Imbedding conditions for λ-matrices

We say that A(λ) is λ-imbeddable in B(λ) whenever B(λ) is equivalent to a λ-matrix having A(λ) as a submatrix. In this paper we solve the problem of finding a necessary and sufficient condition for A(λ) to be λ-imbeddable in B(λ). The solution is given in terms of the invariant polynomials of A(λ) and B(λ). We also solve an analogous problem when A(λ) and B(λ) are required to be equivalent to regular λ-matrices. As a consequence we give a necessary and sufficient condition for the existence of a matrix B, over a field F, with prescribed similarity invariant polynomials and a prescribed principal submatrix A.     

Intersections among Steiner systems

A Steiner system S(l, m, n) is a system of subsets of size m (called blocks) from an n-set S, such that each d-subset from S is contained in precisely one block. Two Steiner systems have intersection k if they share exactly k blocks. The possible intersections among S(5, 6, 12)'s, among S(4, 5, 11)'s, among S(3, 4, 10)'s, and among S(2, 3, 9)'s are determined, together with associated orbits under the action of the automorphism group of an initial Steiner system. The following are results: (i) the maximal number of mutually disjoint S(5, 6, 12)'s is two and any two such pairs are isomorphic; (ii) the maximal number of mutually disjoint S(4, 5, 11)'s is two and any two such pairs are isomorphic; (iii) the maximal number of mutually disjoint S(3, 4, 10)'s is five and any two such sets of five are isomorphic; (iv) a result due to Bays in 1917 that there are exactly two non-isomorphic ways to partition all 3-subsets of a 9-set into seven mutually disjoint S(2, 3, 9)'s.     

The graphs G(n, k) of the Johnson Schemes are unique for n ≥ 20

Let G(n, k) denote the graph of the Johnson Scheme J(n, k), i.e., the graph whose vertices are all k-subsets of a fixed n-set, with two vertices adjacent if and only if their intersection is of size k ? 1. It is known that G(n, k) is a distance regular graph with diameter k. Much work has been devoted to the question of whether a distance regular graph with the parameters of G(n, k) must isomorphic to G(n, k). In this paper, this question is settled affirmatively for n ≥ 20. In fact the result is proved with weaker conditions.     

Hamiltonian uniform subset graphs

The uniform subset graph G(n, k, t) is defined to have all k-subsets of an n-set as vertices and edges joining k-subsets intersecting at t elements. We conjecture that G(n, k, t) is hamiltonian when it is different from the Petersen graph and does possess cycles. We verify this conjecture for k − t = 1, 2, 3 and for suitably large n when t = 0, 1.     

An extremal problem concerning matrices of 0's and 1 's

We determine the smallest number f(n,k) such that every (0,1)-matrix of order n what zero main diagonal which has at least f(n,k) 1's contains an irreducible, principal submatrix of order K. We characterize those matrices with f(n,k)?1 l's having no irreducible, principal submatrix of order k     

On the diameter of generalized Kneser graphs

Let r, k be positive integers, s(<r), a nonnegative integer, and n=2r-s+k. The set of r-subsets of [n]={1,2,…,n} is denoted by [n]^r. The generalized Kneser graph K(n,r,s) is the graph whose vertex-set is [n]^r where two r-subsets A and B are joined by an edge if |A∩B|?s. This note determines the diameter of generalized Kneser graphs. More precisely, the diameter of K(n,r,s) is equal to , which generalizes a result of Valencia-Pabon and Vera [On the diameter of Kneser graphs, Discrete Math. 305 (2005) 383-385].     

