Sequence Of Nonnegative Integers Research Articles

Arithmetical properties of real numbers related to beta-expansions

The main purpose of this paper is to study the arithmetical properties of values \(\sum_{m=0}^{\infty} \beta^{-w(m)}\), where \(\beta\) is a fixed Pisot or Salem number and \(w(m)\) (\(m=0,1,\ldots\)) are distinct sequences of nonnegative integers with \(w(m+1)>w(m)\) for any sufficiently large \(m\). We first introduce the algebraic independence results of such values. Our results are applicable to certain sequences \(w(m)\) (\(m=0,1,\ldots\)) with \(\lim_{m\to\infty}w(m+1)/w(m)=1.\) For example, we prove that two numbers \[\sum_{m=1}^{\infty}\beta^{-\lfloor \varphi(m)\rfloor}, \quad \sum_{m=3}^{\infty}\beta^{-\lfloor a(m)\rfloor}\] are algebraically independent, where \(\varphi(m)=m^{\log m}\) and \(a(m)=m^{\log\log m}\). Moreover, we also give the linear independence results of real numbers. Our results are applicable to the values \(\sum_{m=0}^{\infty}\beta^{-\lfloor m^\rho\rfloor}\), where \(\beta\) is a Pisot or Salem number and \(\rho\) is a real number greater than 1.

On the approximate shape of degree sequences that are not potentially $H$-graphic

A sequence of nonnegative integers $\pi$ is {\it graphic} if it is the degree sequence of some graph $G$. In this case we say that $G$ is a \textit{realization} of $\pi$, and we write $\pi=\pi(G)$. A graphic sequence $\pi$ is {\it potentially $H$-graphic} if there is a realization of $\pi$ that contains $H$ as a subgraph. Given nonincreasing graphic sequences $\pi_1=(d_1,\ldots,d_n)$ and $\pi_2 = (s_1,\ldots,s_n)$, we say that $\pi_1$ {\it majorizes} $\pi_2$ if $d_i \geq s_i$ for all $i$, $1 \leq i \leq n$. In 1970, Erd\H{o}s showed that for any $K_{r+1}$-free graph $H$, there exists an $r$-partite graph $G$ such that $\pi(G)$ majorizes $\pi(H)$. In 2005, Pikhurko and Taraz generalized this notion and showed that for any graph $F$ with chromatic number $r+1$, the degree sequence of an $F$-free graph is, in an appropriate sense, nearly majorized by the degree sequence of an $r$-partite graph. In this paper, we give similar results for degree sequences that are not potentially $H$-graphic. In particular, there is a graphic sequence $\pi^*(H)$ such that if $\pi$ is a graphic sequence that is not potentially $H$-graphic, then $\pi$ is close to being majorized by $\pi^*(H)$. Similar to the role played by complete multipartite graphs in the traditional extremal setting, the sequence $\pi^*(H)$ asymptotically gives the maximum possible sum of a graphic sequence $\pi$ that is not potentially $H$-graphic.

The Generalized Rank of Trace Languages

Given a partially commutative alphabet and a set of words [Formula: see text], the rank of [Formula: see text] expresses the amount of shuffling required to produce a word belonging to [Formula: see text] from two words whose concatenation belongs to the closure of [Formula: see text] with respect to the partial commutation. In this paper, the notion of rank is generalized from concatenations of two words to an arbitrary fixed number of words. In this way, an infinite sequence of non-negative integers and infinity is assigned to every set of words. It is proved that in the case of alphabets defining free commutative monoids, as well as in the more general case of direct products of free monoids, sequences of ranks of regular sets are exactly non-decreasing sequences that are eventually constant. On the other hand, by uncovering a relationship between rank sequences of regular sets and rational series over the min-plus semiring, it is shown that already for alphabets defining free products of free commutative monoids, rank sequences need not be eventually periodic.

