A metric graph is a geometric realization of a finite graph by identifying each edge with a real interval. A divisor on a metric graph $Γ$ is an element of the free abelian group on $Γ$. The rank of a divisor on a metric graph is a concept appearing in the Riemann-Roch theorem for metric graphs (or tropical curves) due to Gathmann and Kerber, and Mikhalkin and Zharkov. We define a \emph{rank-determining set} of a metric graph $Γ$ to be a subset $A$ of $Γ$ such that the rank of a divisor $D$ on $Γ$ is always equal to the rank of $D$ restricted on $A$. We show constructively in this paper that there exist finite rank-determining sets. In addition, we investigate the properties of rank-determining sets in general and formulate a criterion for rank-determining sets. Our analysis is a based on an algorithm to derive the $v_0$-reduced divisor from any effective divisor in the same linear system.
We produce open subsets of the moduli space of metric graphs without separating edges where the dimensions of Brill-Noether loci are larger than the corresponding Brill-Noether numbers. These graphs also have minimal rank determining sets that are larger than expected, giving couterexamples to a conjecture of Luo. Furthermore, limits of these graphs have Brill-Noether loci of the expected dimension, so dimensions of Brill-Noether loci of metric graphs do not vary upper semicontinuously in families. Motivated by these examples, we study a notion of rank for the Brill-Noether locus of a metric g…
A subset U of vertices of a graph G is called a determining set if every automorphism of G is uniquely determined by its action on the vertices of U. A subset W is called a resolving set if every vertex in G is uniquely determined by its distances to the vertices of W. Determining (resolving) sets are said to have the exchange property in G if whenever S and R are minimal determining (resolving) sets for G and r\in R, then there exists s\in S so that S-\{s\}\cup \{r\} is a minimal determining (resolving) set. This work examines graph families in which these sets do, or do not, have the exchang…
A set of vertices $S$ is a \emph{determining set} of a graph $G$ if every automorphism of $G$ is uniquely determined by its action on $S$. The \emph{determining number} of $G$ is the minimum cardinality of a determining set of $G$. This paper studies determining sets of Kneser graphs from a hypergraph perspective. This new technique lets us compute the determining number of a wide range of Kneser graphs, concretely $K_{n:k}$ with $n\geq \frac{k(k+1)}{2}+1$. We also show its usefulness by giving shorter proofs of the characterization of all Kneser graphs with fixed determining number 2, 3 or 4,…
We show that determining Kapranov rank of tropical matrices is not only NP-hard over any infinite field but also if solving Diophantine equations over the rational numbers is undecidable, then determining Kapranov rank over the rational numbers is undecidable. We prove that Kapranov rank of tropical matrices is not bounded in terms of tropical rank, answering a question of Develin, Santos, and Sturmfels.
Bridging the work of Cameron, Harary, and others, we examine the base size set B(G) and determining set D(G) of several families of groups. The base size set is the set of base sizes of all faithful actions of the group G on finite sets. The determining set is the subset of B(G) obtained by restricting the actions of G to automorphism groups of finite graphs. We show that for finite abelian groups, B(G)=D(G)={1,2,...,k} where k is the number of elementary divisors of G. We then characterize B(G) and D(G) for dihedral groups of the form D_{p^k} and D_{2p^k}. Finally, we prove B(G) is not equal …
We generalize work of Erdos and Fishburn to study the structure of finite point sets that determine few distinct triangles. Specifically, we ask for a given $t$, what is the maximum number of points that can be placed in the plane to determine exactly $t$ distinct triangles? Denoting this quantity by $F(t)$, we show that $F(1) = 4$, $F(2) = 5$, and $F(t) < 48(t+1)$ for all $t$. We also completely characterize the optimal configurations for $t = 1, 2$.
We characterize the largest point sets in the plane which define at most 1, 2, and 3 angles. For $P(k)$ the largest size of a point set admitting at most $k$ angles, we prove $P(2)=5$ and $P(3)=5$. We also provide the general bounds of $k+2 \leq P(k) \leq 6k$, although the upper bound may be improved pending progress toward the Weak Dirac Conjecture. Notably, it is surprising that $P(k)=Θ(k)$ since, in the distance setting, the best known upper bound on the analogous quantity is quadratic and no lower bound is well-understood.
In this paper we determine the maximum number of points in $\mathbb{R}^d$ which form exactly $t$ distinct triangles, where we restrict ourselves to the case of $t = 1$. We denote this quantity by $F_d(t)$. It was known from the work of Epstein et al. that $F_2(1) = 4$. Here we show somewhat surprisingly that $F_3(1) = 4$ and $F_d(1) = d + 1$, whenever $d \geq 3$, and characterize the optimal point configurations. This is an extension of a variant of the distinct distance problem put forward by Erdős and Fishburn.
