arXiv - MATH - Combinatorics最新文献_第7页

Interaction graphs of isomorphic automata networks II: universal dynamics 同构自动机网络的交互图 II：通用动力学

arXiv - MATH - Combinatorics Pub Date : 2024-09-12 DOI: arxiv-2409.08041

Florian Bridoux, Aymeric Picard Marchetto, Adrien Richard

{"title":"Interaction graphs of isomorphic automata networks II: universal dynamics","authors":"Florian Bridoux, Aymeric Picard Marchetto, Adrien Richard","doi":"arxiv-2409.08041","DOIUrl":"https://doi.org/arxiv-2409.08041","url":null,"abstract":"An automata network with $n$ components over a finite alphabet $Q$ of size\u0000$q$ is a discrete dynamical system described by the successive iterations of a\u0000function $f:Q^nto Q^n$. In most applications, the main parameter is the\u0000interaction graph of $f$: the digraph with vertex set $[n]$ that contains an\u0000arc from $j$ to $i$ if $f_i$ depends on input $j$. What can be said on the set\u0000$mathbb{G}(f)$ of the interaction graphs of the automata networks isomorphic\u0000to $f$? It seems that this simple question has never been studied. In a\u0000previous paper, we prove that the complete digraph $K_n$, with $n^2$ arcs, is\u0000universal in that $K_nin mathbb{G}(f)$ whenever $f$ is not constant nor the\u0000identity (and $ngeq 5$). In this paper, taking the opposite direction, we\u0000prove that there exists universal automata networks $f$, in that\u0000$mathbb{G}(f)$ contains all the digraphs on $[n]$, excepted the empty one.\u0000Actually, we prove that the presence of only three specific digraphs in\u0000$mathbb{G}(f)$ implies the universality of $f$, and we prove that this forces\u0000the alphabet size $q$ to have at least $n$ prime factors (with multiplicity).\u0000However, we prove that for any fixed $qgeq 3$, there exists almost universal\u0000functions, that is, functions $f:Q^nto Q^n$ such that the probability that a\u0000random digraph belongs to $mathbb{G}(f)$ tends to $1$ as $ntoinfty$. We do\u0000not know if this holds in the binary case $q=2$, providing only partial\u0000results.","PeriodicalId":501407,"journal":{"name":"arXiv - MATH - Combinatorics","volume":"34 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142197594","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

On Christoffel words & their lexicographic array 关于 Christoffel 词及其词典阵列

arXiv - MATH - Combinatorics Pub Date : 2024-09-12 DOI: arxiv-2409.07974

Luca Q. Zamboni

{"title":"On Christoffel words & their lexicographic array","authors":"Luca Q. Zamboni","doi":"arxiv-2409.07974","DOIUrl":"https://doi.org/arxiv-2409.07974","url":null,"abstract":"By a Christoffel matrix we mean a $ntimes n$ matrix corresponding to the\u0000lexicographic array of a Christoffel word of length $n.$ If $R$ is an integral\u0000domain, then the product of two Christoffel matrices over $R$ is commutative\u0000and is a Christoffel matrix over $R.$ Furthermore, if a Christoffel matrix over\u0000$R$ is invertible, then its inverse is a Christoffel matrix over $R.$\u0000Consequently, the set $GC_n(R)$ of all $ntimes n$ invertible Christoffel\u0000matrices over $R$ forms an abelian subgroup of $GL_n(R).$ The subset of\u0000$GC_n(R)$ consisting all invertible Christoffel matrices having some element\u0000$a$ on the diagonal and $b$ elsewhere (with $a,b in R$ distinct) forms a\u0000subgroup $H$ of $GC_n(R).$ If $R$ is a field, then the quotient $GC_n(R)/H$ is\u0000isomorphic to $(Z/nZ)^times,$ the multiplicative group of integers modulo\u0000$n.$ It follows from this that for each finite field $F$ and each finite\u0000abelian group $G,$ there exists $ngeq 2$ and a faithful representation\u0000$Grightarrow GL_n(F)$ consisting entirely of $ntimes n$ (invertible)\u0000Christoffel matrices over $F.$ We find that $GC_n(F_2) simeq (Z/nZ)^times$\u0000for $n$ odd and $GC_n(F_2) simeq Z/2Z times (Z/nZ)^times$ for $n$ even.\u0000As an application, we define an associative and commutative binary operation on\u0000the set of all ${0,1}$-Christoffel words of length $n$ which in turn induces\u0000an associative and commutative binary operation on ${0,1}$-central words of\u0000length $n-2.$","PeriodicalId":501407,"journal":{"name":"arXiv - MATH - Combinatorics","volume":"27 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142197595","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