Generalizations of some results about the regularity properties of an additive representation function

Let $${A = \{a_{1},a_{2},\dots{} \}}$$ $${(a_{1} < a_{2} < \cdots )}$$ be an infinite sequence of nonnegative integers, and let $${R_{A,2}(n)}$$ denote the number of solutions of $${a_{x}+a_{y}=n}$$ $${(a_{x},a_{y} \in A)}$$ . P. Erdős, A. Sarkozy and V. T. Sos proved that if $${\lim_{N\to\infty}\frac{B(A,N)}{\sqrt{N}}=+\infty}$$ then $${|\Delta_{1}(R_{A,2}(n))|}$$ cannot be bounded, where $${B(A,N)}$$ denotes the number of blocks formed by consecutive integers in A up to N and $${\Delta_{l}}$$ denotes the l-th difference. Their result was extended to $${\Delta_{l}(R_{A,2}(n))}$$ for any fixed $${l\ge2}$$ . In this paper we give further generalizations of this problem.

Enumerating sparse uniform hypergraphs with given degree sequence and forbidden edges

For n≥3 and r=r(n)≥3, let k=k(n)=(k1,…,kn) be a sequence of non-negative integers with sum M(k)=∑j=1nkj. We assume that M(k) is divisible by r for infinitely many values of n, and restrict our attention to these values. Let X=X(n) be a simple r-uniform hypergraph on the vertex set V={v1,v2,…,vn} with t edges. We denote by Hr(k) the set of all simple r-uniform hypergraphs on the vertex set V with degree sequence k, and let Hr(k,X) be the set of all hypergraphs in Hr(k) which contain no edge of X. We give an asymptotic enumeration formula for the size of Hr(k,X). This formula holds when r4kmax3=o(M(k)), tkmax3=o(M(k)2) and rtkmax4=o(M(k)3). Our proof involves the switching method.As a corollary, we obtain an asymptotic formula for the number of hypergraphs in Hr(k) which contain every edge of X. We apply this result to find asymptotic expressions for the expected number of perfect matchings and loose Hamilton cycles in a random hypergraph in Hr(k) in the regular case.

A Characterization of Box-bounded Degree Sequences of Graphs

Let $$A_n=(a_1,a_2,\ldots ,a_n)$$ and $$B_n=(b_1,b_2,\ldots ,b_n)$$ be nonnegative integer sequences with $$A_n\le B_n$$ and $$b_i\ge b_{i+1},a_i+b_i\ge a_{i+1}+b_{i+1}, i=1,2\ldots , n-1$$ . The purpose of this note is to give a good characterization for $$A_n$$ and $$B_n$$ such that every integer sequence $$\pi =(d_1,d_2,\ldots d_n)$$ with even sum and $$A_n\le \pi \le B_n$$ is graphic. This improves related results of Guo and Yin and generalizes the Erdős–Gallai theorem.

Generating Infinite Random Graphs

AbstractWe define a growing model of random graphs. Given a sequence of non-negative integers {dn}n=0∞ with the property that di≤i, we construct a random graph on countably infinitely many vertices v0, v1… by the following process: vertex vi is connected to a subset of {v0, …, vi−1} of cardinality di chosen uniformly at random. We study the resulting probability space. In particular, we give a new characterization of random graphs, and we also give probabilistic methods for constructing infinite random trees.

Bigraphic pairs with an [formula omitted]-connected realization

Let S=(a1,…,am;b1,…,bn), where a1,…,am and b1,…,bn are two nonincreasing sequences of nonnegative integers. The pair S=(a1,…,am;b1,…,bn) is said to be a bigraphic pair if there is a simple bipartite graph G=(X∪Y,E) such that a1,…,am and b1,…,bn are the degrees of the vertices in X and Y, respectively. Let A be an (additive) Abelian group. We define σ(A,m,n) to be the minimum integer k such that every bigraphic pair S=(a1,…,am;b1,…,bn) with am,bn≥2 and σ(S)=a1+⋯+am≥k has an A-connected realization. In this paper, we determine the values of σ(A,m,n) for |A|=k and m≥n≥2, where k≥5 is an odd integer.

