A note on k-generalized projections

In this note, we investigate characterizations for k-generalized projections (i.e., A^k = A^*) on Hilbert spaces. The obtained results generalize those for generalized projections on Hilbert spaces in [Hong-Ke Du, Yuan Li, The spectral characterization of generalized projections, Linear Algebra Appl. 400 (2005) 313–318] and those for matrices in [J. Benítez, N. Thome, Characterizations and linear combinations of k-generalized projectors, Linear Algebra Appl. 410 (2005) 150–159].     

Proof of a conjecture of Fiedler and Markham

Let A be an n×n nonsingular M-matrix. For the Hadamard product AA⁻¹, M. Fiedler and T.L. Markham conjectured in [Linear Algebra Appl. 10l (1988) 1] that q(AA⁻¹)2/n, where q(AA⁻¹) is the smallest eigenvalue (in modulus) of AA⁻¹. We considered this conjecture in [Linear Algebra Appl. 288 (1999) 259] having observed an incorrect proof in [Linear Algebra Appl. 144 (1991) 171] and obtained that q(AA⁻¹)(2/n)(n−1)/n. The present paper gives a proof for this conjecture.     

On some properties of the series ∑k=0 kx and the Stirling numbers of the second kind

We partially characterize the rational numbers x and integers n 0 for which the sum ∑_k=0^∞ kⁿx^k assumes integers. We prove that if ∑_k=0^∞ kⁿx^k is an integer for x = 1 − a/b with a, b> 0 integers and gcd(a,b) = 1, then a = 1 or 2. Partial results and conjectures are given which indicate for which b and n it is an integer if a = 2. The proof is based on lower bounds on the multiplicities of factors of the Stirling number of the second kind, S(n,k). More specifically, we obtain for all integers k, 2 k n, and a 3, provided a is odd or divisible by 4, where v_a(m) denotes the exponent of the highest power of a which divides m, for m and a> 1 integers.

New identities are also derived for the Stirling numbers, e.g., we show that ∑_k=0²ⁿk! S(2n, k) , for all integers n 1.     

A note on the determinant formulas computation of generalized inverse matrix Padé approximation

In this paper, the computation of two special determinants which appear in the construction of a generalized inverse matrix Padé approximation of type [n/2k] (described in [Linear Algebra Appl. 322 (2001) 141]) for a given power series is investigated. Here a common computational approach of determinant can not be used. The main tool to be used to do the two special determinants is the well-known Schur complement theorem.     

Maximum number of edges in a critically k-connected graph

A k-connected graph G is said to be critically k-connected if G−v is not k-connected for any vV(G). We show that if n,k are integers with k4 and nk+2, and G is a critically k-connected graph of order n, then |E(G)|n(n−1)/2−p(n−k)+p²/2, where p=(n/k)+1 if n/k is an odd integer and p=n/k otherwise. We also characterize extremal graphs.     

The proof on the conjecture of extremal graphs for the kth eigenvalues of trees

We consider the only remaining unsolved case n≡0 (mod k) for the largest kth eigenvalue λ_k.of trees with n vertices. In this paper, the conjecture for this problem in [Shao Jia-yu, On the largest kth eignevalues of trees, Linear Algebra Appl. 221 (1995) 131] is proved and (from this) the complete solution to this problem, the best upper bound and the extremal trees of λ_k, is given in general cases above.     

Sperner systems containing at most k sets of every cardinality

We prove using a direct construction that one can choose n − 2 subsets of an n-element set with different cardinality such that none of them contains any other. As a generalization, we prove that if for any j we can have at most k subsets containing exactly j elements (k> 1), then for n 5 we can choose at most k(n − 3) subsets from an n-element set such that they form a Sperner system. Moreover, we prove that this can be achieved if n is large enough, and give a construction for n 8k − 4.     