We introduce the notion of Levenshtein graphs, an analog to Hamming graphs but using the edit distance instead of the Hamming distance; in particular, Levenshtein graphs allow for underlying strings (nodes) of different lengths. We characterize various properties of these graphs, including a necessary and sufficient condition for their geodesic distance to be identical to the edit distance, their automorphism group and determining number, and an upper bound on their metric dimension. Regarding the latter, we construct a resolving set composed of two-run strings and an algorithm that computes t…
In this paper we investigate partial spreads of $H(2n-1,q^2)$ through the related notion of partial spread sets of hermitian matrices, and the more general notion of constant rank-distance sets. We prove a tight upper bound on the maximum size of a linear constant rank-distance set of hermitian matrices over finite fields, and as a consequence prove the maximality of extensions of symplectic semifield spreads as partial spreads of $H(2n-1,q^2)$. We prove upper bounds for constant rank-distance sets for even rank, construct large examples of these, and construct maximal partial spreads of $H(3,…
In this paper we investigate connections between linear sets and subspaces of linear maps. We give a geometric interpretation of the results of [18, Section 5] on linear sets on a projective line. We extend this to linear sets in arbitrary dimension, giving the connection between two constructions for linear sets defined in [9]. Finally, we then exploit this connection by using the MacWilliams identities to obtain information about the possible weight distribution of a linear set of rank n on a projective line $PG(1, q^n)$.
The intersection of two linear sets of rank 3 in PG(1,q^3)
School of Mathematical Sciences, Adelaide University（阿德莱德大学数学科学学院）；School of Mathematical and Statistical Sciences, Clemson University（克莱姆森大学数学与统计科学学院）
We take a graph theoretic approach to the problem of finding generators for those prime ideals of $\mathcal{O}_q(\mathcal{M}_{m,n}(\mathbb{K}))$ which are invariant under the torus action ($\mathbb{K}^*)^{m+n}$. Launois \cite{launois3} has shown that the generators consist of certain quantum minors of the matrix of canonical generators of $\mathcal{O}_q(\mathcal{M}_{m,n}(\mathbb{K}))$ and in \cite{launois2} gives an algorithm to find them. In this paper we modify a classic result of Lindström \cite{lind} and Gessel-Viennot~\cite{gv} to show that a quantum minor is in the generating set for a p…
A set $S\subseteq 2^E$ of subsets of a finite set $E$ is \emph{powerful} if, for all $X\subseteq E$, the number of subsets of $X$ in $S$ is a power of 2. Each powerful set is associated with a non-negative integer valued function, which we call the rank function. Powerful sets were introduced by Farr and Wang as a generalisation of binary matroids, as the cocircuit space of a binary matroid gives a powerful set with the corresponding matroid rank function. In this paper we investigate how structural properties of a powerful set can be characterised in terms of its rank function. Powerful sets …
This paper deals with the maximum value of the difference between the determining number and the metric dimension of a graph as a function of its order. Our technique requires to use locating-dominating sets, and perform an independent study on other functions related to these sets. Thus, we obtain lower and upper bounds on all these functions by means of very diverse tools. Among them are some adequate constructions of graphs, a variant of a classical result in graph domination and a polynomial time algorithm that produces both distinguishing sets and determining sets. Further, we consider sp…
We consider the concept of rank as a measure of the vertical levels and positions of elements of partially ordered sets (posets). We are motivated by the need for algorithmic measures on large, real-world hierarchically-structured data objects like the semantic hierarchies of ontological databases. These rarely satisfy the strong property of gradedness, which is required for traditional rank functions to exist. Representing such semantic hierarchies as finite, bounded posets, we recognize the duality of ordered structures to motivate rank functions which respect verticality both from the botto…
A transversal matroid $M$ of rank $r$ on $[n]$ can be associated to a family of binary matrices corresponding to different presentations of $M$. We describe those matrices which arise from unique maximal presentations of size $r$- giving a complete characterization when $r=2$-, give an algorithm for producing such a matrix, and show that determining whether a matrix corresponds to a maximal presentation is related to determining the tropical rank of arbitrary matrices.
In this extended abstract we announce a proof that, in a Coxeter group of rank 3, low elements are in bijection with small inversion sets. This gives a partial confirmation of Conjecture 2 in [Dyer, Hohlweg '16]. That same article provides the main ingredient: the bipodality of the set of small roots is used to propagate information on the vertices of inversion polytopes.