Tournaments, oriented graphs and football sequences

Abstract Consider the result of a soccer league competition where n teams play each other exactly once. A team gets three points for each win and one point for each draw. The total score obtained by each team vi is called the f-score of vi and is denoted by fi. The sequences of all f-scores [ f i ] i = 1 n $\left[ {{\rm{f}}_{\rm{i}} } \right]_{{\rm{i}} = 1}^{\rm{n}} $ arranged in non-decreasing order is called the f-score sequence of the competition. We raise the following problem: Which sequences of non-negative integers in non-decreasing order is a football sequence, that is the outcome of a soccer league competition. We model such a competition by an oriented graph with teams represented by vertices in which the teams play each other once, with an arc from team u to team v if and only if u defeats v. We obtain some necessary conditions for football sequences and some characterizations under restrictions.

Decomposition of a Complete Bipartite Multigraph into Arbitrary Cycle Sizes

In a graph G, let $$\mu _G(xy)$$ denote the number of edges between x and y in G. Let $$\lambda K_{v,u}$$ be the graph $$(V\cup U,E)$$ with $$|V|=v$$ , $$|U|=u$$ , and $$\begin{aligned} \mu _G{(xy)}={\left\{ \begin{array}{ll} \lambda &{}\text{ if }\,\, x\in \text{ and }\,\, y\in V \text{ or } \text{ if }\,\, x\in V \text{ and }\,\, y\in U 0 &{}\text{ otherwise. } \end{array}\right. } \end{aligned}$$ Let M be the sequence of non-negative integers $$m_1,m_2,\ldots ,m_n$$ . An (M)-cycle decomposition of a graph G is a partition of the edge set into cycles of lengths $$m_1,m_2,\ldots ,m_n$$ . In this paper, we establish some necessary and some sufficient conditions for the existence of an (M)-cycle decomposition of $$\lambda K_{v,u}$$ .

Solution to an extremal problem on bigraphic pairs with a Z 3-connected realization

Let S = (a 1, …, a m ; b 1, …, b n ), where a 1, …, a m and b 1, …, b n are two nonincreasing sequences of nonnegative integers. The pair S = (a 1, …, a m ; b 1, …, b n ) is said to be a bigraphic pair if there is a simple bipartite graph G = (X ∪ Y, E) such that a 1, …, a m and b 1, …, b n are the degrees of the vertices in X and Y, respectively. Let Z 3 be the cyclic group of order 3. Define σ(Z 3, m, n) to be the minimum integer k such that every bigraphic pair S = (a 1, …, a m ; b 1, …, b n ) with a m , b n ≥ 2 and σ(S) = a 1 + ⋯ + a m ≥ k has a Z 3-connected realization. For n = m, Yin [Discrete Math., 339, 2018—2026 (2016)] recently determined the values of σ(Z 3, m, m) for m ≥ 4. In this paper, we completely determine the values of σ(Z 3, m, n) for m ≥ n ≥ 4.

