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A Pieri rule for Demazure characters of the general linear groupAug 22 2019The Pieri rule is a nonnegative, multiplicity-free formula for the Schur function expansion of the product of an arbitrary Schur function with a single row Schur function. Key polynomials are characters of Demazure modules for the general linear group ... More
Improved bounds for the sunflower lemmaAug 22 2019A sunflower with $r$ petals is a collection of $r$ sets so that the intersection of each pair is equal to the intersection of all. Erd\H{o}s and Rado proved the sunflower lemma: for any fixed $r$, any family of sets of size $w$, with at least about $w^w$ ... More
Bounding Game Temperature Using Confusion IntervalsAug 22 2019For combinatorial games, temperature is a measure of the volatility, that is, by how much the advantage can change. Typically, the temperature has been measured for individual positions within specific games. In this paper, we give the first general upper ... More
New Nonexistence Results on Circulant Weighing MatricesAug 22 2019A circulant weighing matrix $W = (w_{i,j}) \in CW(n,k)$ is a square matrix of order $n$ and entries $w_{i,j}$ in $\{-1, 0, +1\}$ such that $WW^T=kI_n$. In his thesis, Strassler gave tables of known results on such matrices with $n \leq 200$ and $k \leq ... More
On the cohomology of line bundles over certain flag schemesAug 22 2019Let $G$ be the group scheme $\operatorname{SL}_{d+1}$ over $\mathbb{Z}$ and let $Q$ be the parabolic subgroup scheme corresponding to the simple roots $\alpha_{2},\cdots,\alpha_{d-1}$. Then $G/Q$ is the $\mathbb{Z} $-scheme of partial flags $\{D_{1}\subset ... More
On the cohomology of line bundles over certain flag schemes IIAug 22 2019Over a field $K$ of characteristic $p$, let $Z$ be the incidence variety in $\mathbb{P}^d \times (\mathbb{P}^d)^*$ and let $\mathcal{L}$ be the restriction to $Z$ of the line bundle $\mathcal{O}(-n-d) \boxtimes \mathcal{O}(n)$, where $n = p+f$ with $0 ... More
Visualizing the Support of Kostant's Weight Multiplicity Formula for the Rank Two Lie AlgebrasAug 22 2019The multiplicity of a weight in a finite-dimensional irreducible representation of a simple Lie algebra g can be computed via Kostant's weight multiplicity formula. This formula consists of an alternating sum over the Weyl group (a finite group) and involves ... More
An Efficient Algorithm for Latin Squares in a Bipartite Min-Max-Plus SystemAug 22 2019In this paper, we consider the eigenproblems for Latin squares in a bipartite min-max-plus system. The focus is upon developing a new algorithm to compute the eigenvalue and eigenvectors (trivial and non-trivial) for Latin squares in a bipartite min-max-plus ... More
Spatial and Spatiotemporal GARCH Models -- A Unified ApproachAug 22 2019In time-series analyses and particularly in finance, generalised autoregressive conditional heteroscedasticity (GARCH) models are widely applied statistical tools for modelling volatility clusters (i.e. periods of increased or decreased risks). In contrast, ... More
Torus orbit closures in flag varieties and retractions on Weyl groupsAug 22 2019In this manuscript, we define three kinds of retractions on Weyl groups or finite Coxeter groups and study the relations among them. The first is what we call a \textit{geometric retraction} $\mathcal{R}^g_Y$ associated to a torus orbit closure $Y$ in ... More
Flag complexes and homologyAug 22 2019We prove several relations on the $f$-vectors and Betti numbers of flag complexes. For every flag complex $\Delta$, we show that there exists a balanced complex with the same $f$-vector as $\Delta$, and whose top-dimensional Betti number is at least that ... More
