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 fat partitioning, fat covering and the union size of polygons

The complexity of the contour of the union of simple polygons with n vertices in total can be O(n²) in general. A notion of fatness for simple polygons is introduced that extends most of the existing fatness definitions. It is proved that a set of fat polygons with n vertices in total has union complexity O(n log log n), which is a generalization of a similar result for fat triangles (Matou ek et al., 1994). Applications to several basic problems in computational geometry are given, such as efficient hidden surface removal, motion planning, injection molding, and more. The result is based on a new method to partition a fat simple polygon P with n vertices into O(n) fat convex quadrilaterals, and a method to cover (but not partition) a fat convex quadrilateral with O(l) fat triangles. The maximum overlap of the triangles at any point is two, which is optimal for any exact cover of a fat simple polygon by a linear number of fat triangles.     

Irreducible components of L2 functions on hopf bundles

This paper provides an explicit decomposition of the L² function space on the unit sphereS^dn-1 for d = 1,2 and 4, into irreducible representations under the action of the Lie Groups K= SO(n) × SO(1)S(U(n) × U(1)), and Sp(n) × Sp(l), respectively. The decomposition is realized as the eigenspaces of the Laplacian acting on homogeneous polynomials over the reals, complex numbers and quaternions. For the quaternionic case, an additional differential operator that commutes with the Laplacian is used to find the decomposition.     

Some rationally looking faces of Ωn having irrational minimum permanents

Let Ω_n denote the convex polytope consisting of all n × n doubly stochasiic matrices. We determine the minimum permanents which may or may not be rational and the permanent-minimizing matrices over some rationally looking faces of Ω_n We also discuss the barycentricity of the (0, l)-matrices with which we consider the permanent-minimization problem.     

Area-efficient algorithms for straight-line tree drawings

We investigate several straight-line drawing problems for bounded-degree trees in the integer grid without edge crossings under various types of drawings: (1) upward drawings whose edges are drawn as vertically monotone chains, a sequence of line segments, from a parent to its children, (2) order-preserving drawings which preserve the left-to-right order of the children of each vertex, and (3) orthogonal straight-line drawings in which each edge is represented as a single vertical or horizontal segment.

Main contribution of this paper is a unified framework to reduce the upper bound on area for the straight-line drawing problems from O(nlogn) (Crescenzi et al., 1992) to O(nloglogn). This is the first solution of an open problem stated by Garg et al. (1993). We also show that any binary tree admits a small area drawing satisfying any given aspect ratio in the orthogonal straight-line drawing type.

Our results are briefly summarized as follows. Let T be a bounded-degree tree with n vertices. Firstly, we show that T admits an upward straight-line drawing with area O(nloglogn). If T is binary, we can obtain an O(nloglogn)-area upward orthogonal drawing in which each edge is drawn as a chain of at most two orthogonal segments and which has O(n/logn) bends in total. Secondly, we present O(nloglogn)-area (respectively, -volume) orthogonal straight-line drawing algorithms for binary trees with arbitrary aspect ratios in 2-dimension (respectively, 3-dimension). Finally, we present some experimental results which shows the area requirements, in practice, for (order-preserving) upward drawing are much smaller than theoretical bounds obtained through analysis.     

Subsets of an interval whose product is a power

We form squares from the product of integers in a short interval [n, n + t_n], where we include n in the product. If p is prime, p|n, and (₂^p) > n, we prove that p is the minimum t_n. If no such prime exists, we prove t_n √5n when n> 32. If n = p(2p − 1) and both p and 2p − 1 are primes, then t_n = 3p> 3 √n/2. For n(n + u) a square > n², we conjecture that a and b exist where n < a < b < n + u and nab is a square (except n = 8 and N = 392). Let g₂(n) be minimal such that a square can be formed as the product of distinct integers from [n, g₂(n)] so that no pair of consecutive integers is omitted. We prove that g₂(n) 3n − 3, and list or conjecture the values of g₂(n) for all n. We describe the generalization to kth powers and conjecture the values for large n.     

On the permanent of doubly stochastic matrices with zero diagonal

The permanent function on the set of n×n doubly stochastic matrices with zero main diagonaln≤4, attains its minimum uniquely at the matrix whose off-diagonal entries are all equal to l/(n-1).     

