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Ricci-flat cubic graphs with girth fiveFeb 08 2018We classify all connected, simple, 3-regular graphs with girth at least 5 that are Ricci-flat. We use the definition of Ricci curvature on graphs given in Lin-Lu-Yau, Tohoku Math., 2011, which is a variation of Ollivier, J. Funct. Anal., 2009. A graph ... More
Vanishing ideals of binary Hamming spheresFeb 08 2018We show how to efficiently obtain the Algebraic Normal Form of Boolean functions vanishing on Hamming spheres centred at zero. By exploiting the symmetry of the problem we obtain formulas for particular cases, and a computational method to address the ... More
On certain combinatorial expansions of descent polynomials and the change of grammarsFeb 08 2018In this paper, we study certain combinatorial expansions of descent polynomials by using the change of context-free grammars method. We provide a unified approach to study the gamma-positivity and the partial gamma-positivity of the descent polynomials ... More
On the Algebraic and Arithmetic structure of the monoid of Product-one sequences IIFeb 08 2018Let $G$ be a finite group. By a sequence over $G$, we mean a finite unordered sequence of terms from $G$, where repetition is allowed, and we say that it is a product-one sequence if its terms can be ordered such that their product equals the identity ... More
Convolutions of sets with bounded VC-dimension are uniformly continuousFeb 08 2018We introduce a notion of VC-dimension for subsets of groups, defining this for a set $A$ to be the VC-dimension of the family $\{ A \cap(xA) : x \in A\cdot A^{-1} \}$. We show that if a finite subset $A$ of an abelian group has bounded VC-dimension, then ... More
Gaussian binomial coefficients with negative argumentsFeb 08 2018Loeb showed that a natural extension of the usual binomial coefficient to negative (integer) entries continues to satisfy many of the fundamental properties. In particular, he gave a uniform binomial theorem as well as a combinatorial interpretation in ... More
Upper bound for a minimal quantifier depth of a Monadic Second-Order formula without asymptotic probabilityFeb 07 2018In this paper we found an upper bound for the minimal quantifier depth of a monadic second-order sentence without asymptotic probability described by Jerzy Tyszkiewicz, which express the extension grid axiom in the Erd}os-R\H{o}enyi model of random graphs ... More
Minor preserving deletable edges in graphsFeb 07 2018The simple 3-connected graphs without a prism minor were completely determined by Dirac in 1963. Suppose $G$ is a simple 3-connected rank $r$ graph with a prism minor. Call $G$ a {\it rank $r$ generator} if $\lfloor\frac{3r}{2}\rfloor +2 \le |E(G)|\le ... More
Cubic Preferences and the Character Admissibility ProblemFeb 07 2018In multiple-question referendum elections, the separability problem occurs when a voter's preferences on some questions or proposals depend on the predicted outcomes of others. The notion of separability formalizes the study of interdependence in multidimensional ... More
Factors of generalised polynomials and automatic sequencesFeb 07 2018The aim of this short note is to generalise the result of Rampersad--Shallit saying that an automatic sequence and a Sturmian sequence cannot have arbitrarily long common factors. We show that the same result holds if a Sturmian sequence is replaced by ... More
Combinatorial views on persistent characters in phylogeneticsFeb 07 2018The so-called binary perfect phylogeny with persistent characters has recently been thoroughly studied in computational biology as it is less restrictive than the well known binary perfect phylogeny. Here, we focus on the notion of (binary) persistent ... More