Coloring Hasse diagrams and disjointness graphs of curvesAug 22 2019Given a family of curves $\mathcal{C}$ in the plane, its disjointness graph is the graph whose vertices correspond to the elements of $\mathcal{C}$, and two vertices are joined by an edge if and only if the corresponding sets are disjoint. We prove that ... More
On small balanceable, strongly-balanceable and omnitonal graphsAug 22 2019In classical Ramsey theory for graphs we are given a graph $G$ and we are required to find the least $n_0$ such that, for any $n\geq n_0$, any red/blue colouring of the edges of $K_n$ gives a subgraph $G$ all of whose edges are blue or all are red. Here ... More
Chromatic symmetric function of graphs from Borcherds algebrasAug 22 2019Let $\mathfrak g$ be a Borcherds algebra with the associated graph $G$. We prove that the chromatic symmetric function of $G$ can be recovered from the Weyl denominator identity of $\mathfrak g$ and this gives a Lie theoretic proof of Stanley's expression ... More
The agreement distance of unrooted phylogenetic networksAug 22 2019A rearrangement operation makes a small graph-theoretical change to a phylogenetic network to transform it into another one. For unrooted phylogenetic trees and networks, popular rearrangement operations are tree bisection and reconnection (TBR) and prune ... More
Constructive Method for Finding the Coefficients of a Divided SymmetrizationAug 22 2019We consider a type of divided symmetrization $\overrightarrow{D}_{\lambda,G}$ where $\lambda$ is a nonincreasing partition on $n$ and where $G$ is a graph. We discover that in the case where $\lambda$ is a hook shape partition with first part equal to ... More
Schröder Coloring and ApplicationsAug 21 2019We present several bijections, in terms of combinatorial objects counted by the Schr\"oder numbers, that are then used (via coloring) for the construction and enumeration of rational Schr\"oder paths with integer slope, ordered rooted trees, and simple ... More
The spectra of generalized Paley graphs and their associated irreducible cyclic codesAug 21 2019For $q=p^m$ with $p$ prime and $k\mid q-1$, we consider the generalized Paley graph $\Gamma(k,q) = Cay(\mathbb{F}_q, R_k)$, with $R_k=\{ x^k : x \in \mathbb{F}_q^* \}$, and the irreducible $p$-ary cyclic code $\mathcal{C}(k,q) = \{(\textrm{Tr}_{q/p}(\gamma ... More
A central limit theorem for the two-sided descent statistic on Coxeter groupsAug 21 2019We study the asymptotic behaviour of the statistic (des+ides) which assigns to an element w of a finite Coxeter group W the number of descents of w plus the number of descents of its inverse. Our main result is a central limit theorem for the probability ... More
Real polynomials with constrained real divisors. I. Fundamental groupsAug 21 2019In the late 80s, V.~Arnold and V.~Vassiliev initiated the study of the topology of the space of real univariate polynomials of a given degree d and with no real roots of multiplicity exceeding a given positive integer. Expanding their studies, we consider ... More
On cyclic Schur-positive sets of permutationAug 21 2019We introduce a notion of {\em cyclic Schur-positivity} for sets of permutations, which naturally extends the classical notion of Schur-positivity, and it involves the existence of a bijection from permutations to standard Young tableaux that preserves ... More
New Classes of Multicone Graphs Determinable by Their SpectraAug 21 2019A multicone graph is defined to be the join of a complete graph and a regular graph. A graph $\Gamma$ is determined by its spectrum if every graph cospectral to it is in fact isomorphic to it. It seems hard to prove a graph to be determined by its spectrum. ... More
Tropical Ehrhart Theory and Tropical VolumeAug 21 2019We introduce a novel intrinsic volume concept in tropical geometry. This is achieved by developing the foundations of a tropical analog of lattice point counting in polytopes. We exhibit the basic properties and compare it to existing measures. Our exposition ... More