Embedding of circulant graphs and generalized Petersen graphs on projective plane

Both the circulant graph and the generalized Petersen graph are important types of graphs in graph theory. In this paper, the structures of embeddings of circulant graph C(2n + 1; {1, n}) on the projective plane are described, the number of embeddings of C(2n + 1; {1, n}) on the projective plane follows, then the number of embeddings of the generalized Petersen graph P(2n +1, n) on the projective plane is deduced from that of C(2n +1; {1, n}), because C(2n + 1;{1, n}) is a minor of P(2n + 1, n), their structures of embeddings have relations. In the same way, the number of embeddings of the generalized Petersen graph P(2n, 2) on the projective plane is also obtained.     

Computing rectangle enclosures

The rectangle enclosure problem is the problem of determining the subset of n iso-oriented planar rectangles that enclose a query rectangle Q. In this paper, we use a three layered data structure which is a combination of Range and Priority search trees and answers both the static and dynamic cases of the problem. Both the cases use O(n> log² n) space. For the static case, the query time is O(log² n log log n + K). The dynamic case is supported in O(log³ n + K) query time using O(log³ n) amortized time per update. K denotes the size of the answer. For the d-dimensional space the results are analogous. The query time is O(log^2d-2 n log log n + K) for the static case and O(log^2d-1 n + K) for the dynamic case. The space used is O(n> log^2d-2 n) and the amortized time for an update is O(log^2d-1 n). The existing bounds given for a class of problems which includes the present one, are O(log^2d n + K) query time, O(log^2d n) time for an insertion and O(log^2d-1 n) time for a deletion.     

On embedding an outer-planar graph in a point set

Given an n-vertex outer-planar graph G and a set P of n points in the plane, we present an O(nlog³n) time and O(n) space algorithm to compute a straight-line embedding of G in P, improving upon the algorithm in [8,12] that requires O(n²) time. Our algorithm is near-optimal as there is an Ω(nlogn) lower bound for the problem [4]. We present a simpler O(nd) time and O(n) space algorithm to compute a straight-line embedding of G in P where lognd2n is the length of the longest vertex disjoint path in the dual of G. Therefore, the time complexity of the simpler algorithm varies between O(nlogn) and O(n²) depending on the value of d. More efficient algorithms are presented for certain restricted cases. If the dual of G is a path, then an optimal Θ(nlogn) time algorithm is presented. If the given point set is in convex position then we show that O(n) time suffices.     

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.     

Further results on irregular, critical perfect systems of difference sets II: systems without splits

An (m, n; u, v; c)-system is a collection of components, m of valency u−1 and n of valency v−1, whose difference sets form a perfect system with threshold c. If there is an (m, n; 3, 6; c)-system, then m2c−1; and if there is a (2c−1, n; 3, 6; c)-system, then 2c−1n. For all sufficiently large c, there are (2c−1, n; 3, 6; c)-systems with a split at 3c+6n−1 at least when n=1, 5, 6 and 7, but such systems do not exist for n=2, 3 or 4.

We describe here a general method of construction for (2c−1, n; 3, 6; c)-systems and use it to show that there are such systems for 2n4 and certain values of c depending on n. We also discuss the limitations of this method.     

Label placement by maximum independent set in rectangles

Motivated by the problem of labeling maps, we investigate the problem of computing a large non-intersecting subset in a set of n rectangles in the plane. Our results are as follows. In O(n log n) time, we can find an O(log n)-factor approximation of the maximum subset in a set of n arbitrary axis-parallel rectangles in the plane. If all rectangles have unit height, we can find a 2-approximation in O(n log n) time. Extending this result, we obtain a (1 + 1/k)-approximation in time O(n log n + n^2k−1) time, for any integer k ≥ 1.     

Linear operators preserving unitarily invariant norms of matrices

Let F_{m × n} be the set of all m × n matrices over the field F = C or R Denote by U_n(F) the group of all n × n unitary or orthogonal matrices according as F = C or F-R. A norm N() on Fm ×n, is unitarily invariant if N(UAV) = N(A): for all A ∈ F _m×n U ∈ U_m(F). and V ∈ U_n(F). We characterize those linear operators TF_{m × n} → F_{m × n}which satisfy N (T(A)) = N(A)for all A ∈ F_{m × n}

for a given unitarily invariant norm N(). It is shown that the problem is equivalent to characterizing those operators which preserve certain subsets in F_{m × n} To develop the theory we prove some results concerning unitary operators on F_{m × n} which are of independent interest.     

