Abstract. In this paper, more generalized q-factorial coefficients are examined by a natural exte... more Abstract. In this paper, more generalized q-factorial coefficients are examined by a natural extension of the q-factorial on a sequence of any numbers. This immediately leads to the notions of the extended q-Stirling numbers of both kinds and the extended q-Lah numbers. All results described in this paper may be reduced to well-known results when we set q = 1 or use special sequences. 1.
This report documents the program and the outcomes of Dagstuhl Seminar 13082 "Communica-tion... more This report documents the program and the outcomes of Dagstuhl Seminar 13082 "Communica-tion Complexity, Linear Optimization, and lower bounds for the nonnegative rank of matrices". The nonnegative rank is a measure of the complexity of a matrix that has applications ranging from Communication Complexity to Combinatorial Optimization. At the time of the proposal of the seminar, known lower bounds for the nonnegative rank were either trivial (rank lower bound) or known not to work in many important cases (bounding the nondeterministic communication complexity of the support of the matrix). Over the past couple of years in Combinatorial Optimization, there has been a surge of interest in lower bounds on the sizes of Linear Programming formulations. A number of new methods have been developed, for example characterizing nonnegative rank as a variant of randomized communication complexity. The link between communication complexity and nonnegative rank was also instrumental rec...
The possible dimensions of spaces of matrices over GF(2) whose nonzero elements all have rank 2 a... more The possible dimensions of spaces of matrices over GF(2) whose nonzero elements all have rank 2 are investigated.
A rank 1 matrix has a factorization as uvt for vectors u and v of some orders. The arctic rank of... more A rank 1 matrix has a factorization as uvt for vectors u and v of some orders. The arctic rank of a rank 1 matrix is the half number of nonzero entries in u and v. A matrix of rank k can be expressed as the sum of k rank 1 matrices, a rank 1 decomposition. The arctic rank of a matrix A of rank k is the minimum of the sums of arctic ranks of the rank 1 matrices over all rank 1 decomposition of A. In this paper we obtain characterizations of the linear operators that strongly preserve the symmetric arctic ranks of symmetric matrices over nonnegative reals.
A (0, 1)-labelling of a set is said to be friendly if approximately one half the elements of the ... more A (0, 1)-labelling of a set is said to be friendly if approximately one half the elements of the set are labelled 0 and one half labelled 1. Let g be a labelling of the edge set of a graph that is induced by a labelling f of the vertex set. If both g and f are friendly then g is said to be a cordial labelling of the graph. We extend this concept to directed graphs and investigate the cordiality of sets of directed graphs. We investigate a specific type of cordiality on digraphs, a restriction of quasigroup-cordiality called (2, 3)-cordiality. A directed graph is (2, 3)-cordial if there is a friendly labelling f of the vertex set which induces a (1,−1, 0)-labelling of the arc set g such that about one third of the arcs are labelled 1, about one third labelled -1 and about one third labelled 0. In particular we determine which tournaments are (2, 3)-cordial, which orientations of the n-wheel are (2, 3)-cordial, and which orientations of the n−fan are (2, 3)-cordial.
This report documents the program and the outcomes of Dagstuhl Seminar 13082 "Communication ... more This report documents the program and the outcomes of Dagstuhl Seminar 13082 "Communication Complexity, Linear Optimization, and lower bounds for the nonnegative rank of matrices".
Let A be a Boolean {0, 1} matrix. The isolation number of A is the maximum number of ones in A su... more Let A be a Boolean {0, 1} matrix. The isolation number of A is the maximum number of ones in A such that no two are in any row or any column (that is they are independent), and no two are in a 2 × 2 submatrix of all ones. The isolation number of A is a lower bound on the Boolean rank of A. A linear operator on the set of m× n Boolean matrices is a mapping which is additive and maps the zero matrix, O, to itself. A mapping strongly preserves a set, S, if it maps the set S into the set S and the complement of the set S into the complement of the set S. We investigate linear operators that preserve the isolation number of Boolean matrices. Specifically, we show that T is a Boolean linear operator that strongly preserves isolation number k for any 1 6 k 6 min{m,n} if and only if there are fixed permutation matrices P and Q such that for X ∈ Mm,n(B) T (X) = PXQ or, m = n and T (X) = PXQ where X is the transpose of X.
