Pub Date : 2024-11-16DOI: 10.1016/j.disc.2024.114326
Thiago Assis, Gabriel Coutinho, Emanuel Juliano
The Grundy number of a graph is the minimum number of colors needed to properly color the graph using the first-fit greedy algorithm regardless of the initial vertex ordering. Computing the Grundy number of a graph is an NP-Hard problem. There is a characterization in terms of induced subgraphs: a graph has a Grundy number at least k if and only if it contains a k-atom. In this paper, using properties of the matching polynomial, we determine the smallest possible largest eigenvalue of a k-atom. With this result, we present an upper bound for the Grundy number of a graph in terms of the largest eigenvalue of its adjacency matrix. We also present another upper bound using the largest eigenvalue and the size of the graph. Our bounds are asymptotically tight for some infinite families of graphs and provide improvements on the known bounds for the Grundy number of sparse random graphs.
图形的格兰迪数是指在不考虑初始顶点排序的情况下，使用第一拟合贪婪算法为图形正确着色所需的最少颜色数。计算图形的格兰迪数是一个 NP-Hard（近乎困难）问题。从诱导子图的角度来看，有这样一个特征：当且仅当一个图包含一个 k 原子时，该图的格兰迪数至少为 k。在本文中，我们利用匹配多项式的特性，确定了 k 原子的最小最大特征值。有了这一结果，我们根据图的邻接矩阵的最大特征值提出了图的格兰迪数上限。我们还利用最大特征值和图的大小提出了另一个上限。对于某些无限图族，我们的上界是渐近紧密的，并且改进了稀疏随机图的格兰迪数的已知上界。
{"title":"Spectral upper bounds for the Grundy number of a graph","authors":"Thiago Assis, Gabriel Coutinho, Emanuel Juliano","doi":"10.1016/j.disc.2024.114326","DOIUrl":"10.1016/j.disc.2024.114326","url":null,"abstract":"<div><div>The Grundy number of a graph is the minimum number of colors needed to properly color the graph using the first-fit greedy algorithm regardless of the initial vertex ordering. Computing the Grundy number of a graph is an NP-Hard problem. There is a characterization in terms of induced subgraphs: a graph has a Grundy number at least k if and only if it contains a <em>k</em>-atom. In this paper, using properties of the matching polynomial, we determine the smallest possible largest eigenvalue of a <em>k</em>-atom. With this result, we present an upper bound for the Grundy number of a graph in terms of the largest eigenvalue of its adjacency matrix. We also present another upper bound using the largest eigenvalue and the size of the graph. Our bounds are asymptotically tight for some infinite families of graphs and provide improvements on the known bounds for the Grundy number of sparse random graphs.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 3","pages":"Article 114326"},"PeriodicalIF":0.7,"publicationDate":"2024-11-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142661897","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-11-16DOI: 10.1016/j.disc.2024.114330
Daniel R. Hawtin
A t-fold packing of a projective space is a collection of line-spreads such that each line of occurs in precisely t spreads in . A t-fold packing is transitive if a subgroup of preserves and acts transitively on . We give a construction for a transitive -fold packing of , where , for any odd positive integers n and k, such that . This generalises a construction of Baker from 1976 for the case .
