Florian Bridoux, Aymeric Picard Marchetto, Adrien Richard
An automata network with $n$ components over a finite alphabet $Q$ of size $q$ is a discrete dynamical system described by the successive iterations of a function $f:Q^nto Q^n$. In most applications, the main parameter is the interaction graph of $f$: the digraph with vertex set $[n]$ that contains an arc from $j$ to $i$ if $f_i$ depends on input $j$. What can be said on the set $mathbb{G}(f)$ of the interaction graphs of the automata networks isomorphic to $f$? It seems that this simple question has never been studied. In a previous paper, we prove that the complete digraph $K_n$, with $n^2$ arcs, is universal in that $K_nin mathbb{G}(f)$ whenever $f$ is not constant nor the identity (and $ngeq 5$). In this paper, taking the opposite direction, we prove that there exists universal automata networks $f$, in that $mathbb{G}(f)$ contains all the digraphs on $[n]$, excepted the empty one. Actually, we prove that the presence of only three specific digraphs in $mathbb{G}(f)$ implies the universality of $f$, and we prove that this forces the alphabet size $q$ to have at least $n$ prime factors (with multiplicity). However, we prove that for any fixed $qgeq 3$, there exists almost universal functions, that is, functions $f:Q^nto Q^n$ such that the probability that a random digraph belongs to $mathbb{G}(f)$ tends to $1$ as $ntoinfty$. We do not know if this holds in the binary case $q=2$, providing only partial results.
{"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}
By a Christoffel matrix we mean a $ntimes n$ matrix corresponding to the lexicographic array of a Christoffel word of length $n.$ If $R$ is an integral domain, then the product of two Christoffel matrices over $R$ is commutative and is a Christoffel matrix over $R.$ Furthermore, if a Christoffel matrix over $R$ is invertible, then its inverse is a Christoffel matrix over $R.$ Consequently, the set $GC_n(R)$ of all $ntimes n$ invertible Christoffel matrices over $R$ forms an abelian subgroup of $GL_n(R).$ The subset of $GC_n(R)$ consisting all invertible Christoffel matrices having some element $a$ on the diagonal and $b$ elsewhere (with $a,b in R$ distinct) forms a subgroup $H$ of $GC_n(R).$ If $R$ is a field, then the quotient $GC_n(R)/H$ is isomorphic to $(Z/nZ)^times,$ the multiplicative group of integers modulo $n.$ It follows from this that for each finite field $F$ and each finite abelian group $G,$ there exists $ngeq 2$ and a faithful representation $Grightarrow GL_n(F)$ consisting entirely of $ntimes n$ (invertible) Christoffel matrices over $F.$ We find that $GC_n(F_2) simeq (Z/nZ)^times$ for $n$ odd and $GC_n(F_2) simeq Z/2Z times (Z/nZ)^times$ for $n$ even. As an application, we define an associative and commutative binary operation on the set of all ${0,1}$-Christoffel words of length $n$ which in turn induces an associative and commutative binary operation on ${0,1}$-central words of length $n-2.$
{"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}
An oriented multigraph is a directed multigraph without directed 2-cycles. Let ${rm fas}(D)$ denote the minimum size of a feedback arc set in an oriented multigraph $D$. The degree of a vertex is the sum of its out- and in-degrees. In several papers, upper bounds for ${rm fas}(D)$ were obtained for oriented multigraphs $D$ with maximum degree upper-bounded by a constant. Hanauer (2017) conjectured that ${rm fas}(D)le 2.5n/3$ for every oriented multigraph $D$ with $n$ vertices and maximum degree at most 5. We prove a strengthening of the conjecture: ${rm fas}(D)le m/3$ holds for every oriented multigraph $D$ with $m$ arcs and maximum degree at most 5. This bound is tight and improves a bound of Berger and Shor (1990,1997). It would be interesting to determine $c$ such that ${rm fas}(D)le cn$ for every oriented multigraph $D$ with $n$ vertices and maximum degree at most 5 such that the bound is tight. We show that $frac{5}{7}le c le frac{24}{29} < frac{2.5}{3}$.