The term rank of a matrix A is the least number of lines (rows or columns) needed to include all ... more The term rank of a matrix A is the least number of lines (rows or columns) needed to include all the nonzero entries in A, and is a well-known upper bound for many standard and non-standard matrix ranks, and is one of the most important combinatorially. In this paper, we obtain a characterization of linear operators that preserve term ranks of matrices over antinegative semirings. That is, we show that a linear operator T on a matrix space over antinegative semirings preserves term rank if and only if T preserves any two term ranks k and l if and only if T strongly preserves any one term rank k. 2010 Mathematics Subject Classiflcation : 15A86, 15A03 and 15A04.
There are two forms of preserver problems, one fixes a function, set, or relation and asks what l... more There are two forms of preserver problems, one fixes a function, set, or relation and asks what linear transformations leave it invariant. This is the most common form studied. The other form fixes the structure of a linear transformation and asks what functions, sets, and relations are fixed by it. The first form of problem is well studied, the second is not as easy and few full characterizations are known. We consider the first form first.
Recently L. B. Beasley introduced (2, 3)-cordial labelings of di∗AMS Classification number: 05C20... more Recently L. B. Beasley introduced (2, 3)-cordial labelings of di∗AMS Classification number: 05C20, 05C38, 05C78 †
Let A be an m × n matrix over nonnegative integers. The isolation number of A is the maximum numb... more Let A be an m × n matrix over nonnegative integers. The isolation number of A is the maximum number of isolated entries in A. We investigate linear operators that preserve the isolation number of matrices over nonnegative integers. We obtain that T is a linear operator that strongly preserve isolation number k for 1 ≤ k ≤ min{m,n} if and only if T is a (P,Q)-operator, that is, for fixed permutation matrices P and Q, T (A) = PAQ or, m = n and T (A) = PAtQ for any m× n matrix A, where At is the transpose of A.
L. B. Beasley recently defined a digraph labeling called (2, 3)cordial in [1]. Digraphs for which... more L. B. Beasley recently defined a digraph labeling called (2, 3)cordial in [1]. Digraphs for which a (2, 3)-cordial labeling can be applied are called (2, 3)-cordial digraphs. Herein, we consider the existence and identification of (2, 3)-cordial oriented hypercubes. We demonstrate that for every nonzero dimension, there exists a (2, 3)cordial oriented hypercube. Additionally, we demonstrate that not all oriented hypercubes of nonzero dimension are (2, 3)-cordial. Finally, we present preliminary results regarding the identification of (2, 3)-cordial oriented hypercubes, particularly for dimension 3.
Let $m$ and $n$ be positive integers, and let $R =(r_1, \ldots, r_m)$ and $S =(s_1,\ldots, s_n)$... more Let $m$ and $n$ be positive integers, and let $R =(r_1, \ldots, r_m)$ and $S =(s_1,\ldots, s_n)$ be nonnegative integral vectors. Let $A(R,S)$ be the set of all $m \times n$ $(0,1)$-matrices with row sum vector $R$ and column vector $S$. Let $R$ and $S$ be nonincreasing, and let $F(R)$ be the $m \times n$ $(0,1)$-matrix where for each $i$, the $i^{th}$ row of $F(R,S)$ consists of $r_i$ 1's followed by $n-r_i$ 0's. Let $A\in A(R,S)$. The discrepancy of A, $disc(A)$, is the number of positions in which $F(R)$ has a 1 and $A$ has a 0. In this paper, we investigate the possible discrepancy of $A^t$ versus the discrepancy of $A$. We show that if the discrepancy of $A$ is $\ell$, then the discrepancy of the transpose of $A$ is at least $\frac{\ell}{2}$ and at most $2\ell$. These bounds are tight.
If a graph can be embedded in a smooth orientable surface of genus g without edge crossings and c... more If a graph can be embedded in a smooth orientable surface of genus g without edge crossings and can not be embedded on one of genus g − 1 without edge crossings, then we say that the graph has genus g. We consider a mapping on the set of graphs with m vertices into itself. The mapping is called a linear operator if it preserves a union of graphs and it also preserves the empty graph. On the set of graphs with m vertices, we consider and investigate those linear operators which map graphs of genus g to graphs of genus g and graphs of genus g + j to graphs of genus g + j for j ≤ g and m sufficiently large. We show that such linear operators are necessarily vertex permutations.