The intrinsic geometry of coarse median spaces and their intervalsFeb 07 2018Sholander showed that median algebras can be characterised in terms of the intervals arising from the median operator. In this paper we develop a coarse analogue of Sholander's approach, characterising coarse median spaces in terms of their intervals. ... More
On the Stability of Independence PolynomialsFeb 07 2018The independence polynomial of a graph is the generating polynomial for the number of independent sets of each size, and its roots are called {\em independence roots}. We investigate the stability of such polynomials, that is, conditions under which the ... More
On the gaps between consecutive primesFeb 07 2018Let $p_n$ denote the $n$-th prime. For any $m\geq 1$, there exist infinitely many $n$ such that $p_{n}-p_{n-m}\leq C_m$ for some large constant $C_m>0$, and $$p_{n+1}-p_n\geq \frac{c_m\log n\log\log n\log\log\log\log n}{\log\log\log n}, $$ for some small ... More
Field extensions, Derivations, and Matroids over Skew HyperfieldsFeb 07 2018We show that a field extension $K\subseteq L$ in positive characteristic $p$ and elements $x_e\in L$ for $e\in E$ gives rise to a matroid $M^\sigma$ on ground set $E$ with coefficients in a certain skew hyperfield $L^\sigma$. This skew hyperfield $L^\sigma$ ... More
Tropicalized quartics and canonical embeddings for tropical curves of genus 3Feb 07 2018Brodsky, Joswig, Morrison and Sturmfels showed that not all abstract tropical curves of genus $3$ can be realized as a tropicalization of a quartic in the euclidean plane. In this article, we focus on the interior of the maximal cones in the moduli space ... More
The $b$-branching problem in digraphsFeb 07 2018In this paper, we introduce the concept of $b$-branchings in digraphs, which is a generalization of branchings serving as a counterpart of $b$-matchings. Here $b$ is a positive integer vector on the vertex set of a digraph, and a $b$-branching is defined ... More
Four-coloring $P_6$-free graphs. II. Finding an excellent precoloringFeb 07 2018Feb 08 2018This is the second paper in a series of two. The goal of the series is to give a polynomial time algorithm for the $4$-coloring problem and the $4$-precoloring extension problem restricted to the class of graphs with no induced six-vertex path, thus proving ... More
Four-coloring $P_6$-free graphs. I. Extending an excellent precoloringFeb 07 2018Feb 08 2018This is the first paper in a series whose goal is to give a polynomial time algorithm for the $4$-coloring problem and the $4$-precoloring extension problem restricted to the class of graphs with no induced six-vertex path, thus proving a conjecture of ... More
On the polynomial Szemerédi's theorem in finite fieldsFeb 06 2018Let $P_1,\dots,P_m\in\mathbb{Z}[y]$ be any linearly independent polynomials with zero constant term. We show that there exists a $\gamma>0$ such that any subset of $\mathbb{F}_q$ of size at least $q^{1-\gamma}$ contains a nontrivial polynomial progression ... More
Fixing monotone Boolean networks asynchronouslyFeb 06 2018The asynchronous automaton associated with a Boolean network $f:\{0,1\}^n\to\{0,1\}^n$ is considered in many applications. It is the finite deterministic automaton with set of states $\{0,1\}^n$, alphabet $\{1,\dots,n\}$, where the action of letter $i$ ... More
Updated estimate of the number of edges in induced subgraphs of a special distance graphFeb 06 2018In this article, the author proposes a new approach for the estimating of the number of edges in induced subgraphs of a special distance graph. Author significantly improves previous estimates and suggests a new approach to obtaining better ones.