In $\PG(2,q^3)$, let $π$ be a subplane of order $q$ that is exterior to $\li$. The exterior splash of $π$ is defined to be the set of $q^2+q+1$ points on $\li$ that lie on a line of $π$. This article investigates properties of an exterior \orsp\ and its exterior splash. We show that the following objects are projectively equivalent: exterior splashes, covers of the circle geometry $CG(3,q)$, Sherk surfaces of size $q^2+q+1$, and $\GF(q)$-linear sets of rank 3 and size $q^2+q+1$. We compare our construction of exterior splashes with the projection construction of a linear set. We give a geometr…
We address an analog of a problem introduced by Erdős and Fishburn, itself an inverse formulation of the famous Erdős distance problem, in which the usual Euclidean distance is replaced with the metric induced by the $\ell^1$-norm, commonly referred to as the $\textit{taxicab metric}$. Specifically, we investigate the following question: given $d,k\in \mathbb{N}$, what is the maximum size of a subset of $\mathbb{R}^d$ that determines at most $k$ distinct taxicab distances, and can all such optimal arrangements be classified? We completely resolve the question in dimension $d=2$, as well as the…
An n\times n matrix M is called a fooling-set matrix of size n, if its diagonal entries are nonzero, whereas for every k\ne \ell we have M_{k,\ell} M_{\ell,k} = 0. Dietzfelbinger, Hromkovič, and Schnitger (1996) showed that n \le (\rk M)^2, regardless of over which field the rank is computed, and asked whether the exponent on \rk M can be improved. We settle this question for nonzero characteristic by constructing a family of matrices for which the bound is asymptotically tight. The construction uses linear recurring sequences.
We study two possible tropical analogues of Weierstrass semigroups on graphs, called rank and functional Weierstrass sets. We prove that on simple graphs, the first is contained in the second. We completely characterize the subsets of N arising as a functional Weierstrass set of some graph. Finally, we give a sufficient condition for a subset of N to be the rank Weierstrass set of some graph, allowing us to construct examples of rank Weierstrass sets that are not semigroups.
We study quadratic residue difference sets, GMW difference sets, and difference sets arising from monomial hyperovals, all of which are $(2^d-1, 2^{d-1}-1, 2^{d-2}-1)$ cyclic difference sets in the multiplicative group of the finite field $F_{2^d}$ of $2^d$ elements, with $d \geq 2$. We show that, except for a few cases with small $d$, these difference sets are all pairwise inequivalent. This is accomplished in part by examining their 2-ranks. The 2-ranks of all of these difference sets were previously known, except for those connected with the Segre and Glynn hyperovals. We determine the 2-ra…
This work investigates the structure of rank-metric codes in connection with concepts from finite geometry, most notably the $q$-analogues of projective systems and blocking sets. We also illustrate how to associate a classical Hamming-metric code to a rank-metric one, in such a way that various rank-metric properties naturally translate into the homonymous Hamming-metric notions under this correspondence. The most interesting applications of our results lie in the theory of minimal rank-metric codes, which we introduce and study from several angles. Our main contributions are bounds for the p…
We prove that if a subset of $(\mathbb{F}_q^n)^k$ (with $q$ an odd prime power) avoids a full-rank three-point pattern $\vec{x},\vec{x}+M_1\vec{d},\vec{x}+M_2\vec{d}$ then it is exponentially small, having size at most $3 \cdot c_q^{nk}$ where $0.8414 q \leq c_q \leq 0.9184 q$. This generalizes a theorem of Kovauc and complements results of Berger, Sah, Sawhney and Tidor. As a consequence, we prove that if $3$ is a square in $\mathbb{F}_q$ then subsets of $(\mathbb{F}_q^n)^2$ avoiding equilateral triangles are exponentially small.