Let S be an antinegative semiring. The rank of an m × n matrix B over S is the minimal integer r ... more Let S be an antinegative semiring. The rank of an m × n matrix B over S is the minimal integer r such that B is a product of an m × r matrix and an r × n matrix. The isolation number of B is the maximal number of nonzero entries in the matrix such that no two entries are in the same column, in the same row, and in a submatrix of B of the form b i , j b i , l b k , j b k , l with nonzero entries. We know that the isolation number of B is not greater than the rank of it. Thus, we investigate the upper bound of the rank of B and the rank of its support for the given matrix B with isolation number h over antinegative semirings.
A graph has genus k if it can be embedded without edge crossings on a smooth orientable surface o... more A graph has genus k if it can be embedded without edge crossings on a smooth orientable surface of genus k and not on one of genus k − 1 . A mapping of the set of graphs on n vertices to itself is called a linear operator if the image of a union of graphs is the union of their images and if it maps the edgeless graph to the edgeless graph. We investigate linear operators on the set of graphs on n vertices that map graphs of genus k to graphs of genus k and graphs of genus k + 1 to graphs of genus k + 1 . We show that such linear operators are necessarily vertex permutations. Similar results with different restrictions on the genus k preserving operators give the same conclusion.
Let $\S$ denote the set of symmetric matrices over some semiring, $\s$. A line of $A\in\S$ is a r... more Let $\S$ denote the set of symmetric matrices over some semiring, $\s$. A line of $A\in\S$ is a row or a column of $A$. A star of $A$ is the submatrix of $A$ consisting of a row and the corresponding column of $A$. The term rank of $A$ is the minimum number of lines that contain all the nonzero entries of $A$. The star cover number is the minimum number of stars that contain all the nonzero entries of $A$. This paper investigates linear operators that preserve sets of symmetric matrices of specified term rank and sets of symmetric matrices of specific star cover numbers. Several equivalences to the condition that $T$ preserves the term rank of any matrix are given along with characterizations of a couple of types of linear operators that preserve certain sets of matrices defined by the star cover number that do not preserve all term ranks.
The possible dimensions of spaces of matrices over GF2 whose nonzero elements all have rank 2 are... more The possible dimensions of spaces of matrices over GF2 whose nonzero elements all have rank 2 are investigated.
Abstract. In this paper, more generalized q-factorial coefficients are examined by a natural exte... more Abstract. In this paper, more generalized q-factorial coefficients are examined by a natural extension of the q-factorial on a sequence of any numbers. This immediately leads to the notions of the extended q-Stirling numbers of both kinds and the extended q-Lah numbers. All results described in this paper may be reduced to well-known results when we set q = 1 or use special sequences. 1.
This report documents the program and the outcomes of Dagstuhl Seminar 13082 "Communica-tion... more This report documents the program and the outcomes of Dagstuhl Seminar 13082 "Communica-tion Complexity, Linear Optimization, and lower bounds for the nonnegative rank of matrices". The nonnegative rank is a measure of the complexity of a matrix that has applications ranging from Communication Complexity to Combinatorial Optimization. At the time of the proposal of the seminar, known lower bounds for the nonnegative rank were either trivial (rank lower bound) or known not to work in many important cases (bounding the nondeterministic communication complexity of the support of the matrix). Over the past couple of years in Combinatorial Optimization, there has been a surge of interest in lower bounds on the sizes of Linear Programming formulations. A number of new methods have been developed, for example characterizing nonnegative rank as a variant of randomized communication complexity. The link between communication complexity and nonnegative rank was also instrumental rec...
The possible dimensions of spaces of matrices over GF(2) whose nonzero elements all have rank 2 a... more The possible dimensions of spaces of matrices over GF(2) whose nonzero elements all have rank 2 are investigated.
A rank 1 matrix has a factorization as uvt for vectors u and v of some orders. The arctic rank of... more A rank 1 matrix has a factorization as uvt for vectors u and v of some orders. The arctic rank of a rank 1 matrix is the half number of nonzero entries in u and v. A matrix of rank k can be expressed as the sum of k rank 1 matrices, a rank 1 decomposition. The arctic rank of a matrix A of rank k is the minimum of the sums of arctic ranks of the rank 1 matrices over all rank 1 decomposition of A. In this paper we obtain characterizations of the linear operators that strongly preserve the symmetric arctic ranks of symmetric matrices over nonnegative reals.