On the structure of random graphs that are locally indistinguishable from a latticeFeb 06 2018We study the properties of finite graphs in which the ball of radius $r$ around each vertex induces a graph isomorphic to the ball of radius $r$ in some fixed vertex-transitive graph $F$. This is a natural extension of the study of regular graphs. We ... More
On the irregularity of uniform hypergraphsFeb 06 2018Let $H$ be an $r$-uniform hypergraph on $n$ vertices and $m$ edges, and let $d_i$ be the degree of $i\in V(H)$. Denote by $\varepsilon(H)$ the difference of the spectral radius of $H$ and the average degree of $H$. Also, denote \[ s(H)=\sum_{i\in V(H)}\left|d_i-\frac{rm}{n}\right|,~ ... More
Volume growth, curvature, and Buser-type inequalities in graphsFeb 06 2018We study the volume growth of metric balls as a function of the radius in discrete spaces, and focus on the relationship between volume growth and discrete curvature. We improve volume growth bounds under a lower bound on the so-called Ollivier curvature, ... More
Random cliques in random graphsFeb 06 2018We show that for each $r\ge 4$, in a density range extending up to, and slightly beyond, the threshold for a $K_r$-factor, the copies of $K_r$ in the random graph $G(n,p)$ are randomly distributed, in the (one-sided) sense that the hypergraph that they ... More
$q$-Analouges of two Ramanujan-type forumlas for $1/π$Feb 06 2018We give $q$-analouges of the following two Ramanujan-type forumlas for $1/\pi$: \begin{align*} \sum_{k=0}^\infty \frac{(6k+1)(\frac{1}{2})_k^3}{k!^3 4^k} =\frac{4}{\pi}, \quad\text{and}\quad \sum_{k=0}^\infty (-1)^k(6k+1)\frac{(\frac{1}{2})_k^3}{k!^3 ... More
Cantor combinatorics and almost finitenessFeb 06 2018In this survey we give a concise introduction to a continuous version of Borel combinatorics. Our approach will have a certain algorithm-theoretic nature and we will give special emphasis to the notion of almost finiteness introduced by Matui as a continuous ... More
On the width of transitive sets: bounds on matrix coefficients of finite groupsFeb 06 2018We say that a finite subset of the unit sphere in $\mathbf{R}^d$ is transitive if there is a group of isometries which acts transitively on it. We show that the width of any transitive set is bounded above by a constant times $(\log d)^{-1/2}$. This is ... More
Girth-regular graphsFeb 06 2018We introduce a notion of girth-regularity of graphs, which generalises two very different aspects of symmetry in graph theory: that of vertex transitivity and that of distance regularity. We then show some results about this notion.
Transversals in Uniform Linear HypergraphsFeb 06 2018The transversal number $\tau(H)$ of a hypergraph $H$ is the minimum number of vertices that intersect every edge of $H$. A linear hypergraph is one in which every two distinct edges intersect in at most one vertex. A $k$-uniform hypergraph has all edges ... More
Isotropic Subspaces of Schur ModulesFeb 06 2018It is a well-known fact that over the complex numbers and for a fixed $k$ and $n$, a generic $s$ in $Sym^2V^*$ vanishes on some $k$-dimensional subspace of $V$ if and only if $n\geq 2k$. Tevelev found exact conditions for the extension of this statement ... More
Some results on counting linearizations of posetsFeb 05 2018In section 1 we consider a 3-tuple $S=(|S|,\preccurlyeq,E)$ where $|S|$ is a finite set, $\preccurlyeq$ a partial ordering on $|S|,$ and $E$ a set of unordered pairs of distinct members of $|S|,$ and study, as a function of $n\geq 0,$ the number of maps ... More
Bernstein operators and super-Schur functions: combinatorial aspectsFeb 05 2018The Bernstein vertex operators, which can be used to build recursively the Schur functions, are extended to superspace. Four families of super vertex operators are defined, corresponding to the four natural families of Schur functions in superspace. Combinatorial ... More
Bounds for $L_p$-discrepancies of point distributions in compact metric spacesFeb 05 2018We consider finite point subsets (distributions) in compact connected metric measure spaces. The spaces under study are specialized by conditions on the volume of metric balls as a function of radii. These conditions are not hard and hold, particularly, ... More
Solution for a Bipartite Euclidean TSP in one dimensionFeb 05 2018The travelling salesman problem is one of the most studied combinatorial optimization problems, because of the simplicity in its statement and the difficulty in its solution. We characterize the optimal cycle for every convex and increasing cost function ... More
Exponential functions of finite posets and the number of extensions with a fixed set of minimal pointsFeb 05 2018We establish formulas for the number of all downsets (or equivalently, of all antichains) of a finite poset P. Then, using these numbers, we determine recursively and explicitly the number of all posets having a fixed set of minimal points and inducing ... More