Embedding complete trees into the hypercube

We consider embeddings of the complete t-ary trees of depth k (denotation T^k,t) as subgraphs into the hypercube of minimum dimension n. This n, denoted by dim(T^k,t), is known if max{k,t}2. First, we study the next open cases t=3 and k=3. We improve the known upper bound dim(T^k,3)2k+1 up to lim_k→∞dim(T^k,3)/k5/3 and show lim_t→∞dim(T^3,t)/t=227/120. As a co-result, we present an exact formula for the dimension of arbitrary trees of depth 2, as a function of their vertex degrees. These results and new techniques provide an improvement of the known upper bound for dim(T^k,t) for arbitrary k and t.     

A note on feedback cyclicity of C[Y]

Let (A,B) be an n-dimensional linear system with 2-inputs over C[Y], the ring of polynomials in one-variable over the field of complex numbers. We prove the feedback cyclicity of (A,B) under certain conditions on their entries and deduce that (A,B) is feedback cyclic in an exceptional case left open in W. Schmale [Linear Algebra Appl. 275–276 (1998) 551–562].     

k-Subdomination in graphs

For a positive integer k, a k-subdominating function of a graph G=(V,E) is a function f : V→{−1,1} such that ∑_uNG[v]f(u)1 for at least k vertices v of G. The k-subdomination number of G, denoted by γ_ks(G), is the minimum of ∑_vVf(v) taken over all k-subdominating functions f of G. In this article, we prove a conjecture for k-subdomination on trees proposed by Cockayne and Mynhardt. We also give a lower bound for γ_ks(G) in terms of the degree sequence of G. This generalizes some known results on the k-subdomination number γ_ks(G), the signed domination number γ_s(G) and the majority domination number γ_maj(G).     

Covering vertices of a graph by k disjoint cycles

Let d, k and n be three integers with k3, d4k−1 and n3k. We show that if d(x)+d(y)d for each pair of nonadjacent vertices x and y of a graph G of order n, then G contains k vertex-disjoint cycles converting at least min{d,n} vertices of G.     

Minimal least eigenvalue of connected graphs of order n and size m = n + k (5≤k≤8)

The least eigenvalue of a connected graph is the least eigenvalue of its adjacency matrix. We characterize the connected graphs of order n and size n + k (5≤k≤8 and n>k + 5) with the minimal least eigenvalue.     

k-Lucas numbers and associated bipartite graphs

For a positive integer k2, the k-Fibonacci sequence {g_n^(k)} is defined as: g₁^(k)==g_k−2^(k)=0, g_k−1^(k)=g_k^(k)=1 and for n>k2, g_n^(k)=g_n−1^(k)+g_n−2^(k)++g_n−k^(k). Moreover, the k-Lucas sequence {l_n^(k)} is defined as l_n^(k)=g_n−1^(k)+g_n+k−1^(k) for n1. In this paper, we consider the relationship between g_n^(k) and l_n^(k) and 1-factors of a bipartite graph.     

Optimal fault-tolerant routings with small routing tables for k-connected graphs

We study the problem of designing fault-tolerant routings with small routing tables for a k-connected network of n processors in the surviving route graph model. The surviving route graph R(G,ρ)/F for a graph G, a routing ρ and a set of faults F is a directed graph consisting of nonfaulty nodes of G with a directed edge from a node x to a node y iff there are no faults on the route from x to y. The diameter of the surviving route graph could be one of the fault-tolerance measures for the graph G and the routing ρ and it is denoted by D(R(G,ρ)/F). We want to reduce the total number of routes defined in the routing, and the maximum of the number of routes defined for a node (called route degree) as least as possible. In this paper, we show that we can construct a routing λ for every n-node k-connected graph such that n2k², in which the route degree is , the total number of routes is O(k²n) and D(R(G,λ)/F)3 for any fault set F (|F|<k). In particular, in the case that k=2 we can construct a routing λ′ for every biconnected graph in which the route degree is , the total number of routes is O(n) and D(R(G,λ′)/{f})3 for any fault f. We also show that we can construct a routing ρ₁ for every n-node biconnected graph, in which the total number of routes is O(n) and D(R(G,ρ₁)/{f})2 for any fault f, and a routing ρ₂ (using ρ₁) for every n-node biconnected graph, in which the route degree is , the total number of routes is and D(R(G,ρ₂)/{f})2 for any fault f.     