Free self-decomposability and unimodality of the Fuss-Catalan distributionsAug 21 2019We study properties of the Fuss-Catalan distributions $\mu(p,r)$, $p\geq1$, $0<r\leq p$: free infinite divisibility, free self-decomposability, free regularity and unimodality. We show that the Fuss-Catalan distribution $\mu(p,r)$ is freely self-decomposable ... More
Quasi-local Algebras and Asymptotic ExpandersAug 21 2019In this paper, we study the relation between the uniform Roe algebra and the uniform quasi-local algebra associated to a discrete metric space of bounded geometry. In the process, we introduce a weakening of the notion of expanders, called asymptotic ... More
Delete NimAug 21 2019In this paper, we study an impartial game called Delete Nim. In this game, there are two heaps of stones. The player chooses one of the heaps and delete the other heap. Next, she takes away one stone from the chosen heap and optionally splits it into ... More
Existence and hardness of conveyor beltsAug 21 2019An open problem of Manuel Abellanas asks whether every set of disjoint closed unit disks in the plane can be connected by a conveyor belt, which means a tight simple closed curve that touches the boundary of each disk, possibly multiple times. We prove ... More
On graphic arrangement groupsAug 21 2019A finite simple graph $\Gamma$ determines a quotient $P_\Gamma$ of the pure braid group, called a graphic arrangement group. We analyze homomorphisms of these groups defined by deletion of sets of vertices, using methods developed in prior joint work ... More
A Generalization of Parking Functions Allowing Backward MovementAug 21 2019Classical parking functions are defined as the parking preferences for $n$ cars driving (from west to east) down a one-way street containing parking spaces labeled from $1$ to $n$ (from west to east). Cars drive down the street toward their preferred ... More
K-Nearest Neighbor Approximation Via the Friend-of-a-Friend PrincipleAug 20 2019Aug 22 2019Suppose $V$ is an $n$-element set where for each $x \in V$, the elements of $V \setminus \{x\}$ are ranked by their similarity to $x$. The $K$-nearest neighbor graph is a directed graph including an arc from each $x$ to the $K$ points of $V \setminus ... More
K-Nearest Neighbor Approximation Via the Friend-of-a-Friend PrincipleAug 20 2019Suppose $V$ is an $n$-element set where for each $x \in V$, the elements of $V \setminus \{x\}$ are ranked by their similarity to $x$. The $K$-nearest neighbor graph %$G:=(V, E)$ is a directed graph including an arc from each $x$ to the $K$ points of ... More
Bounds On $(t,r)$ Broadcast Domination of $n$-Dimensional GridsAug 20 2019In this paper, we look at a variant of graph domination known as $(t, r)$ broadcast domination, first defined in \cite{BleInsJohMau15}, and we describe some upper and lower bounds on the density of a $(t, r)$ dominating pattern of an infinite grid, as ... More
Bounds On $(t,r)$ Broadcast Domination of $n$-Dimensional GridsAug 20 2019Aug 22 2019In this paper, we look at a variant of graph domination known as $(t, r)$ broadcast domination, first defined by Blessing, Insko, Johnson, and Mauretour in 2015, and we describe some upper and lower bounds on the density of a $(t, r)$ dominating pattern ... More
Noisy Corruption DetectionAug 20 2019We answer a question of Alon, Mossel, and Pemantle about the corruption detection model on graphs in the noisy setting.
Edge vectors on plabic networks in the disk and amalgamation of totally non-negative GrassmanniansAug 20 2019We construct edge vectors on the planar bicolored trivalent directed networks in the disk which parametrize the positroid cells of totally non-negative Grassmannians. We introduce the notion of local winding and local intersection indices using a gauge ... More