Congruences for the Coefficients of the Powers of the Euler ProductFeb 05 2018Let $p_k(n)$ be given by the $k$-th power of the Euler Product $\prod _{n=1}^{\infty}(1-q^n)^k=\sum_{n=0}^{\infty}p_k(n)q^{n}$. By investigating the properties of the modular equations of the second and the third order under the Atkin $U$-operator, we ... More
Listening to the cohomology of graphsFeb 05 2018We prove that the spectrum of the Kirchhoff Laplacian H0 of a finite simple Barycentric refined graph and the spectrum of the connection Laplacian L of G determine each other: we prove that L-L^(-1) is similar to the Hodge Laplacian H of G which is in ... More
Balanced diagonals in frequency squaresFeb 04 2018We say that a diagonal in an array is {\em $\lambda$-balanced} if each entry occurs $\lambda$ times. Let $L$ be a frequency square of type $F(n;\lambda^m)$; that is, an $n\times n$ array in which each entry from $\{1,2,\dots ,m\}$ occurs $\lambda$ times ... More
Some sharp results on the generalized Turán numbersFeb 04 2018For graphs $T, H$, let $ex(n,T,H)$ denote the maximum number of copies of $T$ in an $n$-vertex $H$-free graph. In this paper we prove some sharp results on this generalization of Tur\'an numbers, where our focus is for the graphs $T,H$ satisfying $\chi(T)<\chi(H)$. ... More
About chromatic uniqueness of some complete tripartite graphsFeb 04 2018Let $P(G, x)$ be the chromatic polynomial of a graph $G$. A graph $G$ is called \textit{chromatically unique} if for any graph $H,\, P(G, x) = P(H, x)$ implies that $G$ and $H$ are isomorphic. In this paper we show that full tripartite graph $K(n_1, n_2, ... More
Equitable partitions of Latin-square graphsFeb 03 2018We study equitable partitions of Latin-square graphs, and give a complete classification of those whose quotient matrix does not have an eigenvalue $-3$.
Parametric Presburger Arithmetic: Complexity of Counting and Quantifier EliminationFeb 03 2018We consider an expansion of Presburger arithmetic which allows multiplication by $k$ parameters $t_1,\ldots,t_k$. A formula in this language defines a parametric set $S_\mathbf{t} \subseteq \mathbb{Z}^{d}$ as $\mathbf{t}$ varies in $\mathbb{Z}^k$, and ... More
Stirling Numbers in Braid Matroid Kazhdan-Lusztig PolynomialsFeb 02 2018Restricted Whitney numbers of the first kind appear in the combinatorial recursion for the matroid Kazhdan-Lusztig polynomials. In the special case of braid matroids (the matroid associated to the partition lattice, the complete graph, the type A Coxeter ... More
On the complexity of the outer-connected bondage and the outer-connected reinforcement problemsFeb 02 2018Let $G=(V,E)$ be a graph. A subset $S \subseteq V$ is a dominating set of $G$ if every vertex not in $S$ is adjacent to a vertex in $S$. A set $\tilde{D} \subseteq V$ of a graph $G=(V,E) $ is called an outer-connected dominating set for $G$ if (1) $\tilde{D}$ ... More
Goethals--Seidel difference families with symmetric or skew base blocksFeb 02 2018We single out a class of difference families which is widely used in some constructions of Hadamard matrices and which we call Goethals--Seidel (GS) difference families. They consist of four subsets (base blocks) of a finite abelian group of order $v$, ... More
On indicated coloring of some classes of graphsFeb 01 2018Indicated coloring is a type of game coloring in which two players collectively color the vertices of a graph in the following way. In each round the first player (Ann) selects a vertex, and then the second player (Ben) colors it properly, using a fixed ... More
Total perfect codes in zero-divisor graphsFeb 01 2018Let R be a commutative ring with unity not equal to zero and let G = (V, E) be a simple, undirected graph. A total perfect code denoted by C(G), in G is a subset C(G) of V (G) such that cardinality of the set {N (v) \cap C(G)} is 1, for all v \in V (G), ... More
On the size of the set $AA+A$Jan 31 2018It is established that there exists an absolute constant $c>0$ such that for any finite set $A$ of positive real numbers $$|AA+A| \gg |A|^{\frac{3}{2}+c}.$$ On the other hand, we give an explicit construction of a finite set $A \subset \mathbb R$ such ... More
On the computability of graphonsJan 31 2018We investigate the relative computability of exchangeable binary relational data when presented in terms of the distribution of an invariant measure on graphs, or as a graphon in either $L^1$ or the cut distance. We establish basic computable equivalences, ... More
Positiveness of the permanent of 4-dimensional polystochastic matrices of order 4Jan 31 2018A nonnegative multidimensional matrix is called polystochastic if the sum of its entries over each line is equal to one. We prove that the permanent of every $4$-dimensional polystochastic matrix of order $4$ is positive.