A (0, 1)-labelling of a set is said to be friendly if approximately one half the elements of the ... more A (0, 1)-labelling of a set is said to be friendly if approximately one half the elements of the set are labelled 0 and one half labelled 1. Let g be a labelling of the edge set of a graph that is induced by a labelling f of the vertex set. If both g and f are friendly then g is said to be a cordial labelling of the graph. We extend this concept to directed graphs and investigate the cordiality of sets of directed graphs. We investigate a specific type of cordiality on digraphs, a restriction of quasigroup-cordiality called (2, 3)-cordiality. A directed graph is (2, 3)-cordial if there is a friendly labelling f of the vertex set which induces a (1,−1, 0)-labelling of the arc set g such that about one third of the arcs are labelled 1, about one third labelled -1 and about one third labelled 0. In particular we determine which tournaments are (2, 3)-cordial, which orientations of the n-wheel are (2, 3)-cordial, and which orientations of the n−fan are (2, 3)-cordial.
This report documents the program and the outcomes of Dagstuhl Seminar 13082 "Communication ... more This report documents the program and the outcomes of Dagstuhl Seminar 13082 "Communication Complexity, Linear Optimization, and lower bounds for the nonnegative rank of matrices".
Let A be a Boolean {0, 1} matrix. The isolation number of A is the maximum number of ones in A su... more Let A be a Boolean {0, 1} matrix. The isolation number of A is the maximum number of ones in A such that no two are in any row or any column (that is they are independent), and no two are in a 2 × 2 submatrix of all ones. The isolation number of A is a lower bound on the Boolean rank of A. A linear operator on the set of m× n Boolean matrices is a mapping which is additive and maps the zero matrix, O, to itself. A mapping strongly preserves a set, S, if it maps the set S into the set S and the complement of the set S into the complement of the set S. We investigate linear operators that preserve the isolation number of Boolean matrices. Specifically, we show that T is a Boolean linear operator that strongly preserves isolation number k for any 1 6 k 6 min{m,n} if and only if there are fixed permutation matrices P and Q such that for X ∈ Mm,n(B) T (X) = PXQ or, m = n and T (X) = PXQ where X is the transpose of X.
The term rank of a matrix A is the least number of lines (rows or columns) needed to include all ... more The term rank of a matrix A is the least number of lines (rows or columns) needed to include all the nonzero entries in A, and is a well-known upper bound for many standard and non-standard matrix ranks, and is one of the most important combinatorially. In this paper, we obtain a characterization of linear operators that preserve term ranks of matrices over antinegative semirings. That is, we show that a linear operator T on a matrix space over antinegative semirings preserves term rank if and only if T preserves any two term ranks k and l if and only if T strongly preserves any one term rank k. 2010 Mathematics Subject Classiflcation : 15A86, 15A03 and 15A04.
There are two forms of preserver problems, one fixes a function, set, or relation and asks what l... more There are two forms of preserver problems, one fixes a function, set, or relation and asks what linear transformations leave it invariant. This is the most common form studied. The other form fixes the structure of a linear transformation and asks what functions, sets, and relations are fixed by it. The first form of problem is well studied, the second is not as easy and few full characterizations are known. We consider the first form first.
Recently L. B. Beasley introduced (2, 3)-cordial labelings of di∗AMS Classification number: 05C20... more Recently L. B. Beasley introduced (2, 3)-cordial labelings of di∗AMS Classification number: 05C20, 05C38, 05C78 †
Let A be an m × n matrix over nonnegative integers. The isolation number of A is the maximum numb... more Let A be an m × n matrix over nonnegative integers. The isolation number of A is the maximum number of isolated entries in A. We investigate linear operators that preserve the isolation number of matrices over nonnegative integers. We obtain that T is a linear operator that strongly preserve isolation number k for 1 ≤ k ≤ min{m,n} if and only if T is a (P,Q)-operator, that is, for fixed permutation matrices P and Q, T (A) = PAQ or, m = n and T (A) = PAtQ for any m× n matrix A, where At is the transpose of A.
L. B. Beasley recently defined a digraph labeling called (2, 3)cordial in [1]. Digraphs for which... more L. B. Beasley recently defined a digraph labeling called (2, 3)cordial in [1]. Digraphs for which a (2, 3)-cordial labeling can be applied are called (2, 3)-cordial digraphs. Herein, we consider the existence and identification of (2, 3)-cordial oriented hypercubes. We demonstrate that for every nonzero dimension, there exists a (2, 3)cordial oriented hypercube. Additionally, we demonstrate that not all oriented hypercubes of nonzero dimension are (2, 3)-cordial. Finally, we present preliminary results regarding the identification of (2, 3)-cordial oriented hypercubes, particularly for dimension 3.