Upper bounds on minimum size of feedback arc set of directed multigraphs with bounded degree 有界有向多图反馈弧集最小尺寸的上界

arXiv - MATH - Combinatorics Pub Date : 2024-09-12 DOI: arxiv-2409.07680

Gregory Gutin, Hui Lei, Anders Yeo, Yacong Zhou

引用次数: 0

Anonymized Network Sensing Graph Challenge 匿名网络传感图挑战

arXiv - MATH - Combinatorics Pub Date : 2024-09-12 DOI: arxiv-2409.08115

Hayden Jananthan, Michael Jones, William Arcand, David Bestor, William Bergeron, Daniel Burrill, Aydin Buluc, Chansup Byun, Timothy Davis, Vijay Gadepally, Daniel Grant, Michael Houle, Matthew Hubbell, Piotr Luszczek, Peter Michaleas, Lauren Milechin, Chasen Milner, Guillermo Morales, Andrew Morris, Julie Mullen, Ritesh Patel, Alex Pentland, Sandeep Pisharody, Andrew Prout, Albert Reuther, Antonio Rosa, Gabriel Wachman, Charles Yee, Jeremy Kepner

{"title":"Anonymized Network Sensing Graph Challenge","authors":"Hayden Jananthan, Michael Jones, William Arcand, David Bestor, William Bergeron, Daniel Burrill, Aydin Buluc, Chansup Byun, Timothy Davis, Vijay Gadepally, Daniel Grant, Michael Houle, Matthew Hubbell, Piotr Luszczek, Peter Michaleas, Lauren Milechin, Chasen Milner, Guillermo Morales, Andrew Morris, Julie Mullen, Ritesh Patel, Alex Pentland, Sandeep Pisharody, Andrew Prout, Albert Reuther, Antonio Rosa, Gabriel Wachman, Charles Yee, Jeremy Kepner","doi":"arxiv-2409.08115","DOIUrl":"https://doi.org/arxiv-2409.08115","url":null,"abstract":"The MIT/IEEE/Amazon GraphChallenge encourages community approaches to\u0000developing new solutions for analyzing graphs and sparse data derived from\u0000social media, sensor feeds, and scientific data to discover relationships\u0000between events as they unfold in the field. The anonymized network sensing\u0000Graph Challenge seeks to enable large, open, community-based approaches to\u0000protecting networks. Many large-scale networking problems can only be solved\u0000with community access to very broad data sets with the highest regard for\u0000privacy and strong community buy-in. Such approaches often require\u0000community-based data sharing. In the broader networking community (commercial,\u0000federal, and academia) anonymized source-to-destination traffic matrices with\u0000standard data sharing agreements have emerged as a data product that can meet\u0000many of these requirements. This challenge provides an opportunity to highlight\u0000novel approaches for optimizing the construction and analysis of anonymized\u0000traffic matrices using over 100 billion network packets derived from the\u0000largest Internet telescope in the world (CAIDA). This challenge specifies the\u0000anonymization, construction, and analysis of these traffic matrices. A\u0000GraphBLAS reference implementation is provided, but the use of GraphBLAS is not\u0000required in this Graph Challenge. As with prior Graph Challenges the goal is to\u0000provide a well-defined context for demonstrating innovation. Graph Challenge\u0000participants are free to select (with accompanying explanation) the Graph\u0000Challenge elements that are appropriate for highlighting their innovations.","PeriodicalId":501407,"journal":{"name":"arXiv - MATH - Combinatorics","volume":"29 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142197601","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

Rational exponents for cliques 小群的有理指数

arXiv - MATH - Combinatorics Pub Date : 2024-09-12 DOI: arxiv-2409.08424

Sean English, Anastasia Halfpap, Robert A. Krueger

引用次数: 0

Daniel Litt's Probability Puzzle 丹尼尔-利特的概率谜题

arXiv - MATH - Combinatorics Pub Date : 2024-09-12 DOI: arxiv-2409.08094

Maura B. Paterson, Douglas R. Stinson

引用次数: 0

Deterministic approximation for the volume of the truncated fractional matching polytope 截断分数匹配多面体体积的确定性近似值

arXiv - MATH - Combinatorics Pub Date : 2024-09-11 DOI: arxiv-2409.07283

Heng Guo, Vishvajeet N

引用次数: 0

New boundes for Sombor index of Graphs 图的 Sombor 指数的新界限

arXiv - MATH - Combinatorics Pub Date : 2024-09-11 DOI: arxiv-2409.07099

Maryam Mohammadi, Hasan Barzegar

引用次数: 0

Connected graphs with large multiplicity of $-1$ in the spectrum of the eccentricity matrix 偏心矩阵频谱中具有 $-1$ 大倍率的连通图形

