In this paper, we consider the existence of group divisible designs (GDDs) with block size 4 and ... more In this paper, we consider the existence of group divisible designs (GDDs) with block size 4 and group sizes 4 and 7. We show that there exists a 4-GDD of type 4 t 7 s for all but a finite specified set of feasible values for (t, s).
We prove that each maximal partial Latin cube must have more than 29.289% of its cells filled and... more We prove that each maximal partial Latin cube must have more than 29.289% of its cells filled and show by construction that this is a nearly tight bound. We also prove upper and lower bounds on the number of cells containing a fixed symbol in maximal partial Latin cubes and hypercubes, and we use these bounds to determine for small orders n the numbers k for which there exists a maximal partial Latin cube of order n with exactly k entries. Finally, we prove that maximal partial Latin cubes of order n exist of each size from approximately half-full (n 3 /2 for even n 10 and (n 3 + n)/2 for odd n 21) to completely full, except for when either precisely 1 or 2 cells are empty.
The Moore-Penrose inverse of a real matrix having no square submatrix with two or more diagonals ... more The Moore-Penrose inverse of a real matrix having no square submatrix with two or more diagonals is described in terms of bipartite graphs. For such a matrix, the sign of every entry of the Moore-Penrose inverse is shown to be determined uniquely by the signs of the matrix entries; i.e., the matrix has a signed generalized inverse. Necessary and sufficient conditions on an acyclic bipartite graph are given so that each nonnegative matrix with this graph has a nonnegative Moore-Penrose inverse. Nearly reducible matrices are proved to contain no submatrix having two or more diagonals, implying that a nearly reducible matrix has a signed generalized inverse. Furthermore, it is proved that the term rank and rank are equal for each submatrix of a nearly reducible matrix.
We present new values and bounds on the (normalised) closeness centralityCC of connected graphs a... more We present new values and bounds on the (normalised) closeness centralityCC of connected graphs and on its productlCC with the mean distancel of these graphs. Our main result presents the fundamental bounds 1 ≤lCC < 2. The lower bound is tight and the upper bound is asymptotically tight. Combining the lower bound with known upper bounds on the mean distance, we find ten new lower bounds for the closeness centrality of graphs. We also present explicit expressions forCC andlCC for specific families of graphs. Elegantly and perhaps surprisingly, the asymptotic values nCC Pn and of nCC Ln both equal π, and the asymptotic limits oflCC for these families of graphs are both equal to π/3. We conjecture that the set of valueslCC for all connected graphs is dense in the interval [1, 2).
Mubayi's Conjecture states that if F is a family of k-sized subsets of [n] = {1,. .. , n} which, ... more Mubayi's Conjecture states that if F is a family of k-sized subsets of [n] = {1,. .. , n} which, for k ≥ d ≥ 3 and n ≥ dk d−1 , satisfies A 1 ∩• • •∩A d = ∅ whenever |A 1 ∪• • •∪A d | ≤ 2k for all distinct sets A 1 ,. .. , A d ∈ F , then |F | ≤ n−1 k−1 , with equality occurring only if F is the family of all k-sized subsets containing some fixed element. This paper proves that Mubayi's Conjecture is true for all families that are invariant with respect to shifting; indeed, these families satisfy a stronger version of Mubayi's Conjecture. Relevant to the conjecture, we prove a fundamental bijective duality between what we call (i, j)unstable families and (j, i)-unstable families. Generalising previous intersecting conditions, we introduce the (d, s, t)-conditionally intersecting condition for families of sets and prove general results thereon. We prove fundamental theorems on two (d, s)-conditionally intersecting families that generalise previous intersecting families, and we pose an extension of a previous conjecture by Frankl and Füredi. Finally, we generalise a classical result by Erdős, Ko and Rado by proving tight upper bounds on the size of (2, s)-conditionally intersecting families F ⊆ 2 [n] and by characterising the families that attain these bounds. We extend this theorem for sufficiently large n to families F ⊆ 2 [n] whose members have at most a fixed size u.
Proceedings of the American Mathematical Society, Feb 10, 2021
For trace class operators A, B ∈ B 1 (H) (H a complex, separable Hilbert space), the product form... more For trace class operators A, B ∈ B 1 (H) (H a complex, separable Hilbert space), the product formula for Fredholm determinants holds in the familiar form det H ((I H − A)(I H − B)) = det H (I H − A)det H (I H − B). When trace class operators are replaced by Hilbert-Schmidt operators A, B ∈ B 2 (H) and the Fredholm determinant det H (I H − A), A ∈ B 1 (H), by the 2nd regularized Fredholm determinant det H,2 (I H − A) = det H ((I H − A) exp(A)), A ∈ B 2 (H), the product formula must be replaced by det H,2 ((I H − A)(I H − B)) = det H,2 (I H − A)det H,2 (I H − B) × exp(− tr H (AB)). The product formula for the case of higher regularized Fredholm determinants det H,k (I H −A), A ∈ B k (H), k ∈ N, k 2, does not seem to be easily accessible and hence this note aims at filling this gap in the literature.