{"title":"Upper bounds on minimum size of feedback arc set of directed multigraphs with bounded degree","authors":"Gregory Gutin, Hui Lei, Anders Yeo, Yacong Zhou","doi":"arxiv-2409.07680","DOIUrl":"https://doi.org/arxiv-2409.07680","url":null,"abstract":"An oriented multigraph is a directed multigraph without directed 2-cycles.\u0000Let ${rm fas}(D)$ denote the minimum size of a feedback arc set in an oriented\u0000multigraph $D$. The degree of a vertex is the sum of its out- and in-degrees.\u0000In several papers, upper bounds for ${rm fas}(D)$ were obtained for oriented\u0000multigraphs $D$ with maximum degree upper-bounded by a constant. Hanauer (2017)\u0000conjectured that ${rm fas}(D)le 2.5n/3$ for every oriented multigraph $D$\u0000with $n$ vertices and maximum degree at most 5. We prove a strengthening of the\u0000conjecture: ${rm fas}(D)le m/3$ holds for every oriented multigraph $D$ with\u0000$m$ arcs and maximum degree at most 5. This bound is tight and improves a bound\u0000of Berger and Shor (1990,1997). It would be interesting to determine $c$ such\u0000that ${rm fas}(D)le cn$ for every oriented multigraph $D$ with $n$ vertices\u0000and maximum degree at most 5 such that the bound is tight. We show that\u0000$frac{5}{7}le c le frac{24}{29} < frac{2.5}{3}$.","PeriodicalId":501407,"journal":{"name":"arXiv - MATH - Combinatorics","volume":"30 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142197598","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}
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
The MIT/IEEE/Amazon GraphChallenge encourages community approaches to developing new solutions for analyzing graphs and sparse data derived from social media, sensor feeds, and scientific data to discover relationships between events as they unfold in the field. The anonymized network sensing Graph Challenge seeks to enable large, open, community-based approaches to protecting networks. Many large-scale networking problems can only be solved with community access to very broad data sets with the highest regard for privacy and strong community buy-in. Such approaches often require community-based data sharing. In the broader networking community (commercial, federal, and academia) anonymized source-to-destination traffic matrices with standard data sharing agreements have emerged as a data product that can meet many of these requirements. This challenge provides an opportunity to highlight novel approaches for optimizing the construction and analysis of anonymized traffic matrices using over 100 billion network packets derived from the largest Internet telescope in the world (CAIDA). This challenge specifies the anonymization, construction, and analysis of these traffic matrices. A GraphBLAS reference implementation is provided, but the use of GraphBLAS is not required in this Graph Challenge. As with prior Graph Challenges the goal is to provide a well-defined context for demonstrating innovation. Graph Challenge participants are free to select (with accompanying explanation) the Graph Challenge elements that are appropriate for highlighting their innovations.
{"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}
Sean English, Anastasia Halfpap, Robert A. Krueger
Let $mathrm{ex}(n,H,mathcal{F})$ be the maximum number of copies of $H$ in an $n$-vertex graph which contains no copy of a graph from $mathcal{F}$. Thinking of $H$ and $mathcal{F}$ as fixed, we study the asymptotics of $mathrm{ex}(n,H,mathcal{F})$ in $n$. We say that a rational number $r$ is emph{realizable for $H$} if there exists a finite family $mathcal{F}$ such that $mathrm{ex}(n,H,mathcal{F}) = Theta(n^r)$. Using randomized algebraic constructions, Bukh and Conlon showed that every rational between $1$ and $2$ is realizable for $K_2$. We generalize their result to show that every rational between $1$ and $t$ is realizable for $K_t$, for all $t geq 2$. We also determine the realizable rationals for stars and note the connection to a related Sidorenko-type supersaturation problem.
{"title":"Rational exponents for cliques","authors":"Sean English, Anastasia Halfpap, Robert A. Krueger","doi":"arxiv-2409.08424","DOIUrl":"https://doi.org/arxiv-2409.08424","url":null,"abstract":"Let $mathrm{ex}(n,H,mathcal{F})$ be the maximum number of copies of $H$ in\u0000an $n$-vertex graph which contains no copy of a graph from $mathcal{F}$.\u0000Thinking of $H$ and $mathcal{F}$ as fixed, we study the asymptotics of\u0000$mathrm{ex}(n,H,mathcal{F})$ in $n$. We say that a rational number $r$ is\u0000emph{realizable for $H$} if there exists a finite family $mathcal{F}$ such\u0000that $mathrm{ex}(n,H,mathcal{F}) = Theta(n^r)$. Using randomized algebraic\u0000constructions, Bukh and Conlon showed that every rational between $1$ and $2$\u0000is realizable for $K_2$. We generalize their result to show that every rational\u0000between $1$ and $t$ is realizable for $K_t$, for all $t geq 2$. We also\u0000determine the realizable rationals for stars and note the connection to a\u0000related Sidorenko-type supersaturation problem.","PeriodicalId":501407,"journal":{"name":"arXiv - MATH - Combinatorics","volume":"12 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142255038","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}
In this expository note, we discuss a ``balls-and-urns'' probability puzzle posed by Daniel Litt.