Colored five-vertex models and Lascoux polynomials and atomsAug 20 2019We construct an integrable colored five-vertex model whose partition function is a Lascoux atom based on the five-vertex model of Motegi and Sakai [arXiv:1305.3030] and the colored five-vertex model of Brubaker, the first author, Bump, and Gustafsson ... More
Permutations with few inversions are locally uniformAug 20 2019We prove that permutations with few inversions exhibit a local-global dichotomy in the following sense. Suppose ${\boldsymbol\sigma}$ is a permutation chosen uniformly at random from the set of all permutations of $[n]$ with exactly $m=m(n)\ll n^2$ inversions. ... More
On the growth rate of dichromatic numbers of finite subdigraphsAug 20 2019Chris Lambie-Hanson proved recently that for every function $ f:\mathbb{N}\rightarrow \mathbb{N} $ there is an $ \aleph_1 $-chromatic graph $ G $ of size $ 2^{\aleph_1} $ such that every $ (n+3) $-chromatic subgraph of $ G $ has at least $ f(n) $ vertices. ... More
Sensitivity estimation of conditional value at risk using randomized quasi-Monte CarloAug 20 2019Conditional value at risk (CVaR) is a popular measure for quantifying portfolio risk. Sensitivity analysis of CVaR is very useful in risk management and gradient-based optimization algorithms. In this paper, we study the infinitesimal perturbation analysis ... More
Decoding Downset codes over a finite gridAug 20 2019In a recent paper, Kim and Kopparty (Theory of Computing, 2017) gave a deterministic algorithm for the unique decoding problem for polynomials of bounded total degree over a general grid. We show that their algorithm can be adapted to solve the unique ... More
Melonic Dominance in Subchromatic Sextic Tensor ModelsAug 20 2019We study tensor models based on $O(N)^r$ symmetry groups constructed out of rank-$r$ tensors with order-$q$ interaction vertices. We define \textit{subchromatic} tensor models to be those for which $r<q-1$. We focus most of our attention on sextic ($q=6$) ... More
Extendable shellability for $d$-dimensional complexes on $d+3$ verticesAug 20 2019We prove that for all $d \geq 1$ a shellable $d$-dimensional simplicial complex with at most $d+3$ vertices is extendably shellable. The proof involves considering the structure of `exposed' edges in chordal graphs as well as a connection to linear quotients ... More
Computer Bounds for Kronheimer-Mrowka Foam EvaluationAug 20 2019Kronheimer and Mrowka recently suggested a possible approach towards a new proof of the four color theorem that does not rely on computer calculations. One outgrowth of their approach is the definition of a functor $J^\flat$ from the category of webs ... More
On the 486-vertex distance-regular graphs of Koolen--Riebeek and SoicherAug 19 2019This paper considers three imprimitive distance-regular graphs with 486 vertices and diameter 4: the Koolen--Riebeek graph (which is bipartite), the Soicher graph (which is antipodal), and the incidence graph of a symmetric transversal design obtained ... More
An Omega(n^2) Lower Bound for Random Universal Sets for Planar GraphsAug 19 2019A set $U\subseteq \reals^2$ is $n$-universal if all $n$-vertex planar graphs have a planar straight-line embedding into $U$. We prove that if $Q \subseteq \reals^2$ consists of points chosen randomly and uniformly from the unit square then $Q$ must have ... More
Paley type partial difference sets in abelian groupsAug 19 2019Partial difference sets with parameters $(v,k,\lambda,\mu)=(v, (v-1)/2, (v-5)/4, (v-1)/4)$ are called Paley type partial difference sets. In this note we prove that if there exists a Paley type partial difference set in an abelian group $G$ of an order ... More
No two Jellyfish graphs are L-cospectral and Q-cospectralAug 19 2019In this paper, it is proved that the jellyfish graphs, a natural generalization of sun graphs, are both DLS and DQS.