Cycle extendability in graphs and digraphs

In 1990, Hendry conjectured that all chordal Hamiltonian graphs are cycle extendable, that is, the vertices of each non-Hamiltonian cycle are contained in a cycle of length one greater. Let A be a symmetric (0,1)-matrix with zero main diagonal such that A is the adjacency matrix of a chordal Hamiltonian graph. Hendry’s conjecture in this case is that every k×k principle submatrix of A that dominates a full cycle permutation k×k matrix is a principle submatrix of a (k+1)×(k+1) principle submatrix of A that dominates a (k+1)×(k+1) full cycle permutation matrix. This article generalizes the concept of cycle-extendability to S-extendable; that is, with S⊆{1,2,…,n} and G a graph on n vertices, G is S-extendable if the vertices of every non-Hamiltonian cycle are contained in a cycle length i greater, where i∈S. We investigate this concept in directed graphs and in particular tournaments, i.e., anti-symmetric matrices with zero main diagonal.     

Bijective linear maps on semimodules spanned by Boolean matrices of fixed rank

Let M_m,n(B) be the semimodule of all m×n Boolean matrices where B is the Boolean algebra with two elements. Let k be a positive integer such that 2?k?min(m,n). Let B(m,n,k) denote the subsemimodule of M_m,n(B) spanned by the set of all rank k matrices. We show that if T is a bijective linear mapping on B(m,n,k), then there exist permutation matrices P and Q such that T(A)=PAQ for all A∈B(m,n,k) or m=n and T(A)=PA^tQ for all A∈B(m,n,k). This result follows from a more general theorem we prove concerning the structure of linear mappings on B(m,n,k) that preserve both the weight of each matrix and rank one matrices of weight k². Here the weight of a Boolean matrix is the number of its nonzero entries.     

A note on the probabilistic approach to Turán's problem

The Turán number T(n, l, k) is the smallest possible number of edges in a k-graph on n vertices such that every l-set of vertices contains an edge. Given a k-graph H = (V(H), E(H)), we let X_s(S) equal the number of edges contained in S, for any s-set S?V(H). Turán's problem is equivalent to estimating the expectation E(X_l), given that min(X_l) ≥ 1. The following lower bound on the variance of X_s is proved:

$\text{Var(X}{}_{\mathrm{s}}\text{)?mm}\text{n?2k}\text{s?k}\text{n}\text{s}{}^{\mathrm{?1}}\text{n}\text{k}{}^1$

, where m = |E(H)| and $\text{m}\text{= (}{}_{\mathrm{k}}{}^{\mathrm{n}}\text{) ? m}$. This implies the following: putting t(k, l) = lim_n→∞T(n, l, k)(_kⁿ)^?1 then t(k, l) ≥ T(s, l, k)((_k^s) ? 1)^?1, whenever s ≥ l > k ≥ 2. A connection of these results with the existence of certain t-designs is mentioned.     

On a problem of Katona on minimal separating systems

Let S be an n-element set. In this paper, we determine the smallest number f(n) for which there exists a family of subsets of S{A₁,A₂,…,A_f(n)} with the following property: Given any two elements x, y ∈ S (x ≠ y), there exist k, l such that A_k ∩ A_l= ?, and x ∈ A_k, y ∈ A_l. In particular it is shown that f(n)= 3 log₃n when n is a power of 3.     

Preservers of eigenvalue inclusion sets

For a square matrix A, let S(A) be an eigenvalue inclusion set such as the Gershgorin region, the Brauer region in terms of Cassini ovals, and the Ostrowski region. Characterization is obtained for maps Φ on n×n matrices satisfying S(Φ(A)-Φ(B))=S(A-B) for all matrices A and B. From these results, one can deduce the structure of additive or (real) linear maps satisfying S(A)=S(Φ(A)) for every matrix A.     

Preservers of eigenvalue inclusion sets of matrix products

For a square matrix A, let S(A) be an eigenvalue inclusion set such as the Gershgorin region, the union of Cassini ovals, and the Ostrowski’s set. Characterization is obtained for maps Φ on n×n matrices satisfying S(Φ(A)Φ(B))=S(AB) for all matrices A and B.     