Aspects of semiorders within interval orders

Let s_k(n) be the largest integer such that every n-point interval order with no antichain of more than k points includes an s_k(n)-point semiorder. When k = 1, s₁(n) = n since all interval orders with no two-point antichains are chains. Given (c₁,...,c₅) = (1, 2, 3, 4), it is shown that s₂(n) = c_n for n 4, s₃(n) = c_n for n 5, and for all positive n, s₂ (n+4) =s₂(n)+3, s₃(n+5) = s₃(n)+3. Hence s₂ has a repeating pattern of length 4 [1, 2, 3, 3; 4, 5, 6, 6; 7, 8, 9, 9;...], and s₃ has a repeating pattern of length 5 [1, 2, 3, 3, 4; 4, 5, 6, 6, 7; 7, 8, 9, 9, 10;...].

Let s(n) be the largest integer such that every n-point interval order includes an s(n)-point semiorder. It was proved previously that for even n from 4 to 14, and that s(17) = 9. We prove here that s(15) = s(16) = 9, so that s begins 1, 2, 3, 3, 4, 4,..., 8, 8, 9, 9, 9. Since s(n)/n→0, s cannot have a repeating pattern.     

Popular distances in 3-space

Let m(n) denote the smallest integer m with the property that any set of n points in Euclidean 3-space has an element such that at most m other elements are equidistant from it. We have that cn^1/3 log log n m(n) n^3/5 β(n), where c> 0 is a constant and β(n) is an extremely slowly growing function, related to the inverse of the Ackermann function.     

Inertially arbitrary patterns

An n×n sign pattern matrix A is an inertially arbitrary pattern (IAP) if each non-negative triple (rst) with r+s+t=n is the inertia of a matrix with sign pattern A. This paper considers the n×n(n≥2) skew-symmetric sign pattern S_n with each upper off-diagonal entry positive, the (1,1) entry negative, the (nn) entry positive, and every other diagonal entry zero. We prove that S_n is an IAP.     

On the sum powers of matrices

A theorem of Lagrange says that every natural number is the sum of 4 squares. M. Newman proved that every integral n by n matrix is the sum of 8 (-1)ⁿ squares when n is at least 2. He asked to generalize this to the rings of integers of algebraic number fields. We show that an n by n matrix over a a commutative R with 1 is the sum of squares if and only if its trace reduced modulo 2Ris a square in the ring R/2R. It this is the case (and n is at least 2), then the matrix is the sum of 6 squares (5 squares would do when n is even). Moreover, we obtain a similar result for an arbitrary ring R with 1. Answering another question of M. Newman, we show that every integral n by n matrix is the sum of ten k-th powers for all sufficiently large n. (depending on k).     

Degrees in a digraph whose nodes are graphs

By an f-graph we mean an unlabeled graph having no vertex of degree greater than f. Let D(n, f) denote the digraph whose node set is the set of f-graphs of order n and such that there is an arc from the node corresponding to graph H to the node corresponding to the graph K if and only if K is obtainable from H by the addition of a single edge. In earlier work, algorithms were developed which produce exact results about the structure of D(n, f), nevertheless many open problems remain. For example, the computation of the order and size of D(n, f) for a number of values of n and f have been obtained. Formulas for the order and size for f = 2 have also been derived. However, no closed form formulas have been determined for the order and size of D(n, f) for any value of f. Here we focus on questions concerning the degrees of the nodes in D(n,n − 1) and comment on related questions for D(n,f) for 2 f < n − 1.     

Graphs with prescribed local connectivities

A graph G is locally n-connected (locally n-edge connected) if the neighborhood of each vertex of G is n-connected (n-edge connected). The local connectivity (local edge-connectivity) of G is the maximum n for which G is locally n-connected (locally n-edge connected). It is shown that if k and m are integers with O k < m, then a graph exists which has connectivity m and local connectivity k. Furthermore, such a graph with smallest order is determined. Corresponding results are obtained involving the local connectivity and the local edge-conectivity.