Energy-efficient Deployment of Relay Nodes in Wireless Sensor Networks using Evolutionary TechniquesJan 30 2018Feb 03 2018Random deployment of sensor nodes is susceptible to initial communication hole, even when the network is densely populated. However, eliminating holes using structural deployment poses its difficulties. In either case, the resulting coverage holes can ... More
Standard modules, Jones-Wenzl projectors, and the valenced Temperley-Lieb algebraJan 30 2018This article concerns a generalization of the Temperley-Lieb algebra, motivated by applications to conformal field theory. We call this algebra the valenced Temperley-Lieb algebra. We prove salient facts concerning this algebra and its representation ... More
Extensions of Erdős-Gallai Theorem and Luo's Theorem with ApplicationsJan 30 2018The famous Erd\H{o}s-Gallai Theorem states that every graph with $n$ vertices and $m$ edges contains a path of length at least $\frac{2m}{n}$. In this note, we first establish a simple but novel extension of Erd\H{o}s-Gallai Theorem by proving that every ... More
New characterizations of freeness for hyperplane arrangementsJan 30 2018In this article we describe two new characterizations of freeness for hyperplane arrangements via the study of the generic initial ideal and of the sectional matrix of the Jacobian ideal of arrangements.
Subgraph counts for dense random graphs with specified degreesJan 30 2018Feb 05 2018We prove two estimates for the expectation of the exponential of a complex function of a random permutation or subset. Using this theory, we find asymptotic expressions for the expected number of copies and induced copies of a given graph in a uniformly ... More
Earthmover Resilience and Testing in Ordered StructuresJan 29 2018One of the main challenges in property testing is to characterize those properties that are testable with a constant number of queries. For unordered structures such as graphs and hypergraphs this task has been mostly settled. However, for ordered structures ... More
A note on expansion in prime fieldsJan 29 2018Let $\beta,\epsilon \in (0,1]$, and $k \geq \exp(122 \max\{1/\beta,1/\epsilon\})$. We prove that if $A,B$ are subsets of a prime field $\mathbb{Z}_{p}$, and $|B| \geq p^{\beta}$, then there exists a sum of the form $$S = a_{1}B \pm \ldots \pm a_{k}B, ... More
A Pascal-like Bound for the Number of Necklaces with Fixed DensityJan 29 2018A bound resembling Pascal's identity is presented for binary necklaces with fixed density using Lyndon words with fixed density. The result is generalized to k-ary necklaces and Lyndon words with fixed content. The bound arises in the study of Nichols ... More
A Gale-Berlekamp permutation-switching problem in higher dimensionsJan 28 2018Let an $n\times n$ array $\left( a_{ij}\right) $ of lights be given, each either on (when $a_{ij}=1$) or off (when $a_{ij}=-1$). For each row and each column there is a switch so that if the switch is pulled ($x_{i}=-1$ for row $i$ and $y_{j}=-1$ for ... More
Affine Schubert calculus and double coinvariantsJan 27 2018We first define an action of the double coinvariant algebra $DR_n$ on the homology of the affine flag variety $\widetilde{Fl}_n$ in type $A$, and use affine Schubert calculus to prove that it preserves the image of the homology of the rational $(n,m)$-affine ... More
The totally nonnegative part of G/P is a ballJan 26 2018We show that the totally nonnegative part of a partial flag variety (in the sense of Lusztig) is homeomorphic to a closed ball.