Equivalence constants for certain matrix norms II

We give an almost complete solution of a problem posed by Klaus and Li [A.-L. Klaus, C.-K. Li, Isometries for the vector (p, q) norm and the induced (p, q) norm, Linear and Multilinear Algebra 38 (1995) 315–332]. Klaus and Li’s problem, which arose during their investigations of isometries, was to relate the Frobenius (or Hilbert–Schmidt) norm of a matrix to various operator norms of that matrix. Our methods are based on earlier work of Feng [B.Q. Feng, Equivalence constants for certain matrix norms, Linear Algebra Appl. 374 (2003) 247–253] and Tonge [A. Tonge, Equivalence constants for matrix norms: a problem of Goldberg, Linear Algebra Appl. 306 (2000) 1–13], but introduce as a new ingredient some techniques developed by Hardy and Littlewood [G.H. Hardy, J.E. Littlewood, Bilinear forms bounded in space [p, q], Quart. J. Math. (Oxford) 5 (1934) 241–254].     

Properties of core-EP order in rings with involution

We study properties of a relation in *-rings, called the core-EP (pre)order which was introduced by H. Wang on the set of all n × n complex matrices [Linear Algebra Appl., 2016, 508: 289–300] and has been recently generalized by Y. Gao, J. Chen, and Y. Ke to *-rings [Filomat, 2018, 32: 3073–3085]. We present new characterizations of the core-EP order in *-rings with identity and introduce the notions of the dual core-EP decomposition and the dual core-EP order in-rings.     

On the number of edge-disjoint one factors and the existence of k-factors in complete multipartite graphs

In this paper we use Tutte's f-factor theorem and the method of amalgamations to find necessary and sufficient conditions for the existence of a k-factor in the complete multipartite graph K(p(1), …, p(n)), conditions that are reminiscent of the Erdös-Gallai conditions for the existence of simple graphs with a given degree sequence. We then use this result to investigate the maximum number of edge-disjoint 1-factors in K(p(1), …, p(n)), settling the problem in the case where this number is greater than δ - p(2), where p(1) p(2) … p(n).     

Holes in random graphs

It is shown that for every >0 with the probability tending to 1 as n→∞ a random graph G(n,p) contains induced cycles of all lengths k, 3 ≤ k ≤ (1 − )n log c/c, provided c(n) = (n − 1)p(n)→∞.     

Triangular range counting query in 2D and its application in finding k nearest neighbors of a line segment

We present an efficient algorithm for finding k nearest neighbors of a query line segment among a set of points distributed arbitrarily on a two dimensional plane. Along the way to finding this algorithm, we have obtained an improved triangular range searching technique in 2D. Given a set of n points, we preprocess them to create a data structure using O(n²) time and space, such that given a triangular query region Δ, the number of points inside Δ can be reported in O(logn) time. Finally, this triangular range counting technique is used for finding k nearest neighbors of a query straight line segment in O(log²n+k) time.     

A special k-coloring for a connected k-chromatic graph

For each positive integer k we consider the smallest positive integer f(k) (dependent only on k) such that the following holds: Each connected graph G with chromatic number χ(G) = k can be properly vertex colored by k colors so that for each pair of vertices x_o and x_p in any color class there exist vertices x₁, x₂, …, x_p-1 of the same class with dist(x_i, x_i+1) f(k) for each i, 0 i p − 1. Thus, the graph is k-colorable with the vertices of each color class placed throughout the graph so that no subset of the class is at a distance > f(k) from the remainder of the class.

We prove that f(k) < 12k when the order of the graph is k(k − 2) + 1.