In this paper, we construct a number of 4-GDDs where the group sizes are all congruent to 2 (mod ... more In this paper, we construct a number of 4-GDDs where the group sizes are all congruent to 2 (mod 3). We also show that 4-GDDs of type 2t8s exist for all but a finite number of feasible values of s and t. The largest unknown case has type 24818 and has 152 points. A number of 4-GDDs with at most 50 points are also constructed. These include one of type 4811101, the last feasible type of the form 4s1tn1 with at most 50 points for which no 4-GDD was known
A matrix is free, or generic, if its nonzero entries are algebraically independent. Necessary and... more A matrix is free, or generic, if its nonzero entries are algebraically independent. Necessary and sufficient combinatorial conditions are presented for a complex free matrix to have a free Moore-Penrose inverse. These conditions extend previously known results for square, nonsingular free matrices. The result used to prove this characterization relates the combinatorial structure of a free matrix to that of its Moore-Penrose inverse. Also, it is proved that the bipartite graph or, equivalently, the zero pattern of a free matrix uniquely determines that of its Moore-Penrose inverse, and this mapping is described explicitly. Finally, it is proved that a free matrix contains at most as many nonzero entries as does its Moore-Penrose inverse.
In this paper, we consider the coboundary polynomial for a matroid as a generalization of the wei... more In this paper, we consider the coboundary polynomial for a matroid as a generalization of the weight enumerator of a linear code. By describing properties of this polynomial and of a more general polynomial, we investigate the matroid analogue of the MacWilliams identity. From coding-theoretical approaches, upper bounds are given on the size of circuits and cocircuits of a matroid, which generalizes bounds on minimum Hamming weights of linear codes due to I. Duursma.
The Critical Theorem, due to Henry Crapo and Gian-Carlo Rota, has previously been extended or gen... more The Critical Theorem, due to Henry Crapo and Gian-Carlo Rota, has previously been extended or generalised in a number of different ways. The main result of the present paper is a general form of the Critical Theorem that encompasses many of these results. Applications include generalisations of a theorem by Curtis Greene that describes how the weight enumerator of a linear code is determined by the Tutte polynomial of the associated vector matroid, as well as generalisations of the MacWilliams identity for linear codes.
ABSTRACT We present a construction of demi-matroids, a generalization of matroids, from linear co... more ABSTRACT We present a construction of demi-matroids, a generalization of matroids, from linear codes over finite Frobenius rings, as well as a Greene-type identity for rank generating functions of demi-matroids. We also prove a MacWilliams-type identity for Hamming support enumerators of linear codes over finite Frobenius rings. As a special case, these results give a combinatorial proof of the MacWilliams identity for Hamming weight enumerators of linear codes over finite Frobenius rings.
The purpose of this paper is to provide links between matroid theory and the theory of subcode we... more The purpose of this paper is to provide links between matroid theory and the theory of subcode weights and supports in linear codes. We describe such weights and supports in terms of certain matroids arising from the vector matroids associated to the linear codes. Our results generalize classical results by Whitney, Tutte, Crapo and Rota, Greene, and other authors. As an application of our results, we obtain a new and elegant dual correspondence between the bond union and cycle union cardinalities of a graph.
ABSTRACT This paper presents new connections between designs and matroids. We generalize the Assm... more ABSTRACT This paper presents new connections between designs and matroids. We generalize the Assmus-Mattson theorem and another coding-theoretical theorem with respect to matroids, and thereby present new sufficient conditions for obtaining $t$-designs from matroids. These conditions may be relaxed for some self-dual matroids. We use our results to prove new constructions of $t$-designs from linear codes, including several new extensions and variants of the Assmus-Mattson theorem. We also present weighted $t$-designs which generalize $t$-designs. New $t$-designs are obtained from our results.
We present new constructions of t-designs by considering subcode supports of linear codes over fi... more We present new constructions of t-designs by considering subcode supports of linear codes over finite fields. In particular, we prove an Assmus-Mattson type theorem for such subcodes, as well as an automorphism characterization. We derive new t-designs (t ≤ 5) from our constructions. Keywords t-design • Linear code • Generalized Hamming weight • Higher weight enumerator • Assmus-Mattson Theorem AMS Classifications 94B05 • 05B05 Communicated by Q. Xiang.