Tropical geometryAug 19 2019Tropical mathematics redefines the rules of arithmetic by replacing addition with taking a maximum, and by replacing multiplication with addition. After briefly discussing a tropical version of linear algebra, we study polynomials build with these new ... More
Vector-relation configurations and plabic graphsAug 19 2019We study a simple geometric model for local transformations of bipartite graphs. The state consists of a choice of a vector at each white vertex made in such a way that the vectors neighboring each black vertex satisfy a linear relation. Evolution for ... More
A New Formula of q-Fubini Numbers via Goncharov polynomialsAug 19 2019Connected the generalized Goncharov polynomials associated to a pair ($\partial,\mathcal{Z}$) if a delta operator $\partial$ and an interpolation grid $\mathcal{Z}$, introduced by Lorentz, Tringali and Yan in [7], with the theory of binomial enumeration ... More
Strong G-schemes and strict homomorphismsAug 19 2019Let $\mathfrak{P}_r$ be a representation system of the non-isomorphic finite posets, and let ${\cal H}(P,Q)$ be the set of order homomorphisms from $P$ to $Q$. For finite posets $R$ and $S$, we write $R \sqsubseteq_G S$ iff, for every $P \in \mathfrak{P}_r$, ... More
Addition-deletion results for the minimal degree of logarithmic derivations of arrangementsAug 19 2019We study the change of the minimal degree of a logarithmic derivation of a hyperplane arrangement under the addition or the deletion of a hyperplane, and give a number of applications. In particular, starting with Ziegler's example of a pair of arrangements ... More
Impact of Some Graph Operations on Double Roman Domination NumberAug 19 2019Given a graph $G=(V,E)$, a function $f:V\rightarrow \{0,1,2,3\}$ having the property that if $f(v)=0$, then there exist $ v_{1},v_{2}\in N(v)$ such that $f(v_{1})=f(v_{2})=2$ or there exists $ w \in N(v)$ such that $f(w)=3$, and if $f(v)=1$, then there ... More
On the Double Roman Domination Number of Generalized Sierpinski GraphsAug 19 2019In this paper, we study the double Roman domination number of generalized Sierpi\'{n}ski graphs $S(G,t)$. More precisely, we obtain a bound for the double Roman domination number of $S(G, t)$. We also find the exact value of $\gamma_{dR}(S(K_{n}, 2))$. ... More
Linear representations of finite geometries and associated LDPC codesAug 19 2019The {\it linear representation} of a subset of a finite projective space is an incidence system of affine points and lines determined by the subset. In this paper we use character theory to show that the rank of the incidence matrix has a direct geometric ... More
An Efficient Algorithm to Test Potentially Bipartiteness of Graphical Degree SequencesAug 19 2019As a partial answer to a question of Rao, a deterministic and customizable efficient algorithm is presented to test whether an arbitrary graphical degree sequence has a bipartite realization. The algorithm can be configured to run in polynomial time, ... More
Fast multi-precision computation of some Euler productsAug 19 2019For every modulus $q\ge3$, we define a family of subsets $\mathcal{A}$ of the multiplicative group $(\mathbb{Z}/{q}\mathbb{Z})^\times$ for which the Euler product $\prod_{p\text{mod}q\in\mathcal{A}}(1-p^{-s})$ can be computed in double exponential time, ... More
Combinatorial Proof of the Minimal Excludant TheoremAug 19 2019The maximal excludant of a partition $\lambda$, $\rm{mex}(\lambda)$, is defined to be the least gap of $\lambda$. For each positive integer $n$, the function $ \sigma\, \rm{mex}(n)$ is defined to be the sum of the least gaps in all partitions of $n$. ... More
On bin packing with clustering and bin packing with delaysAug 19 2019We continue the study of two recently introduced bin packing type problems, called bin packing with clustering, and online bin packing with delays. A bin packing input consists of items of sizes not larger than 1, and the goal is to partition or pack ... More
Counterexamples to Thomassen's conjecture on decomposition of cubic graphsAug 19 2019We construct an infinite family of counterexamples to Thomassen's conjecture that the vertices of every 3-connected, cubic graph on at least 8 vertices can be colored blue and red such that the blue subgraph has maximum degree at most 1 and the red subgraph ... More