如果 PΓLn+1(q)的一个子群保存并在 P 上起传递作用，那么一个 t 折叠包装 P 就是传递性的。我们给出了一个 PGn(q)的传递性 (q-1)-fold 包装的构造，其中 q=2k, 对于任何奇数正整数 n 和 k，使得 n⩾3 。这概括了贝克 1976 年针对 q=2 情况的构造。
{"title":"Transitive (q − 1)-fold packings of PGn(q)","authors":"Daniel R. Hawtin","doi":"10.1016/j.disc.2024.114330","DOIUrl":"10.1016/j.disc.2024.114330","url":null,"abstract":"<div><div>A <em>t-fold packing</em> of a projective space <span><math><msub><mrow><mi>PG</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><mi>q</mi><mo>)</mo></math></span> is a collection <span><math><mi>P</mi></math></span> of line-spreads such that each line of <span><math><msub><mrow><mi>PG</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><mi>q</mi><mo>)</mo></math></span> occurs in precisely <em>t</em> spreads in <span><math><mi>P</mi></math></span>. A <em>t</em>-fold packing <span><math><mi>P</mi></math></span> is <em>transitive</em> if a subgroup of <span><math><msub><mrow><mi>P</mi><mi>Γ</mi><mi>L</mi></mrow><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>(</mo><mi>q</mi><mo>)</mo></math></span> preserves and acts transitively on <span><math><mi>P</mi></math></span>. We give a construction for a transitive <span><math><mo>(</mo><mi>q</mi><mo>−</mo><mn>1</mn><mo>)</mo></math></span>-fold packing of <span><math><msub><mrow><mi>PG</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><mi>q</mi><mo>)</mo></math></span>, where <span><math><mi>q</mi><mo>=</mo><msup><mrow><mn>2</mn></mrow><mrow><mi>k</mi></mrow></msup></math></span>, for any odd positive integers <em>n</em> and <em>k</em>, such that <span><math><mi>n</mi><mo>⩾</mo><mn>3</mn></math></span>. This generalises a construction of Baker from 1976 for the case <span><math><mi>q</mi><mo>=</mo><mn>2</mn></math></span>.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 3","pages":"Article 114330"},"PeriodicalIF":0.7,"publicationDate":"2024-11-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142661898","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-11-14DOI: 10.1016/j.disc.2024.114319
Cristina Ballantine , Brooke Feigon
The work of Andrews and Merca on the truncated Euler's pentagonal number theorem led to a resurgence in research on truncated theta series identities. In particular, Yee proved a truncated version of the Jacobi Triple Product (JTP) identity. Recently, Merca conjectured a stronger form of the truncated JTP identity. In this article we prove the first three cases of the conjecture and several related truncated identities. We prove combinatorially an identity related to the JTP identity which in particular cases reduces to identities conjectured by Merca and proved analytically by Krattenthaler, Merca and Radu. Moreover, we introduce a new combinatorial interpretation for the number of distinct 5-regular partitions of n.
{"title":"Truncated theta series related to the Jacobi Triple Product identity","authors":"Cristina Ballantine , Brooke Feigon","doi":"10.1016/j.disc.2024.114319","DOIUrl":"10.1016/j.disc.2024.114319","url":null,"abstract":"<div><div>The work of Andrews and Merca on the truncated Euler's pentagonal number theorem led to a resurgence in research on truncated theta series identities. In particular, Yee proved a truncated version of the Jacobi Triple Product (JTP) identity. Recently, Merca conjectured a stronger form of the truncated JTP identity. In this article we prove the first three cases of the conjecture and several related truncated identities. We prove combinatorially an identity related to the JTP identity which in particular cases reduces to identities conjectured by Merca and proved analytically by Krattenthaler, Merca and Radu. Moreover, we introduce a new combinatorial interpretation for the number of distinct 5-regular partitions of <em>n</em>.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 2","pages":"Article 114319"},"PeriodicalIF":0.7,"publicationDate":"2024-11-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142655715","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-11-13DOI: 10.1016/j.disc.2024.114322
Stefan Mitrović, Tanja Stojadinović
We introduce two classes of graphs - suns and dumbbells, both with few variations and explore their chromatic symmetric function and its e-positivity. We also give many connections of these two classes with other classes of connected graphs.