Intervals in the Hales-Jewett theoremJan 26 2018The Hales-Jewett theorem states that for any $m$ and $r$ there exists an $n$ such that any $r$-colouring of the elements of $[m]^n$ contains a monochromatic combinatorial line. We study the structure of the wildcard set $S \subseteq [n]$ which determines ... More
Antipodal Point Arrangements on Spheres, Classification of Normal Systems and Hyperplane ArrangementsJan 26 2018For any positive integer k, we classify the antipodal point arrangements (refer to Definitions $[3.1,3.2,5.1,5.2]$) on the sphere $\mathbb{PF}^{k+}_{\mathbb{F}}$ over a field $\mathbb{F}$ with $1-ad$ structure (refer to Definition $1.1$), upto isomorphism, ... More
New bounds on the dimensions of planar distance setsJan 26 2018We prove new bounds on the dimensions of distance sets and pinned distance sets of planar sets. Among other results, we show that if $A\subset\mathbb{R}^2$ is a Borel set of Hausdorff dimension $s>1$, then its distance set has Hausdorff dimension at least ... More
An improved upper bound for the size of the multiplicative 3-Sidon setsJan 26 2018We say that a set is a multiplicative 3-Sidon set if the equation $s_1s_2s_3=t_1t_2t_3$ does not have a solution consisting of distinct elements taken from this set. In this paper we show that the size of a multiplicative 3-Sidon subset of $\{1,2,\dots,n\}$ ... More
On the Coherent Labelling Inequalities of a Polyhedron in Three DimensionsJan 26 2018In this article we consider open Question $1$ about coherent labelling inequalities arising from a polyhedron (refer to Definition $1$). We exhibit some examples like pyramids and bi-pyramids where coherent labelling is possible and we also exhibit a ... More
Absolutely split metacyclic groups and weak metacirculantsJan 26 2018Let $m,n,r$ be positive integers, and let $G=\langle a\rangle: \langle b\rangle \cong \mathbb{Z}_n: \mathbb{Z}_m$ be a split metacyclic group such that $b^{-1}ab=a^r$. We say that $G$ is {\em absolutely split with respect to $\langle a\rangle$} provided ... More
Individual testing is optimal for nonadaptive group testing in the linear regimeJan 25 2018We consider nonadaptive probabilistic group testing in the linear regime, where each of n items is defective independently with probability p in (0,1), where p is a constant independent of n. We show that testing each item individually is optimal, in ... More
On the algorithmic complexity of decomposing graphs into regular/irregular structuresJan 25 2018Jan 29 2018A locally irregular graph is a graph whose adjacent vertices have distinct degrees, a regular graph is a graph where each vertex has the same degree and a locally regular graph is a graph where for every two adjacent vertices u, v, their degrees are equal. ... More
On Sidorenko's conjecture for determinants and Gaussian Markov random fieldsJan 25 2018Feb 07 2018We study a class of determinant inequalities that are closely related to Sidorenko's famous conjecture (Also conjectured by Erd\H os and Simonovits in a different form). Our main result can also be interpreted as an entropy inequality for Gaussian Markov ... More
On a New Ring Invariant - Towards a Framework for Rank IdentitiesJan 24 2018For a square matrix $A$ over a field $F$, it is known that $\mathrm{rank}(A) + \mathrm{rank}(I-A) = \mathrm{rank}(I) + \mathrm{rank}(A-A^2)$. The main goal of this article is to start a program to characterize and generalize such rank identities by constructing ... More
A recurrence relation for Wronskian Hermite polynomialsJan 24 2018We consider polynomials that are defined as Wronskians of certain sets of Hermite polynomials. Our main result is a recurrence relation for these polynomials in terms of those of one or two degrees smaller, which generalizes the well-known three term ... More
On the Hamilton-Waterloo Problem with cycle lengths of distinct paritiesJan 23 2018Let $K_v^*$ denote the complete graph $K_v$ if $v$ is odd and $K_v-I$, the complete graph with the edges of a 1-factor removed, if $v$ is even. Given non-negative integers $v, M, N, \alpha, \beta$, the Hamilton-Waterloo problem asks for a $2$-factorization ... More
Stable gonality is computableJan 23 2018Stable gonality is a multigraph parameter that measures the complexity of a graph. It is defined using maps to trees. Those maps, in some sense, divide the edges equally over the edges of the tree; stable gonality asks for the map with the minimum number ... More
Connections between rank and dimension for subspaces of bilinear formsJan 23 2018Let $K$ be a field and let $V$ be a vector space of dimension $n$ over $K$. Let $M$ be a subspace of bilinear forms defined on $V\times V$. Let $r$ be the number of different non-zero ranks that occur among the elements of $M$. Our aim is to obtain an ... More
Incidence bicomodules, Möbius inversion, and a Rota formula for infinity adjunctionsJan 23 2018In the same way decomposition spaces, also known as unital 2-Segal spaces, have incidence (co)algebras, and certain relative decomposition spaces have incidence (co)modules, we identify the structures that have incidence bi(co)modules: they are certain ... More
Toll number of the strong product of graphsJan 23 2018A tolled walk $T$ between two non-adjacent vertices $u$ and $v$ in a graph $G$ is a walk, in which $u$ is adjacent only to the second vertex of $T$ and $v$ is adjacent only to the second-to-last vertex of $T$. A toll interval between $u,v\in V(G)$ is ... More
The sum of nonsingular matrices is often nonsingularJan 23 2018Jan 24 2018If $M$ is a set of nonsingular $k\times k$ matrices then for many pairs of matrices, $A,B\in M,$ the sum is nonsingular, $\det(A+B)\neq 0.$ We prove a more general statement on nonsingular sums with an application.