Balanced Schnyder woods for planar triangulations: an experimental study with applications to graph drawing and graph separatorsAug 19 2019In this work we consider balanced Schnyder woods for planar graphs, which are Schnyder woods where the number of incoming edges of each color at each vertex is balanced as much as possible. We provide a simple linear-time heuristic leading to obtain well ... More
The 3-way flower intersection problem for Steiner triple systemsAug 19 2019The flower at a point x in a Steiner triple system (X; B) is the set of all triples containing x. Denote by J3F(r) the set of all integers k such that there exists a collection of three STS(2r+1) mutually intersecting in the same set of k + r triples, ... More
Safe sets in digraphsAug 19 2019A non-empty subset $S$ of the vertices of a digraph $D$ is called a {\it safe set} if \begin{itemize} \item[(i)] for every strongly connected component $M$ of $D-S$, there exists a strongly connected component $N$ of $D[S]$ such that there exists an arc ... More
Proving two conjectural series for $ζ(7)$ and discovering more series for $ζ(7)$Aug 19 2019We give a proof of two identities involving binomial sums at infinity conjectured by Z-W Sun. In order to prove these identities, we use a recently presented method i.e. we view the series as specializations of generating series and derive integral representations. ... More
Chromatic nonsymmetric polynomials of Dyck graphs are slide-positiveAug 19 2019Motivated by the study of Macdonald polynomials, J. Haglund and A. Wilson introduced a nonsymmetric polynomial analogue of the chromatic quasisymmetric function called the \emph{chromatic nonsymmetric polynomial} of a Dyck graph. We give a positive expansion ... More
The index of Lie poset algebrasAug 19 2019We provide general closed-form formulas for the index of type-A Lie poset algebras corresponding to posets of restricted height. Furthermore, we provide a combinatorial recipe for constructing all posets corresponding to type-A Frobenius Lie poset algebras ... More
Energized simplicial complexesAug 19 2019For a simplicial complex with n sets, let W^-(x) be the set of sets in G contained in x and W^+(x) the set of sets in G containing x. An integer-valued function h on G defines for every A subset G an energy E[A]=sum_x in A h(x). The function energizes ... More
Some results on concatenating bipartite graphsAug 19 2019Aug 21 2019We consider two functions $\phi$ and $\psi$, defined as follows. Let $x,y \in (0,1]$ and let $A,B,C$ be disjoint nonempty subsets of a graph $G$, where every vertex in $A$ has at least $x|B|$ neighbors in $B$, and every vertex in $B$ has at least $y|C|$ ... More
Some results on concatenating bipartite graphsAug 19 2019We consider two functions $\phi$ and $\psi$, defined as follows. Let $x,y \in (0,1]$ and let $A,B,C$ be disjoint nonempty subsets of a graph $G$, where every vertex in $A$ has at least $x|B|$ neighbors in $B$, and every vertex in $B$ has at least $y|C|$ ... More
The Landscape of Minimum Label Cut (Hedge Connectivity) ProblemAug 19 2019Minimum Label Cut (or Hedge Connectivity) problem is defined as follows: given an undirected graph $G=(V, E)$ with $n$ vertices and $m$ edges, in which, each edge is labeled (with one or multiple labels) from a label set $L=\{\ell_1,\ell_2, ..., \ell_{|L|}\}$, ... More
The Landscape of Minimum Label Cut (Hedge Connectivity) ProblemAug 19 2019Aug 20 2019Minimum Label Cut (or Hedge Connectivity) problem is defined as follows: given an undirected graph $G=(V, E)$ with $n$ vertices and $m$ edges, in which, each edge is labeled (with one or multiple labels) from a label set $L=\{\ell_1,\ell_2, ..., \ell_{|L|}\}$, ... More
Computing Estimators of Dantzig Selector type via Column and Constraint GenerationAug 18 2019We consider a class of linear-programming based estimators in reconstructing a sparse signal from linear measurements. Specific formulations of the reconstruction problem considered here include Dantzig selector, basis pursuit (for the case in which the ... More
Positional Voting, Doubly Stochastic Matrices, and the Braid ArrangementAug 18 2019We provide elementary proofs of results from \cite{Saari} and \cite{DEMO} regarding the possible outcomes arising from a fixed profile within the class of positional voting systems. Our arguments enable a simple and explicit construction of paradoxical ... More