The condition for a sequence to be potentially $A_{L, M}$- graphic

The set of all non-increasing non-negative integer sequences $pi=(d_1‎, ‎d_2,ldots,d_n)$ is denoted by $NS_n$‎. ‎A sequence $piin NS_{n}$ is said to be graphic if it is the degree sequence of a simple graph $G$ on $n$ vertices‎, ‎and such a graph $G$ is called a realization of $pi$‎. ‎The set of all graphic sequences in $NS_{n}$ is denoted by $GS_{n}$‎. ‎The complete product split graph on $L‎ + ‎M$ vertices is denoted by $overline{S}_{L‎, ‎M}=K_{L} vee overline{K}_{M}$‎, ‎where $K_{L}$ and $K_{M}$ are complete graphs respectively on $L = sumlimits_{i = 1}^{p}r_{i}$ and $M = sumlimits_{i = 1}^{p}s_{i}$ vertices with $r_{i}$ and $s_{i}$ being integers‎. ‎Another split graph is denoted by $S_{L‎, ‎M} = overline{S}_{r_{1}‎, ‎s_{1}} veeoverline{S}_{r_{2}‎, ‎s_{2}} vee cdots vee overline{S}_{r_{p}‎, ‎s_{p}}= (K_{r_{1}} vee overline{K}_{s_{1}})vee (K_{r_{2}} vee overline{K}_{s_{2}})vee cdots vee (K_{r_{p}} vee overline{K}_{s_{p}})$‎. ‎A sequence $pi=(d_{1}‎, ‎d_{2},ldots,d_{n})$ is said to be potentially $S_{L‎, ‎M}$-graphic (respectively $overline{S}_{L‎, ‎M}$)-graphic if there is a realization $G$ of $pi$ containing $S_{L‎, ‎M}$ (respectively $overline{S}_{L‎, ‎M}$) as a subgraph‎. ‎If $pi$ has a realization $G$ containing $S_{L‎, ‎M}$ on those vertices having degrees $d_{1}‎, ‎d_{2},ldots,d_{L+M}$‎, ‎then $pi$ is potentially $A_{L‎, ‎M}$-graphic‎. ‎A non-increasing sequence of non-negative integers $pi = (d_{1}‎, ‎d_{2},ldots,d_{n})$ is potentially $A_{L‎, ‎M}$-graphic if and only if it is potentially $S_{L‎, ‎M}$-graphic‎. ‎In this paper‎, ‎we obtain the sufficient condition for a graphic sequence to be potentially $A_{L‎, ‎M}$-graphic and this result is a generalization of that given by J‎. ‎H‎. ‎Yin on split graphs‎.

An extremal problem on bigraphic pairs with an [formula omitted]-connected realization

Let S=(a1,…,am;b1,…,bn), where a1,…,am and b1,…,bn are two nonincreasing sequences of nonnegative integers. The pair S=(a1,…,am;b1,…,bn) is said to be a bigraphic pair if there is a simple bipartite graph G=(X∪Y,E) such that a1,…,am and b1,…,bn are the degrees of the vertices in X and Y, respectively. Let A be an (additive) Abelian group. We define σ(A,m,n) to be the minimum integer k such that every bigraphic pair S=(a1,…,am;b1,…,bn) with am,bn≥2 and σ(S)=a1+⋯+am≥k has an A-connected realization. In this paper, we determine the values of σ(Z3,m,m) for m≥4.

On the maximum values of the additive representation functions

Let [Formula: see text] and [Formula: see text] be infinite sequences of nonnegative integers. For a positive integer [Formula: see text] let [Formula: see text] denote the number of representations of [Formula: see text] as the sum of two terms from [Formula: see text]. Let [Formula: see text] denote the maximum value of [Formula: see text] up to [Formula: see text] and [Formula: see text] denote the distance of the sequences [Formula: see text] and [Formula: see text]. In this paper, we study the connection between [Formula: see text], [Formula: see text] and [Formula: see text]. We improve a result of Haddad and Helou about the Erdős–Turán conjecture.

Step graphs and their application to the organization of commodity flows in networks

The concepts of step graphs and networks in which the edges are divided into ordered classes (shortage classes) are introduced. Each path in such graphs is assigned a tuple (an ordered sequence of nonnegative integers). The tuples are compared lexicographically. The problem of finding the path with the minimum tuple according to this ordering is stated. Functionals are compared using the lexicographic rule. An algorithm for finding the optimal path is described, and the relationship between step networks and weighted networks is investigated. The proposed formalism is applied to the problem of filling a network subject to constraints imposed on communication line capacities with communication flows under the condition that requests for the organization of such flows arrive to the network sequentially in time and the filling strategy must maximize the number of satisfied requests. The idea of an algorithm for the approximate solution of the integer multicommodity problem that uses the concepts of step networks and sequential algorithms is proposed.