Exploring a Delta Schur ConjectureJan 23 2018In \cite{HRW15}, Haglund, Remmel, Wilson state a conjecture which predicts a purely combinatorial way of obtaining the symmetric function $\Delta_{e_k}e_n$. It is called the Delta Conjecture. It was recently proved in \cite{GHRY} that the Delta Conjecture ... More
The Eulerian distribution on the involutions of the hyperoctahedral group is unimodalJan 22 2018The Eulerian distribution on the involutions of the symmetric group is unimodal, as shown by Guo and Zeng. In this paper we prove that the Eulerian distribution on the involutions of the hyperoctahedral group, when viewed as a colored permutation group, ... More
A coding theoretic approach to the uniqueness conjecture for projective planes of prime orderJan 22 2018An outstanding folklore conjecture asserts that, for any prime $p$, up to isomorphism the projective plane $PG(2,\mathbb{F}_p)$ over the field $\mathbb{F}_p := \mathbb{Z}/p\mathbb{Z}$ is the unique projective plane of order $p$. Let $\pi$ be any projective ... More
Fractional Powers of the Generating Function for the Partition FunctionJan 22 2018Jan 30 2018Let $p_{k}(n)$ be the coefficient of $q^n$ in the series expansion of $(q;q)_{\infty}^{k}$. It is known that the partition function $p(n)$, which corresponds to the case when $k=-1$, satisfies congruences such as $p(5n+4)\equiv 0\pmod{5}$. In this article, ... More
The extremal functions for triangle-free graphs with excluded minorsJan 21 2018We prove two results: 1. A graph $G$ on at least seven vertices with a vertex $v$ such that $G-v$ is planar and $t$ triangles satisfies $|E(G)| \leq 3|V(G)|- 9 + t/3$. 2. For $p=2,3,\ldots,9$, a triangle-free graph $G$ on at least $2p-5$ vertices with ... More
Generalized Laminar MatroidsJan 21 2018Nested matroids were introduced by Crapo in 1965 and have appeared frequently in the literature since then. A flat of a matroid $M$ is Hamiltonian if it has a spanning circuit. A matroid $M$ is nested if and only if its Hamiltonian flats form a chain ... More
Approximation of Banzhaf indices and its application to voting gamesJan 21 2018In this paper, we propose an improved version of the power index related to the Banzhaf power index for weighted voting systems. This index now takes into account the mutual persuasion power matrix(PPM) existing among the voters. This improved index is ... More
Efficient algorithms for computing a minimal homology basisJan 21 2018Efficient computation of shortest cycles which form a homology basis under $\mathbb{Z}_2$-additions in a given simplicial complex $\mathcal{K}$ has been researched actively in recent years. When the complex $\mathcal{K}$ is a weighted graph with $n$ vertices ... More
The Slow-coloring Game on Outerplanar, Planar, and $k$-degenerate GraphsJan 21 2018The \emph{slow-coloring game} is played by Lister and Painter on a graph $G$. Initially, all vertices of $G$ are uncolored. In each round, Lister marks a non-empty set $M$ of uncolored vertices, and Painter colors a subset of $M$ that is independent in ... More
Linear programming methods for exponential dominationJan 19 2018For a graph $G,$ we consider $D \subset V(G)$ to be a porous exponential dominating set if $1 \le \sum_{d \in D}$ $\left( \tfrac{1}{2} \right)^{dist(d,v) -1}$ for every $v \in V(G),$ where $dist(d,v)$ denotes the length of the shortest $dv$ path. The ... More