On the local structure of oriented graphs -- a case study in flag algebrasAug 18 2019Let $G$ be an $n$-vertex oriented graph. Let $t(G)$ (respectively $i(G)$) be the probability that a random set of $3$ vertices of $G$ spans a transitive triangle (respectively an independent set). We prove that $t(G) + i(G) \geq \frac{1}{9}-o_n(1)$. Our ... More
On the 2-colored crossing numberAug 18 2019Let $D$ be a straight-line drawing of a graph. The rectilinear 2-colored crossing number of $D$ is the minimum number of crossings between edges of the same color, taken over all possible 2-colorings of the edges of $D$. First, we show lower and upper ... More
On sets of $n$ points in general position that determine lines that can be pierced by $n$ pointsAug 18 2019Let $P$ be a set of $n$ points in general position in the plane. Let $R$ be a set of $n$ points disjoint from $P$ such that for every $x,y \in P$ the line through $x$ and $y$ contains a point in $R$ outside of the segment delimited by $x$ and $y$. We ... More
Configurations related to combinatorial Veronesians representing a skew perspectiveAug 17 2019A combinatorial object representing schemas of, possibly skew, perspectives, called {\em a configuration of skew perspective} has been defined in \cite{klik:binom}, \cite{maszko}. Here we develop the theory of configurations generalizing perspectives ... More
Asymptotic enumeration of linear hypergraphs with given number of vertices and edgesAug 17 2019For $n\geq 3$, let $r=r(n)\geq 3$ be an integer. A hypergraph is $r$-uniform if each edge is a set of $r$ vertices, and is said to be linear if two edges intersect in at most one vertex. In this paper, the number of linear $r$-uniform hypergraphs on $n\to\infty$ ... More
Majorana fermions and the Sensitivity ConjectureAug 17 2019Recently, Hao Huang proved the Sensitivity Conjecture, an important result about complexity measures of Boolean functions. We will discuss how this simple and elegant proof turns out to be closely related to physics concepts of the Jordan-Wigner transformation ... More
Quasirandom quantum channelsAug 17 2019Mixing (or quasirandom) properties of the natural transition matrix associated to a graph can be quantified by its distance to the complete graph. Different mixing properties correspond to different norms to measure this distance. For dense graphs, two ... More
The stable set problem in graphs with bounded genus and bounded odd cycle packing numberAug 17 2019Consider the family of graphs without $ k $ node-disjoint odd cycles, where $ k $ is a constant. Determining the complexity of the stable set problem for such graphs $ G $ is a long-standing problem. We give a polynomial-time algorithm for the case that ... More
Finding Hamiltonian and Longest (s, t)-paths of C-shaped Supergrid Graphs in Linear TimeAug 17 2019A supergrid graph is a finite vertex-induced subgraph of the infinite graph whose vertex set consists of all points of the plane with integer coordinates and in which two vertices are adjacent if the difference of their x or y coordinates is not larger ... More
Discrete and Fast Fourier Transform Made ClearAug 17 2019Fast Fourier transform was included in the Top 10 Algorithms of 20th Century by Computing in Science & Engineering. In this paper, we provide a new simple derivation of both the discrete Fourier transform and fast Fourier transform by means of elementary ... More
The general spectral radius and majorization theorem of $t$-cone graphs with given degree sequencesAug 17 2019The general spectral radius of a graph $G$, denoted by $\Theta(G,\alpha)$, is the maximal eigenvalue of $M_{\alpha}(G)=A(G)+\alpha D(G)$ $(\alpha\geq 0)$, where $A(G)$ and $D(G)$ are the adjacency matrix and the diagonal matrix of vertex degrees of $G$, ... More
On $(t,r)$ broadcast domination of certain grid graphsAug 16 2019Let $G=( V(G), E(G) )$ be a connected graph with vertex set $V(G)$ and edge set $E(G)$. We say a subset $D$ of $V(G)$ dominates $G$ if every vertex in $V \setminus D$ is adjacent to a vertex in $D$. A generalization of this concept is $(t,r)$ broadcast ... More
Antimagic orientations of graphs with large maximum degreeAug 16 2019Given a digraph $D$ with $m $ arcs, a bijection $\tau: A(D)\rightarrow \{1, 2, \ldots, m\}$ is an antimagic labeling of $D$ if no two vertices in $D$ have the same vertex-sum, where the vertex-sum of a vertex $u $ in $D$ under $\tau$ is the sum of labels ... More
Lower bounds in the polynomial Szemerédi theoremAug 16 2019We construct large subsets of the first $N$ positive integers which avoid certain arithmetic configurations. In particular, we construct a set of order $N^{0.7685}$ lacking the configuration $\{x,x+y,x+y^2\},$ surpassing the $N^{3/4}$ limit of Ruzsa's ... More
