n?15, then if and only if , where the equality holds if and only if T is a tree in the Class (C6). At the same time we give a complete ordering of the trees in these six classes by their algebraic connectivity. In particular, we show that α(T_i)>α(T_j) if 1?i<j?6 and T_i is any tree in the Class (Ci) and T_j is any tree in the Class (Cj). We also give the values of the algebraic connectivity of the trees in these six classes. As a technique used in the proofs of the above mentioned results, we also give a complete characterization of the equality case of a well-known relation between the algebraic connectivity of a tree T and the Perron value of the bottleneck matrix of a Perron branch of T.     

Asymptotic enumeration of sparse 0-1 matrices with irregular row and column sums

Let s=(s₁,…,s_m) and t=(t₁,…,t_n) be vectors of non-negative integer-valued functions with equal sum . Let N(s,t) be the number of m×n matrices with entries from {0,1} such that the ith row has row sum s_i and the jth column has column sum t_j. Equivalently, N(s,t) is the number of labelled bipartite graphs with degrees of the vertices in one side of the bipartition given by s and the degrees of the vertices in the other side given by t. We give an asymptotic formula for N(s,t) which holds when S→∞ with 1?st=o(S^2/3), where and . This extends a result of McKay and Wang [Linear Algebra Appl. 373 (2003) 273-288] for the semiregular case (when s_i=s for 1?i?m and t_j=t for 1?j?n). The previously strongest result for the non-semiregular case required 1?max{s,t}=o(S^1/4), due to McKay [Enumeration and Design, Academic Press, Canada, 1984, pp. 225-238].     

A version of Chevet's theorem for stable processes

Let $\text{X}\text{}\text{?}\text{bo}\text{Y}$ be the injective tensor product of the separable Banach spaces X and Y and let S_X, S_Y and $\text{S}{}_{\mathrm{<mtext>X\; </mtext><mtext>\backslash </mtext><mtext>?</mtext><mtext>bo\; Y</mtext>}}$ be the unit spheres of these spaces. The tensor product of two symmetric finite measures η₁ on S_X and η₂ on S_Y, η₁?η₂, is defined in a natural way as a measure on $\text{S}{}_{\mathrm{<mtext>X\; </mtext><mtext>\backslash </mtext><mtext>?</mtext><mtext>bo\; Y</mtext>}}$. It is shown that η₁? η₂ is the spectral measure of a p-stable random variable W on $\text{X}\text{}\text{?}\text{bo}\text{Y}$, 0 <p < 2, if and only if η₁ and η₂ are the spectral measures of p-stable random variables U and V on X and Y, respectively. Actually upper and lower bounds for $\text{(E∥ W∥}{}^{\mathrm{r}}\text{)}{}^{\mathrm{<mtext>1</mtext><mtext>r</mtext>}}$ in terms of the random variables U and V are obtained. When X = C(S), Y = C(T) with S, T compact metric spaces, and η₁, and η₂ are discrete, our results imply that if θ_i, θ_ij are i.i.d. standard symmetric real valued p-stable random variables, 0 < p <2, x_i?C(S), and y_i?C(T), then the series ∑_ijθ_ijx_i(s) y_j(t) converges uniformly a.s. iff the series ∑_iθ_ix_i(s) and ∑_iθ_iy_i(t) both converge uniformly a.s. When p = 2 this follows from Chevet's theorem on Gaussian processes. Several examples are given. One of them requires an interesting upper bound on the probability distribution of the maximum of i.i.d. p-stable random variables taking values in a general Banach space.     

The Sperner Capacity of Linear and Nonlinear Codes for the Cyclic Triangle

Shannon introduced the concept of zero-error capacity of a discrete memoryless channel. The channel determines an undirected graph on the symbol alphabet, where adjacency means that symbols cannot be confused at the receiver. The zero-error or Shannon capacity is an invariant of this graph. Gargano, Körner, and Vaccaro have recently extended the concept of Shannon capacity to directed graphs. Their generalization of Shannon capacity is called Sperner capacity. We resolve a problem posed by these authors by giving the first example (the two orientations of the triangle) of a graph where the Sperner capacity depends on the orientations of the edges. Sperner capacity seems to be achieved by nonlinear codes, whereas Shannon capacity seems to be attainable by linear codes. In particular, linear codes do not achieve Sperner capacity for the cyclic triangle. We use Fourier analysis or linear programming to obtain the best upper bounds for linear codes. The bounds for unrestricted codes are obtained from rank arguments, eigenvalue interlacing inequalities and polynomial algebra. The statement of the cyclic q-gon problem is very simple: what is the maximum size N _q(n) of a subset S _n of {0, 1, \(\ldots\) , q?1} ⁿ with the property that for every pair of distinct vectors x = (x _i), y = (y _i) \(\in \) S _n, we have x _j ?y _j ≡ 1(mod q) for some j? For q = 3 (the cyclic triangle), we show N ₃(n)?2 ⁿ . If however S _n is a subgroup, then we give a simple proof that \(\left| {S_n } \right| \leqslant \sqrt 3 ^n \) .