我们介绍了两类图--太阳图和哑铃图，它们都有很少的变化，并探讨了它们的色度对称函数及其 e 正性。我们还给出了这两类图与其他连通图的许多联系。
{"title":"The e−positivity of some new classes of graphs","authors":"Stefan Mitrović, Tanja Stojadinović","doi":"10.1016/j.disc.2024.114322","DOIUrl":"10.1016/j.disc.2024.114322","url":null,"abstract":"<div><div>We introduce two classes of graphs - suns and dumbbells, both with few variations and explore their chromatic symmetric function and its <em>e</em>-positivity. We also give many connections of these two classes with other classes of connected graphs.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 3","pages":"Article 114322"},"PeriodicalIF":0.7,"publicationDate":"2024-11-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142661895","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-11-13DOI: 10.1016/j.disc.2024.114317
Qianghui Guo , Yinglie Jin , Lisa Hui Sun , Hang Yang , Jie Yang
Combinatorial enumeration of various RNA secondary structures and protein contact maps is of significant interest for both combinatorialists and computational biologists. Numerous results have been obtained, most of which are in terms of generating functions, recurrences or asymptotic formulas, few are of explicit formulas. This paper is mainly concerned with finding explicit enumeration formulas related to m-regular simple stacks, a classic combinatorial model for RNA secondary structures. By using the theories of noncrossing matching and Dyck path, we obtain explicit enumeration formulas for m-regular simple stacks with statistics on arcs, hairpins, components and visible vertices. The results can reduce to some classic formulas like Schmitt and Waterman's closed form formula for the number of RNA secondary structures. Furthermore, we study the enumeration of enhanced m-regular simple stacks, stimulated by the study of protein contact maps, in which the upper bound of the degrees of the two terminal vertices is relaxed to two, explicit formulas are obtained.
{"title":"Explicit enumeration formulas for m-regular simple stacks","authors":"Qianghui Guo , Yinglie Jin , Lisa Hui Sun , Hang Yang , Jie Yang","doi":"10.1016/j.disc.2024.114317","DOIUrl":"10.1016/j.disc.2024.114317","url":null,"abstract":"<div><div>Combinatorial enumeration of various RNA secondary structures and protein contact maps is of significant interest for both combinatorialists and computational biologists. Numerous results have been obtained, most of which are in terms of generating functions, recurrences or asymptotic formulas, few are of explicit formulas. This paper is mainly concerned with finding explicit enumeration formulas related to <em>m</em>-regular simple stacks, a classic combinatorial model for RNA secondary structures. By using the theories of noncrossing matching and Dyck path, we obtain explicit enumeration formulas for <em>m</em>-regular simple stacks with statistics on arcs, hairpins, components and visible vertices. The results can reduce to some classic formulas like Schmitt and Waterman's closed form formula for the number of RNA secondary structures. Furthermore, we study the enumeration of enhanced <em>m</em>-regular simple stacks, stimulated by the study of protein contact maps, in which the upper bound of the degrees of the two terminal vertices is relaxed to two, explicit formulas are obtained.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 3","pages":"Article 114317"},"PeriodicalIF":0.7,"publicationDate":"2024-11-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142661894","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-11-13DOI: 10.1016/j.disc.2024.114324
E.J. Cheon , S.J. Kim , W. Kuranaka , T. Maruta
A fundamental problem in coding theory is to find the exact value , the minimum length n for which an code exists for given and d. The code of length is called length optimal. Finding length optimal codes presents the most interesting problem in optimal linear codes, because length optimal codes are simultaneously distance optimal and dimension optimal. In this article, we focus on finding 5-dimensional length optimal codes. We prove the nonexistence of 5-dimensional Griesmer code, and it is proved for with and with .