arXiv - MATH - Combinatorics Pub Date : 2024-09-11 DOI: arxiv-2409.07198

Xinghui Zhao, Lihua You

{"title":"Connected graphs with large multiplicity of $-1$ in the spectrum of the eccentricity matrix","authors":"Xinghui Zhao, Lihua You","doi":"arxiv-2409.07198","DOIUrl":"https://doi.org/arxiv-2409.07198","url":null,"abstract":"The eccentricity matrix of a simple connected graph is obtained from the\u0000distance matrix by only keeping the largest distances for each row and each\u0000column, whereas the remaining entries become zero. This matrix is also called\u0000the anti-adjacency matrix, since the adjacency matrix can also be obtained from\u0000the distance matrix but this time by keeping only the entries equal to $1$. It\u0000is known that, for $lambda notin {-1,0}$ and a fixed $iin mathbb{N}$,\u0000there is only a finite number of graphs with $n$ vertices having $lambda$ as\u0000an eigenvalue of multiplicity $n-i$ on the spectrum of the adjacency matrix.\u0000This phenomenon motivates researchers to consider the graphs has a large\u0000multiplicity of an eigenvalue in the spectrum of the eccentricity matrix, for\u0000example, the eigenvalue $-2$ [X. Gao, Z. Stani'{c}, J.F. Wang, Grahps with\u0000large multiplicity of $-2$ in the spectrum of the eccentricity matrix, Discrete\u0000Mathematics, 347 (2024) 114038]. In this paper, we characterize the connected\u0000graphs with $n$ vertices having $-1$ as an eigenvalue of multiplicity $n-i$\u0000$(ileq5)$ in the spectrum of the eccentricity matrix. Our results also become\u0000meaningful in the framework of the median eigenvalue problem [B. Mohar, Median\u0000eigenvalues and the HOMO-LUMO index of graphs, Journal of Combinatorial Theory\u0000Series B, 112 (2015) 78-92].","PeriodicalId":501407,"journal":{"name":"arXiv - MATH - Combinatorics","volume":"11 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142197604","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

Chromatic Ramsey numbers and two-color Turán densities 色度拉姆齐数和双色图兰密度

arXiv - MATH - Combinatorics Pub Date : 2024-09-11 DOI: arxiv-2409.07535

Maria Axenovich, Simon Gaa, Dingyuan Liu

{"title":"Chromatic Ramsey numbers and two-color Turán densities","authors":"Maria Axenovich, Simon Gaa, Dingyuan Liu","doi":"arxiv-2409.07535","DOIUrl":"https://doi.org/arxiv-2409.07535","url":null,"abstract":"Given a graph $G$, its $2$-color Tur'{a}n number $mathrm{ex}^{(2)}(n,G)$ is\u0000the largest number of edges in an $n$-vertex graph whose edges can be colored\u0000with two colors avoiding a monochromatic copy of $G$. Let\u0000$pi^{(2)}(G)=lim_{ntoinfty}mathrm{ex}^{(2)}(n,G)/binom{n}{2}$ be the\u0000$2$-color Tur'{a}n density of $G$. What real numbers in the interval $(0,1)$\u0000are realized as the $2$-color Tur'{a}n density of some graph? It is known that\u0000$pi^{(2)}(G)=1-(R_{chi}(G)-1)^{-1}$, where $R_{chi}(G)$ is the chromatic\u0000Ramsey number of $G$. However, determining specific values of $R_{chi}(G)$ is\u0000challenging. Burr, ErdH{o}s, and Lov'{a}sz showed that\u0000$(k-1)^2+1leqslant{R_{chi}(G)}leqslant{R(k)}$, for any $k$-chromatic graph\u0000$G$, where $R(k)$ is the classical Ramsey number. The upper bound here can be\u0000attained by a clique and the lower bound is achieved by a graph constructed by\u0000Zhu. To the best of our knowledge, there are no other, besides these two, known\u0000values of $R_{chi}(G)$ among $k$-chromatic graphs $G$ for general $k$. In this\u0000paper we prove that there are $Omega(k)$ different values of $R_{chi}(G)$\u0000among $k$-chromatic graphs $G$. In addition, we determine a new value for the\u0000chromatic Ramsey numbers of $4$-chromatic graphs. This sheds more light into\u0000the possible $2$-color Tur'{a}n densities of graphs.","PeriodicalId":501407,"journal":{"name":"arXiv - MATH - Combinatorics","volume":"204 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142197600","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0