In this paper, we consider the existence of group divisible designs (GDDs) with block size 4 and ... more In this paper, we consider the existence of group divisible designs (GDDs) with block size 4 and group sizes 4 and 7. We show that there exists a 4-GDD of type 4 t 7 s for all but a finite specified set of feasible values for (t, s).
We prove that each maximal partial Latin cube must have more than 29.289% of its cells filled and... more We prove that each maximal partial Latin cube must have more than 29.289% of its cells filled and show by construction that this is a nearly tight bound. We also prove upper and lower bounds on the number of cells containing a fixed symbol in maximal partial Latin cubes and hypercubes, and we use these bounds to determine for small orders n the numbers k for which there exists a maximal partial Latin cube of order n with exactly k entries. Finally, we prove that maximal partial Latin cubes of order n exist of each size from approximately half-full (n 3 /2 for even n 10 and (n 3 + n)/2 for odd n 21) to completely full, except for when either precisely 1 or 2 cells are empty.
The Moore-Penrose inverse of a real matrix having no square submatrix with two or more diagonals ... more The Moore-Penrose inverse of a real matrix having no square submatrix with two or more diagonals is described in terms of bipartite graphs. For such a matrix, the sign of every entry of the Moore-Penrose inverse is shown to be determined uniquely by the signs of the matrix entries; i.e., the matrix has a signed generalized inverse. Necessary and sufficient conditions on an acyclic bipartite graph are given so that each nonnegative matrix with this graph has a nonnegative Moore-Penrose inverse. Nearly reducible matrices are proved to contain no submatrix having two or more diagonals, implying that a nearly reducible matrix has a signed generalized inverse. Furthermore, it is proved that the term rank and rank are equal for each submatrix of a nearly reducible matrix.
We present new values and bounds on the (normalised) closeness centralityCC of connected graphs a... more We present new values and bounds on the (normalised) closeness centralityCC of connected graphs and on its productlCC with the mean distancel of these graphs. Our main result presents the fundamental bounds 1 ≤lCC < 2. The lower bound is tight and the upper bound is asymptotically tight. Combining the lower bound with known upper bounds on the mean distance, we find ten new lower bounds for the closeness centrality of graphs. We also present explicit expressions forCC andlCC for specific families of graphs. Elegantly and perhaps surprisingly, the asymptotic values nCC Pn and of nCC Ln both equal π, and the asymptotic limits oflCC for these families of graphs are both equal to π/3. We conjecture that the set of valueslCC for all connected graphs is dense in the interval [1, 2).
Mubayi's Conjecture states that if F is a family of k-sized subsets of [n] = {1,. .. , n} which, ... more Mubayi's Conjecture states that if F is a family of k-sized subsets of [n] = {1,. .. , n} which, for k ≥ d ≥ 3 and n ≥ dk d−1 , satisfies A 1 ∩• • •∩A d = ∅ whenever |A 1 ∪• • •∪A d | ≤ 2k for all distinct sets A 1 ,. .. , A d ∈ F , then |F | ≤ n−1 k−1 , with equality occurring only if F is the family of all k-sized subsets containing some fixed element. This paper proves that Mubayi's Conjecture is true for all families that are invariant with respect to shifting; indeed, these families satisfy a stronger version of Mubayi's Conjecture. Relevant to the conjecture, we prove a fundamental bijective duality between what we call (i, j)unstable families and (j, i)-unstable families. Generalising previous intersecting conditions, we introduce the (d, s, t)-conditionally intersecting condition for families of sets and prove general results thereon. We prove fundamental theorems on two (d, s)-conditionally intersecting families that generalise previous intersecting families, and we pose an extension of a previous conjecture by Frankl and Füredi. Finally, we generalise a classical result by Erdős, Ko and Rado by proving tight upper bounds on the size of (2, s)-conditionally intersecting families F ⊆ 2 [n] and by characterising the families that attain these bounds. We extend this theorem for sufficiently large n to families F ⊆ 2 [n] whose members have at most a fixed size u.
Proceedings of the American Mathematical Society, Feb 10, 2021
For trace class operators A, B ∈ B 1 (H) (H a complex, separable Hilbert space), the product form... more For trace class operators A, B ∈ B 1 (H) (H a complex, separable Hilbert space), the product formula for Fredholm determinants holds in the familiar form det H ((I H − A)(I H − B)) = det H (I H − A)det H (I H − B). When trace class operators are replaced by Hilbert-Schmidt operators A, B ∈ B 2 (H) and the Fredholm determinant det H (I H − A), A ∈ B 1 (H), by the 2nd regularized Fredholm determinant det H,2 (I H − A) = det H ((I H − A) exp(A)), A ∈ B 2 (H), the product formula must be replaced by det H,2 ((I H − A)(I H − B)) = det H,2 (I H − A)det H,2 (I H − B) × exp(− tr H (AB)). The product formula for the case of higher regularized Fredholm determinants det H,k (I H −A), A ∈ B k (H), k ∈ N, k 2, does not seem to be easily accessible and hence this note aims at filling this gap in the literature.