Gale-Ryser型刻划定理的一个推广

设及 是两个由非负整数构成的不增序列。如果存在一个简单X，Y-二部图使得X中的顶点的度分别为 且Y中的顶点的度分别为 ，那么称序列对 是二部可图的。如果二部可图且任何两个来自不同部集的顶点之间最多连有t条边，那么称是t-二部可图的。本文给出一个t-二部可图序列的刻划定理。事实上，该定理是Gale-Ryser型刻划定理的一个推广。 Let and be two non-increasing sequences of nonnegative integers. The pair is said to be bigraphic if there is a simple X,Y-bigraph such that the vertices of X have degrees and the vertices of Y have degrees . is said to be t-bigraphic if it is bigraphic and no two vertices from different partite sets are joined by more than t edges. In this paper, we give a characterization for to be t-bigraphic. In fact, it is a gen-eration of the Gale-Ryser type characterization theorem.

Depth and regularity of powers of sums of ideals

Given arbitrary homogeneous ideals $I$ and $J$ in polynomial rings $A$ and $B$ over a field $k$, we investigate the depth and the Castelnuovo-Mumford regularity of powers of the sum $I+J$ in $A \otimes_k B$ in terms of those of $I$ and $J$. Our results can be used to study the behavior of the depth and regularity functions of powers of an ideal. For instance, we show that such a depth function can take as its values any infinite non-increasing sequence of non-negative integers.

Notes on simplicial rook graphs

The simplicial rook graph $$\mathrm{SR}(m,n)$$SR(m,n) is the graph of which the vertices are the sequences of nonnegative integers of length m summing to n, where two such sequences are adjacent when they differ in precisely two places. We show that $$\mathrm{SR}(m,n)$$SR(m,n) has integral eigenvalues, and smallest eigenvalue $$s = \max \left( -n, -{m \atopwithdelims ()2}\right) $$s=max-n,-m2, and that this graph has a large part of its spectrum in common with the Johnson graph $$J(m+n-1,n)$$J(m+n-1,n). We determine the automorphism group and several other properties.

On additive representation functions

Let 𝒜 = {a1 &lt; a2 &lt; a3 &lt; ⋯ &lt; an &lt; ⋯} be an infinite sequence of nonnegative integers and let R2(n) = |{(i, j) : ai + aj = n; ai, aj ∈ 𝒜; i ≤ j}|. We define [Formula: see text]. We prove that if the L∞-norm of [Formula: see text] is small, then the L1-norm of [Formula: see text] is large.

Connection Coefficients Between Generalized Rising and Falling Factorial Bases

Let $${\mathcal{S} = (s_1, s_2, \ldots)}$$ be any sequence of nonnegative integers and let $${{S_{k} = \sum_{i=1}^k} {s_i}}$$ .We then define the falling (rising) factorials relative to $${\mathcal{S}}$$ by setting $${(x)\downarrow_{k, \mathcal{S}} = (x- S_{1})(x-S_2) \cdots (x-S_k)}$$ and $${(x)\uparrow_{k, \mathcal{S}}= (x+S_1)(x+S_{2}) \cdots (x+S_{k})}$$ if $${k \geq 1}$$ with $${(x)\downarrow_{0,\mathcal{S}}= (x)\uparrow_{0, \mathcal{S}}= 1}$$ . It follows that $${\{(x)\downarrow_{k,\mathcal{S}}\}_{k > 0}}$$ and $${\{(x)\uparrow_{k,\mathcal{S}}\}_{k > 0}}$$ are bases for the polynomial ring $${\mathbb{Q}[x]}$$ . We use a rook theory model due to Miceli and Remmel to give combinatorial interpretations for the connection coefficients between any two of the bases $${\{(x)\downarrow_{k,\mathcal{S}}\}_{k \geq 0}, \{(x)\uparrow_{k,\mathcal{S}}\}_{k \geq 0}, \{(x)\downarrow_{k,\mathcal{T}}\}_{k \geq 0}}$$ , and $${\{(x)\uparrow_{k, \mathcal{T}}\}_{k \geq 0}}$$ for any two sequences of nonnegative integers $${\mathcal{S} =(s_1, s_2, \ldots)}$$ and $${\mathcal{T} = (t_1, t_2, \ldots)}$$ . We also give two different q-analogues of such coefficients. Moreover, we use this rook model to give an alternative combinatorial interpretation of such coefficients in terms of certain pairs of colored permutations and set partitions with restricted insertion patterns.