Minimum Coprime Labelings of Generalized Petersen and Prism GraphsAug 16 2019A coprime labeling of a graph of order $n$ is an assignment of distinct positive integer labels in which adjacent vertices have relatively prime labels. Restricting labels to only the set $1$ to $n$ results in a prime labeling. In this paper, we consider ... More
A finitary structure theorem for vertex-transitive graphs of polynomial growthAug 16 2019We prove a quantitative, finitary version of Trofimov's result that a connected, locally finite vertex-transitive graph G of polynomial growth admits a quotient with finite fibres on which the action of Aut(G) is virtually nilpotent with finite vertex ... More
Higher Connectivity of TropicalizationsAug 16 2019We show that the tropicalization of an irreducible d-dimensional variety over a field of characteristic 0is (d-l)-connected through codimension one, where l is the dimension of the lineality space of the tropicalization. From this we obtain a higher connectivity ... More
Turán Problems for Vertex Disjoint Cliques in Multi-partite HypergraphsAug 16 2019Let $s, r, k, n_1, \ldots, n_r$ be integers satisfying $2\leq s\leq r$ and $n_1\leq n_2\leq \cdots\leq n_r$. For two $s$-uniform hypergraphs $H$ and $F$, the Tur\'{a}n number $ex_s(H,F)$ is the maximum number of edges in an $F$-free subgraph of $H$. De ... More
LaserTank is NP-completeAug 16 2019We show that the classical game LaserTank is $\mathrm{NP}$-complete, even when the tank movement is restricted to a single column and the only blocks appearing on the board are mirrors and solid blocks. We show this by reducing $3$-SAT instances to LaserTank ... More
Sum-Essential Graphs of ModulesAug 16 2019The sum-essential graph $ \mathcal{S}_R(M) $ of a left $R$-module $M$ is a graph whose vertices are all nontrivial submodules of $M$ and two distinct submodules are adjacent iff their sum is an essential submodule of $M$. Properties of the graph $\mathcal{S}_R(M)$ ... More
Multiset Dimensions of TreesAug 16 2019Let $G$ be a connected graph and $W$ be a set of vertices of $G$. The representation multiset of a vertex $v$ with respect to $W$, $r_m (v|W)$, is defined as a multiset of distances between $v$ and the vertices in $W$. If $r_m (u |W) \neq r_m(v|W)$ for ... More
A Unified Framework for Constructing Centralized Coded Caching SchemesAug 16 2019In caching system, we prefer to design a coded caching scheme with the rate $R$ and the packet number $F$ as small as possible since the efficiency of transmission in the peak traffic times increases with the decreasing of $R$ and the realizing complexity ... More
Flag-tranitive block designs and finite exceptional simple groups of Lie typeAug 16 2019In this article, we study $2$-designs whose replication number is coprime to the parameter $\lambda$ and admitting a flag-transitive almost simple automorphism group with socle a finite exceptional simple group of Lie type. We obtain four infinite families ... More
Flag-transitive block designs and finite exceptional simple groups of Lie typeAug 16 2019Aug 20 2019In this article, we study $2$-designs whose replication number is coprime to the parameter $\lambda$ and admitting a flag-transitive almost simple automorphism group with socle a finite exceptional simple group of Lie type. We obtain four infinite families ... More
Holonomy Lie algebra of a fiber-type arrangementAug 16 2019We prove that the holonomy Lie algebra of a fiber-type arrangement is an iterated almost-direct product of a series of free Lie algebras with ranks the exponents of the arrangement. This is a Lie algebra version analogue of the well-known result of Falk-Randell ... More
Invariant synchrony subspaces of sets of matricesAug 15 2019A synchrony subspace of R^n is defined by setting certain components of the vectors equal according to an equivalence relation. Synchrony subspaces invariant under a given set of square matrices form a lattice. Applications of these invariant synchrony ... More
How many ways to color the map of America?Aug 15 2019Although the Four Color Conjecture originated in cartography, surprisingly, there is nothing in the literature on the number of ways to color an actual geographic map with four or fewer colors. In this paper, we compute these numbers, with exponentially ... More