在本说明中，我们将讨论丹尼尔-利特提出的一个 "瓮中捉鳖 "的概率难题。
{"title":"Daniel Litt's Probability Puzzle","authors":"Maura B. Paterson, Douglas R. Stinson","doi":"arxiv-2409.08094","DOIUrl":"https://doi.org/arxiv-2409.08094","url":null,"abstract":"In this expository note, we discuss a ``balls-and-urns'' probability puzzle\u0000posed by Daniel Litt.","PeriodicalId":501407,"journal":{"name":"arXiv - MATH - Combinatorics","volume":"68 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142197599","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}
We give a deterministic polynomial-time approximation scheme (FPTAS) for the volume of the truncated fractional matching polytope for graphs of maximum degree $Delta$, where the truncation is by restricting each variable to the interval $[0,frac{1+delta}{Delta}]$, and $deltale frac{C}{Delta}$ for some constant $C>0$. We also generalise our result to the fractional matching polytope for hypergraphs of maximum degree $Delta$ and maximum hyperedge size $k$, truncated by $[0,frac{1+delta}{Delta}]$ as well, where $deltale CDelta^{-frac{2k-3}{k-1}}k^{-1}$ for some constant $C>0$. The latter result generalises both the first result for graphs (when $k=2$), and a result by Bencs and Regts (2024) for the truncated independence polytope (when $Delta=2$). Our approach is based on the cluster expansion technique.
{"title":"Deterministic approximation for the volume of the truncated fractional matching polytope","authors":"Heng Guo, Vishvajeet N","doi":"arxiv-2409.07283","DOIUrl":"https://doi.org/arxiv-2409.07283","url":null,"abstract":"We give a deterministic polynomial-time approximation scheme (FPTAS) for the\u0000volume of the truncated fractional matching polytope for graphs of maximum\u0000degree $Delta$, where the truncation is by restricting each variable to the\u0000interval $[0,frac{1+delta}{Delta}]$, and $deltale frac{C}{Delta}$ for\u0000some constant $C>0$. We also generalise our result to the fractional matching\u0000polytope for hypergraphs of maximum degree $Delta$ and maximum hyperedge size\u0000$k$, truncated by $[0,frac{1+delta}{Delta}]$ as well, where $deltale\u0000CDelta^{-frac{2k-3}{k-1}}k^{-1}$ for some constant $C>0$. The latter result\u0000generalises both the first result for graphs (when $k=2$), and a result by\u0000Bencs and Regts (2024) for the truncated independence polytope (when\u0000$Delta=2$). Our approach is based on the cluster expansion technique.","PeriodicalId":501407,"journal":{"name":"arXiv - MATH - Combinatorics","volume":"54 23 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142197612","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}
In this paper, we find some bounds for the Sombor index of the graph G by triangle inequality, arithmetic index, geometric index, forgotten index (F(G)), arithmetic-geometric (AG) index, geometric-arithmetic (GA) index, symmetric division deg index (SDD(G)) and some central and dispersion indices. The bounds could state estimated values and error intervals of the Sombor index to show limits of accuracy. The error intervals are written as inequalities.