Let $m$ and $n$ be positive integers, and let $R =(r_1, \ldots, r_m)$ and $S =(s_1,\ldots, s_n)$... more Let $m$ and $n$ be positive integers, and let $R =(r_1, \ldots, r_m)$ and $S =(s_1,\ldots, s_n)$ be nonnegative integral vectors. Let $A(R,S)$ be the set of all $m \times n$ $(0,1)$-matrices with row sum vector $R$ and column vector $S$. Let $R$ and $S$ be nonincreasing, and let $F(R)$ be the $m \times n$ $(0,1)$-matrix where for each $i$, the $i^{th}$ row of $F(R,S)$ consists of $r_i$ 1's followed by $n-r_i$ 0's. Let $A\in A(R,S)$. The discrepancy of A, $disc(A)$, is the number of positions in which $F(R)$ has a 1 and $A$ has a 0. In this paper, we investigate the possible discrepancy of $A^t$ versus the discrepancy of $A$. We show that if the discrepancy of $A$ is $\ell$, then the discrepancy of the transpose of $A$ is at least $\frac{\ell}{2}$ and at most $2\ell$. These bounds are tight.
If a graph can be embedded in a smooth orientable surface of genus g without edge crossings and c... more If a graph can be embedded in a smooth orientable surface of genus g without edge crossings and can not be embedded on one of genus g − 1 without edge crossings, then we say that the graph has genus g. We consider a mapping on the set of graphs with m vertices into itself. The mapping is called a linear operator if it preserves a union of graphs and it also preserves the empty graph. On the set of graphs with m vertices, we consider and investigate those linear operators which map graphs of genus g to graphs of genus g and graphs of genus g + j to graphs of genus g + j for j ≤ g and m sufficiently large. We show that such linear operators are necessarily vertex permutations.
Let S be an antinegative semiring. The rank of an m × n matrix B over S is the minimal integer r ... more Let S be an antinegative semiring. The rank of an m × n matrix B over S is the minimal integer r such that B is a product of an m × r matrix and an r × n matrix. The isolation number of B is the maximal number of nonzero entries in the matrix such that no two entries are in the same column, in the same row, and in a submatrix of B of the form b i , j b i , l b k , j b k , l with nonzero entries. We know that the isolation number of B is not greater than the rank of it. Thus, we investigate the upper bound of the rank of B and the rank of its support for the given matrix B with isolation number h over antinegative semirings.
A graph has genus k if it can be embedded without edge crossings on a smooth orientable surface o... more A graph has genus k if it can be embedded without edge crossings on a smooth orientable surface of genus k and not on one of genus k − 1 . A mapping of the set of graphs on n vertices to itself is called a linear operator if the image of a union of graphs is the union of their images and if it maps the edgeless graph to the edgeless graph. We investigate linear operators on the set of graphs on n vertices that map graphs of genus k to graphs of genus k and graphs of genus k + 1 to graphs of genus k + 1 . We show that such linear operators are necessarily vertex permutations. Similar results with different restrictions on the genus k preserving operators give the same conclusion.
Let $\S$ denote the set of symmetric matrices over some semiring, $\s$. A line of $A\in\S$ is a r... more Let $\S$ denote the set of symmetric matrices over some semiring, $\s$. A line of $A\in\S$ is a row or a column of $A$. A star of $A$ is the submatrix of $A$ consisting of a row and the corresponding column of $A$. The term rank of $A$ is the minimum number of lines that contain all the nonzero entries of $A$. The star cover number is the minimum number of stars that contain all the nonzero entries of $A$. This paper investigates linear operators that preserve sets of symmetric matrices of specified term rank and sets of symmetric matrices of specific star cover numbers. Several equivalences to the condition that $T$ preserves the term rank of any matrix are given along with characterizations of a couple of types of linear operators that preserve certain sets of matrices defined by the star cover number that do not preserve all term ranks.
The possible dimensions of spaces of matrices over GF2 whose nonzero elements all have rank 2 are... more The possible dimensions of spaces of matrices over GF2 whose nonzero elements all have rank 2 are investigated.
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