{"title":"On the minimum length of linear codes of dimension 5","authors":"E.J. Cheon , S.J. Kim , W. Kuranaka , T. Maruta","doi":"10.1016/j.disc.2024.114324","DOIUrl":"10.1016/j.disc.2024.114324","url":null,"abstract":"<div><div>A fundamental problem in coding theory is to find the exact value <span><math><msub><mrow><mi>n</mi></mrow><mrow><mi>q</mi></mrow></msub><mo>(</mo><mi>k</mi><mo>,</mo><mi>d</mi><mo>)</mo></math></span>, the minimum length <em>n</em> for which an <span><math><msub><mrow><mo>[</mo><mi>n</mi><mo>,</mo><mi>k</mi><mo>,</mo><mi>d</mi><mo>]</mo></mrow><mrow><mi>q</mi></mrow></msub></math></span> code exists for given <span><math><mi>q</mi><mo>,</mo><mi>k</mi></math></span> and <em>d</em>. The code of length <span><math><msub><mrow><mi>n</mi></mrow><mrow><mi>q</mi></mrow></msub><mo>(</mo><mi>k</mi><mo>,</mo><mi>d</mi><mo>)</mo></math></span> is called length optimal. Finding length optimal codes presents the most interesting problem in optimal linear codes, because length optimal codes are simultaneously distance optimal and dimension optimal. In this article, we focus on finding 5-dimensional length optimal codes. We prove the nonexistence of 5-dimensional Griesmer code, and it is proved <span><math><msub><mrow><mi>n</mi></mrow><mrow><mi>q</mi></mrow></msub><mo>(</mo><mn>5</mn><mo>,</mo><mi>d</mi><mo>)</mo><mo>=</mo><msub><mrow><mi>g</mi></mrow><mrow><mi>q</mi></mrow></msub><mo>(</mo><mn>5</mn><mo>,</mo><mi>d</mi><mo>)</mo><mo>+</mo><mn>1</mn></math></span> for <span><math><mn>3</mn><msup><mrow><mi>q</mi></mrow><mrow><mn>4</mn></mrow></msup><mo>−</mo><mn>4</mn><msup><mrow><mi>q</mi></mrow><mrow><mn>3</mn></mrow></msup><mo>−</mo><mi>a</mi><mi>q</mi><mo>+</mo><mn>1</mn><mo>≤</mo><mi>d</mi><mo>≤</mo><mn>3</mn><msup><mrow><mi>q</mi></mrow><mrow><mn>4</mn></mrow></msup><mo>−</mo><mn>4</mn><msup><mrow><mi>q</mi></mrow><mrow><mn>3</mn></mrow></msup><mo>−</mo><mi>q</mi></math></span> with <span><math><mn>1</mn><mo>≤</mo><mi>a</mi><mo>≤</mo><mo>⌊</mo><mfrac><mrow><mn>2</mn></mrow><mrow><mn>3</mn></mrow></mfrac><mi>q</mi><mo>+</mo><mn>1</mn><mo>⌋</mo></math></span> and <span><math><mn>2</mn><msup><mrow><mi>q</mi></mrow><mrow><mn>4</mn></mrow></msup><mo>−</mo><mn>2</mn><msup><mrow><mi>q</mi></mrow><mrow><mn>3</mn></mrow></msup><mo>−</mo><mn>2</mn><msup><mrow><mi>q</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>−</mo><mi>q</mi><mo>+</mo><mn>1</mn><mo>≤</mo><mi>d</mi><mo>≤</mo><mn>2</mn><msup><mrow><mi>q</mi></mrow><mrow><mn>4</mn></mrow></msup><mo>−</mo><mn>2</mn><msup><mrow><mi>q</mi></mrow><mrow><mn>3</mn></mrow></msup><mo>−</mo><mn>2</mn><msup><mrow><mi>q</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span> with <span><math><mi>q</mi><mo>≥</mo><mn>5</mn></math></span>.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 3","pages":"Article 114324"},"PeriodicalIF":0.7,"publicationDate":"2024-11-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142661896","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-11-12DOI: 10.1016/j.disc.2024.114309
Jan Kurkofka , Emily Nevinson
We propose an open question that seeks to generalise the Four Colour Theorem from two to three dimensions. As an appetiser, we show that 12 instead of four colours are both sufficient and necessary to colour every 2-complex that embeds in a prescribed 3-manifold. However, our example of a 2-complex that requires 12 colours is not simplicial.