We consider the problem of determining rank loss conditions for a concatenation of full-rank matrices, such that each row of the composing matrices is scaled by a random coefficient. This problem has applications in wireless interference management and recommendation systems. We determine necessary and sufficient conditions for the design of each matrix, such that the random ensemble will almost surely lose rank by a certain amount. The result is proved by converting the problem to determining rank loss conditions for the union of some specific matroids, and then using tools from matroid and g…
The rank of a scattered $\mathbb{F}_q$-linear set of $\mathrm{PG}(r-1,q^n)$, $rn$ even, is at most $rn/2$ as it was proved by Blokhuis and Lavrauw. Existence results and explicit constructions were given for infinitely many values of $r$, $n$, $q$ ($rn$ even) for scattered $\mathbb{F}_q$-linear sets of rank $rn/2$. In this paper we prove that the bound $rn/2$ is sharp also in the remaining open cases. Recently Sheekey proved that scattered $\mathbb{F}_q$-linear sets of $\mathrm{PG}(1,q^n)$ of maximum rank $n$ yield $\mathbb{F}_q$-linear MRD-codes with dimension $2n$ and minimum distance $n-1$.…
In order theory, a rank function measures the vertical "level" of a poset element. It is an integer-valued function on a poset which increments with the covering relation, and is only available on a graded poset. Defining a vertical measure to an arbitrary finite poset can be accomplished by extending a rank function to be interval-valued. This establishes an order homomorphism from a base poset to a poset over real intervals, and a standard (canonical) specific interval rank function is available as an extreme case. Various ordering relations are available over intervals, and we begin in this…
Linear sets on the projective line have attracted a lot of attention because of their link with blocking sets, KM-arcs and rank-metric codes. In this paper, we study linear sets having two points of complementary weight, that is with two points for which the sum of their weights equals the rank of the linear set. As a special case, we study those linear sets having exactly two points of weight greater than one, by showing new examples and studying their equivalence issue. Also we determine some linearized polynomials defining the linear sets recently introduced by Jena and Van de Voorde (2021)…
An $n\times n$ matrix $M$ is called a \textit{fooling-set matrix of size $n$} if its diagonal entries are nonzero and $M_{k,\ell} M_{\ell,k} = 0$ for every $k\ne \ell$. Dietzfelbinger, Hromkovi{č}, and Schnitger (1996) showed that $n \le (\mbox{rk} M)^2$, regardless of over which field the rank is computed, and asked whether the exponent on $\mbox{rk} M$ can be improved. We settle this question. In characteristic zero, we construct an infinite family of rational fooling-set matrices with size $n = \binom{\mbox{rk} M+1}{2}$. In nonzero characteristic, we construct an infinite family of matrices…
The rank of a point-line geometry G is usually defined as the generating rank of G, namely the minimal cardinality of a generating set. However, when the subspace lattice of G satisfies the Exchange Property we can also try a different definition: consider all chains of subspaces of G and take the least upper bound of their lengths as the rank of G. If G is finitely generated then these two definitions yield the same number. On the other hand, as we shall show in this paper, if infinitely many points are needed to generate G then the rank as defined in the latter way is often (perhaps always) …
Let $\mathcal{C}\subseteq \mathbb{F}_{q^m}^n$ be an $\mathbb{F}_{q^m}$-linear non-degenerate rank metric code with dimension $k$. In this paper we investigate the problem of determining the number $M(\mathcal{C})$ of codewords in $\mathcal{C}$ with maximum weight, that is $\min\{m,n\}$, and to characterize those with the maximum and the minimum values of $M(\mathcal{C})$.
The distinguishing number $D(G)$ of a graph $G$ is the least integer $d$ such that $G$ has an vertex labeling with $d$ labels that is preserved only by a trivial automorphism. The minimum size of a label class in such a labeling of $G$ with $D(G) = d$ is called the cost of $d$-distinguishing $G$ and is denoted by $ρ_d(G)$. A set of vertices $S\subseteq V(G)$ is a determining set for $G$ if every automorphism of $G$ is uniquely determined by its action on $S$. The determining number of $G$, Det(G), is the minimum cardinality of determining sets of $G$. In this paper we obtain some general upper…
Minimal rank-metric codes or, equivalently, linear cutting blocking sets are characterized in terms of the second generalized rank weight, via their connection with evasiveness properties of the associated $q$-system. Using this result, we provide the first construction of a family of $\mathbb{F}_{q^m}$-linear MRD codes of length $2m$ that are not obtained as a direct sum of two smaller MRD codes. In addition, such a family has better parameters, since its codes possess generalized rank weights strictly larger than those of the previously known MRD codes. This shows that not all the MRD codes …
In this paper we introduce a new technique to analyze families of rankings focused on the study of structural properties of a new type of graphs. Given a finite number of elements and a family of rankings of those elements, we say that two elements compete when they exchange their relative positions in at least two rankings. This allows us to define an undirected graph by connecting elements that compete. We call this graph a competitivity graph. We study the relationship of competitivity graphs with other well-known families of graphs, such as permutation graphs, comparability graphs and chor…
The quantum chromatic number of a graph $G$ is sandwiched between its chromatic number and its clique number, which are well known NP-hard quantities. We restrict our attention to the rank-1 quantum chromatic number $χ_q^{(1)}(G)$, which upper bounds the quantum chromatic number, but is defined under stronger constraints. We study its relation with the chromatic number $χ(G)$ and the minimum dimension of orthogonal representations $ξ(G)$. It is known that $ξ(G) \leq χ_q^{(1)}(G) \leq χ(G)$. We answer three open questions about these relations: we give a necessary and sufficient condition to ha…
We consider the problem of determining the maximum cardinality of a subset containing no arithmetic progressions of length $k$ in a given set of size $n$. It is proved that it is sufficient, in a certain sense, to consider the interval $[1,\dots, n]$. The study continues the work of Komlós, Sulyok, and Szemerédi.