Difference sets and power residuesJan 19 2018Jan 31 2018Let $p\geq 3$ be a prime and $n\geq 1$ be an integer. Let $K\subseteq {\mathbb Z_p}$ denote a fixed subset with $0\in K$. Let $A\subseteq ({\mathbb Z_p})^n$ be an arbitrary subset such that $$ \{ \mathbf{a}-\mathbf{b}:~\mathbf{a},\mathbf{b}\in A,\mathbf{a}\neq ... More
Scattered classes of graphsJan 18 2018For a class $\mathcal C$ of graphs $G$ equipped with functions $f_G$ defined on subsets of $E(G)$ or $V(G)$, we say that $\mathcal{C}$ is $k$-scattered with respect to $f_G$ if there exists a constant $\ell$ such that for every graph $G\in \mathcal C$, ... More
Efficient Computation of the 8-point DCT via Summation by PartsJan 17 2018This paper introduces a new fast algorithm for the 8-point discrete cosine transform (DCT) based on the summation-by-parts formula. The proposed method converts the DCT matrix into an alternative transformation matrix that can be decomposed into sparse ... More
Algorithms for Computing Wiener Indices of Acyclic and Unicyclic GraphsJan 16 2018Let $G=(V(G),E(G))$ be a molecular graph, where $V(G)$ and $E(G)$ are the sets of vertices (atoms) and edges (bonds). A topological index of a molecular graph is a numerical quantity which helps to predict the chemical/physical properties of the molecules. ... More
Total dominator coloring of central graphsJan 16 2018A total dominator coloring of a graph $G$ is a proper coloring of $G$ in which each vertex of the graph is adjacent to every vertex of some color class. The total dominator chromatic number $\chi_d^t(G)$ of $G$ is the minimum number of color classes in ... More
Changing and unchanging of the domination number of a graph: Path addition numbersJan 15 2018Given a graph $G = (V,E)$ and two its distinct vertices $u$ and $v$. The $(u,v)$-$P_k$-{\em addition graph} of $G$ is the graph $G_{u,v,k-2}$ obtained from disjoint union of $G$ and a path $P_k: x_0,x_1,..,x_{k-1}$, $k \geq 2$, by identifying the vertices ... More
Directed Strongly Regular Cayley Graphs on Dihedral groupsJan 15 2018In this paper,\;we characterize some certain directed strongly regular Cayley graphs on Dihedral groups $D_{p^\alpha}$,\;where $p$ is a prime and $\alpha\geqslant 1$ is a positive integer.\;We achieve this result by using some tools from Representation ... More
Partial geodesics on symmetric groups endowed with breakpoint distanceJan 15 2018The notion of partial geodesic was introduced by Jamshidpey et al. in "Sets of medians in the non-geodesic pseudometric space of unsigned genomes with breakpoints", 2014. In this paper, we study the density of points on non-trivial partial geodesics between ... More
An Elementary Dyadic Riemann HypothesisJan 15 2018The connection zeta function of a finite abstract simplicial complex G is defined as zeta_L(s)=sum_x 1/lambda_x^s, where lambda_x are the eigenvalues of the connection Laplacian L defined by L(x,y)=1 if x and y intersect and 0 else. (I) As a consequence ... More
Remarks on GraphonsJan 14 2018The notion of the graphon (a symmetric measurable fuzzy set of $[0, 1]^2$) was introduced by L. Lov\'asz and B. Szegedy in 2006 to describe limit objects of convergent sequences of dense graphs. In their investigation the integral \[t(F,W)=\int _{[0, ... More