本文通过三角形不等式、算术指数、几何指数、遗忘指数（F(G)）、算术几何指数（AG）、几何算术指数（GA）、对称分割度指数（SDD(G)）以及一些中心指数和离散指数，为图 G 的松博指数找到了一些约束。边界可以说明松博指数的估计值和误差区间，以显示精确度的极限。误差区间用不等式表示。
{"title":"New boundes for Sombor index of Graphs","authors":"Maryam Mohammadi, Hasan Barzegar","doi":"arxiv-2409.07099","DOIUrl":"https://doi.org/arxiv-2409.07099","url":null,"abstract":"In this paper, we find some bounds for the Sombor index of the graph G by\u0000triangle inequality, arithmetic index, geometric index, forgotten index (F(G)),\u0000arithmetic-geometric (AG) index, geometric-arithmetic (GA) index, symmetric\u0000division deg index (SDD(G)) and some central and dispersion indices. The bounds\u0000could state estimated values and error intervals of the Sombor index to show\u0000limits of accuracy. The error intervals are written as inequalities.","PeriodicalId":501407,"journal":{"name":"arXiv - MATH - Combinatorics","volume":"2 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142197606","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}
The eccentricity matrix of a simple connected graph is obtained from the distance matrix by only keeping the largest distances for each row and each column, whereas the remaining entries become zero. This matrix is also called the anti-adjacency matrix, since the adjacency matrix can also be obtained from the distance matrix but this time by keeping only the entries equal to $1$. It is known that, for $lambda notin {-1,0}$ and a fixed $iin mathbb{N}$, there is only a finite number of graphs with $n$ vertices having $lambda$ as an eigenvalue of multiplicity $n-i$ on the spectrum of the adjacency matrix. This phenomenon motivates researchers to consider the graphs has a large multiplicity of an eigenvalue in the spectrum of the eccentricity matrix, for example, the eigenvalue $-2$ [X. Gao, Z. Stani'{c}, J.F. Wang, Grahps with large multiplicity of $-2$ in the spectrum of the eccentricity matrix, Discrete Mathematics, 347 (2024) 114038]. In this paper, we characterize the connected graphs with $n$ vertices having $-1$ as an eigenvalue of multiplicity $n-i$ $(ileq5)$ in the spectrum of the eccentricity matrix. Our results also become meaningful in the framework of the median eigenvalue problem [B. Mohar, Median eigenvalues and the HOMO-LUMO index of graphs, Journal of Combinatorial Theory Series B, 112 (2015) 78-92].
简单连通图的偏心矩阵是从距离矩阵中得到的，方法是只保留每行和每列的最大距离，而其余条目为零。这个矩阵也被称为反邻接矩阵，因为邻接矩阵也可以从距离矩阵中得到，但这次只保留等于 1$ 的条目。众所周知，对于$lambda notin {-1,0}$和固定的$iin mathbb{N}$来说，只有有限数量的具有$n$顶点的图具有$lambda$作为邻接矩阵谱上乘数为$n-i$的特征值。这一现象促使研究人员考虑在偏心矩阵谱中具有大倍率特征值的图，例如，特征值 $-2$ [X. Gao, Z. Stanich.Gao, Z. Stani'{c}, J.F. Wang, Grahps withlarge multiplicity of $-2$ in the spectrum of the eccentricity matrix, DiscreteMathematics, 347 (2024) 114038]。在本文中，我们描述了在偏心矩阵谱中具有$-1$作为乘数$n-i$$(ileq5)$特征值的具有$n$顶点的连通图的特征。我们的结果在中值特征值问题的框架中也变得有意义[B. Mohar, Medianeigenvalue [中值特征值]]。Mohar, Medianeigenvalues and the HOMO-LUMO index of graphs, Journal of Combinatorial TheorySeries B, 112 (2015) 78-92].
{"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}
Given a graph $G$, its $2$-color Tur'{a}n number $mathrm{ex}^{(2)}(n,G)$ is the largest number of edges in an $n$-vertex graph whose edges can be colored with two colors avoiding a monochromatic copy of $G$. Let $pi^{(2)}(G)=lim_{ntoinfty}mathrm{ex}^{(2)}(n,G)/binom{n}{2}$ be the $2$-color Tur'{a}n density of $G$. What real numbers in the interval $(0,1)$ are realized as the $2$-color Tur'{a}n density of some graph? It is known that $pi^{(2)}(G)=1-(R_{chi}(G)-1)^{-1}$, where $R_{chi}(G)$ is the chromatic Ramsey number of $G$. However, determining specific values of $R_{chi}(G)$ is challenging. Burr, ErdH{o}s, and Lov'{a}sz showed that $(k-1)^2+1leqslant{R_{chi}(G)}leqslant{R(k)}$, for any $k$-chromatic graph $G$, where $R(k)$ is the classical Ramsey number. The upper bound here can be attained by a clique and the lower bound is achieved by a graph constructed by Zhu. To the best of our knowledge, there are no other, besides these two, known values of $R_{chi}(G)$ among $k$-chromatic graphs $G$ for general $k$. In this paper we prove that there are $Omega(k)$ different values of $R_{chi}(G)$ among $k$-chromatic graphs $G$. In addition, we determine a new value for the chromatic Ramsey numbers of $4$-chromatic graphs. This sheds more light into the possible $2$-color Tur'{a}n densities of graphs.
{"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}
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