In this paper, we construct a number of 4-GDDs where the group sizes are all congruent to 2 (mod ... more In this paper, we construct a number of 4-GDDs where the group sizes are all congruent to 2 (mod 3). We also show that 4-GDDs of type 2t8s exist for all but a finite number of feasible values of s and t. The largest unknown case has type 24818 and has 152 points. A number of 4-GDDs with at most 50 points are also constructed. These include one of type 4811101, the last feasible type of the form 4s1tn1 with at most 50 points for which no 4-GDD was known
A matrix is free, or generic, if its nonzero entries are algebraically independent. Necessary and... more A matrix is free, or generic, if its nonzero entries are algebraically independent. Necessary and sufficient combinatorial conditions are presented for a complex free matrix to have a free Moore-Penrose inverse. These conditions extend previously known results for square, nonsingular free matrices. The result used to prove this characterization relates the combinatorial structure of a free matrix to that of its Moore-Penrose inverse. Also, it is proved that the bipartite graph or, equivalently, the zero pattern of a free matrix uniquely determines that of its Moore-Penrose inverse, and this mapping is described explicitly. Finally, it is proved that a free matrix contains at most as many nonzero entries as does its Moore-Penrose inverse.
In this paper, we consider the coboundary polynomial for a matroid as a generalization of the wei... more In this paper, we consider the coboundary polynomial for a matroid as a generalization of the weight enumerator of a linear code. By describing properties of this polynomial and of a more general polynomial, we investigate the matroid analogue of the MacWilliams identity. From coding-theoretical approaches, upper bounds are given on the size of circuits and cocircuits of a matroid, which generalizes bounds on minimum Hamming weights of linear codes due to I. Duursma.
The Critical Theorem, due to Henry Crapo and Gian-Carlo Rota, has previously been extended or gen... more The Critical Theorem, due to Henry Crapo and Gian-Carlo Rota, has previously been extended or generalised in a number of different ways. The main result of the present paper is a general form of the Critical Theorem that encompasses many of these results. Applications include generalisations of a theorem by Curtis Greene that describes how the weight enumerator of a linear code is determined by the Tutte polynomial of the associated vector matroid, as well as generalisations of the MacWilliams identity for linear codes.
ABSTRACT We present a construction of demi-matroids, a generalization of matroids, from linear co... more ABSTRACT We present a construction of demi-matroids, a generalization of matroids, from linear codes over finite Frobenius rings, as well as a Greene-type identity for rank generating functions of demi-matroids. We also prove a MacWilliams-type identity for Hamming support enumerators of linear codes over finite Frobenius rings. As a special case, these results give a combinatorial proof of the MacWilliams identity for Hamming weight enumerators of linear codes over finite Frobenius rings.
The purpose of this paper is to provide links between matroid theory and the theory of subcode we... more The purpose of this paper is to provide links between matroid theory and the theory of subcode weights and supports in linear codes. We describe such weights and supports in terms of certain matroids arising from the vector matroids associated to the linear codes. Our results generalize classical results by Whitney, Tutte, Crapo and Rota, Greene, and other authors. As an application of our results, we obtain a new and elegant dual correspondence between the bond union and cycle union cardinalities of a graph.
ABSTRACT This paper presents new connections between designs and matroids. We generalize the Assm... more ABSTRACT This paper presents new connections between designs and matroids. We generalize the Assmus-Mattson theorem and another coding-theoretical theorem with respect to matroids, and thereby present new sufficient conditions for obtaining $t$-designs from matroids. These conditions may be relaxed for some self-dual matroids. We use our results to prove new constructions of $t$-designs from linear codes, including several new extensions and variants of the Assmus-Mattson theorem. We also present weighted $t$-designs which generalize $t$-designs. New $t$-designs are obtained from our results.
We present new constructions of t-designs by considering subcode supports of linear codes over fi... more We present new constructions of t-designs by considering subcode supports of linear codes over finite fields. In particular, we prove an Assmus-Mattson type theorem for such subcodes, as well as an automorphism characterization. We derive new t-designs (t ≤ 5) from our constructions. Keywords t-design • Linear code • Generalized Hamming weight • Higher weight enumerator • Assmus-Mattson Theorem AMS Classifications 94B05 • 05B05 Communicated by Q. Xiang.
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