Irreducibility of associated matrices

For an n by n matrix A, let K(A) be the associated matrix corresponding to a permutation group (of degree m) and one of its characters. Let D_r(A) be the coefficient of x^m?r in K(A+xI). If A is reducible, then D_r(A) is reducible. If A is irreducible and the character is identically one, then D₁(A) is irreducible. If A is row stochastic and the character is identically one, then D_r(A) is essentially row stochastic. Finally, the results motivate the definition of group induced diagraphs.     

F-Sets in graphs

A subset S of the vertex set of a graph G is called an F-set if every α?Γ(G), the automorphism group of G, is completely specified by specifying the images under α of all the points of S, and S has a minimum number of points. The number of points, k(G), in an F-set is an invariant of G, whose properties are studied in this paper. For a finite group Γ we define k(Γ) = max{k(G) | Γ(G) = Γ}. Graphs with a given Abelian group and given k-value (k ≤ k(Γ)) have been constructed. Graphs with a given group and k-value 1 are constructed which give simple proofs to the theorems of Frucht and Bouwer on the existence of graphs with given abstract/permutation groups.     

On a problem of rota

Let S(n, k) denote Stirling numbers of the second kind; for each n, let K_n be such that S(n, K_n) ? S(n, k) for all k. Also, let P(n) denote the lattice of partitions of an n-element set. Say that a collection of partitions from P(n) is incomparable if no two are related by refinement. Rota has asked if for all n, the largest possible incomparable collection in P(n) contains S(n, K_n) partitions. In this paper, we construct for all n sufficiently large an incomparable collection in P(n) containing strictly more than S(n, K_n) partitions. We also estimate how large n must be for this construction to work.     

Nonnegative factorization of completely positive matrices

Let A be a real symmetric n × n matrix of rank k, and suppose that A = BB′ for some real n × m matrix B with nonnegative entries (for some m). (Such an A is called completely positive.) It is shown that such a B exists with $\text{m?}\text{1}\text{2}\text{k(k+1)?N}$, where 2N is the maximal number of (off-diagonal) entries which equal zero in a nonsingular principal submatrix of A. An example is given where the least m which works is $\text{(k+1)}{}^2\text{4}$ (k odd),$\text{k(k+2)}\text{4}$ (k even).     

An Almost Accurate Location of the Maximum Stirling Number(s) of the Second Kind

The Stirling number of the second kind S(n, k) is the number of ways of partitioning a set of n elements into k nonempty subsets. It is well known that the numbers S(n, k) are unimodal in k, and there are at most two consecutive values K _n such that (for fixed n) S(n, K _n ) is maximal. We determine asymptotic bounds for K _n , which are unexpectedly good and improve earlier results. The method used here shows a possible strategy for obtaining numerical bounds such that in almost all cases K _n can be uniquely determined.     

An interlacing theorem for matrices whose graph is a given tree

Let A and B be (n×n)-matrices. For an index set S ⊂ {1, …, n}, denote by A(S) the principal submatrix that lies in the rows and columns indexed by S. Denote by S′ the complement of S and define η(A, B) = det A(S) det B(S′), where the summation is over all subsets of {1, …, n} and, by convention, det A(∅) = det B(∅) = 1. C. R. Johnson conjectured that if A and B are Hermitian and A is positive semidefinite, then the polynomial η(λA,-B) has only real roots. G. Rublein and R. B. Bapat proved that this is true for n ⩽ 3. Bapat also proved this result for any n with the condition that both A and B are tridiagonal. In this paper, we generalize some little-known results concerning the characteristic polynomials and adjacency matrices of trees to matrices whose graph is a given tree and prove the conjecture for any n under the additional assumption that both A and B are matrices whose graph is a tree. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 10, No. 3, pp. 245–254, 2004.     

Norms and inequalities for condition numbers,III

The main results provide comparisons between condition numbers (based on unitarily invariant norms) of (i) positive definite (Hermitian) matrices A, B and of A + B, (ii) a positive definite matrix and its principal submatrix, and (iii) a matrix and an augmented form of the matrix.