{"title":"On the edge-chromatic number of 2-complexes","authors":"Jan Kurkofka , Emily Nevinson","doi":"10.1016/j.disc.2024.114309","DOIUrl":"10.1016/j.disc.2024.114309","url":null,"abstract":"<div><div>We propose an open question that seeks to generalise the Four Colour Theorem from two to three dimensions. As an appetiser, we show that 12 instead of four colours are both sufficient and necessary to colour every 2-complex that embeds in a prescribed 3-manifold. However, our example of a 2-complex that requires 12 colours is not simplicial.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 2","pages":"Article 114309"},"PeriodicalIF":0.7,"publicationDate":"2024-11-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142655323","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-11-12DOI: 10.1016/j.disc.2024.114320
Diego Arcis , Camilo González , Sebastián Márquez
In 2004, Rosas and Sagan developed the theory of symmetric functions in noncommuting variables, achieving results analogous to classical symmetric functions. On the other hand, the same year, Desrosiers, Lapointe and Mathieu introduced the theory of symmetric functions in superspace, involving both commuting and anticommuting variables, extending the classic theory. Here, we introduce symmetric functions in noncommuting variables in superspace. We extend the classical symmetric functions in noncommuting variables to superspace: monomial, power sum, elementary and complete homogeneous functions. These functions generalize both those studied by Rosas and Sagan and those studied by Desrosiers, Lapointe, and Mathieu. Additionally, we define Schur–type functions in noncommuting variables in superspace.
{"title":"Symmetric functions in noncommuting variables in superspace","authors":"Diego Arcis , Camilo González , Sebastián Márquez","doi":"10.1016/j.disc.2024.114320","DOIUrl":"10.1016/j.disc.2024.114320","url":null,"abstract":"<div><div>In 2004, Rosas and Sagan developed the theory of symmetric functions in noncommuting variables, achieving results analogous to classical symmetric functions. On the other hand, the same year, Desrosiers, Lapointe and Mathieu introduced the theory of symmetric functions in superspace, involving both commuting and anticommuting variables, extending the classic theory. Here, we introduce symmetric functions in noncommuting variables in superspace. We extend the classical symmetric functions in noncommuting variables to superspace: monomial, power sum, elementary and complete homogeneous functions. These functions generalize both those studied by Rosas and Sagan and those studied by Desrosiers, Lapointe, and Mathieu. Additionally, we define Schur–type functions in noncommuting variables in superspace.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 2","pages":"Article 114320"},"PeriodicalIF":0.7,"publicationDate":"2024-11-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142655325","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-11-12DOI: 10.1016/j.disc.2024.114321
Paul Terwilliger
In 2023 we obtained a Q-polynomial structure for the projective geometry . In the present paper, we display a more general Q-polynomial structure for . Our new Q-polynomial structure is defined using a free parameter φ that takes any positive real value. For we recover the original Q-polynomial structure. We interpret the new Q-polynomial structure using the quantum group in the equitable presentation. We use the new Q-polynomial structure to obtain analogs of the four split decompositions that appear in the theory of Q-polynomial distance-regular graphs.
{"title":"Projective geometries, Q-polynomial structures, and quantum groups","authors":"Paul Terwilliger","doi":"10.1016/j.disc.2024.114321","DOIUrl":"10.1016/j.disc.2024.114321","url":null,"abstract":"<div><div>In 2023 we obtained a <em>Q</em>-polynomial structure for the projective geometry <span><math><msub><mrow><mi>L</mi></mrow><mrow><mi>N</mi></mrow></msub><mo>(</mo><mi>q</mi><mo>)</mo></math></span>. In the present paper, we display a more general <em>Q</em>-polynomial structure for <span><math><msub><mrow><mi>L</mi></mrow><mrow><mi>N</mi></mrow></msub><mo>(</mo><mi>q</mi><mo>)</mo></math></span>. Our new <em>Q</em>-polynomial structure is defined using a free parameter <em>φ</em> that takes any positive real value. For <span><math><mi>φ</mi><mo>=</mo><mn>1</mn></math></span> we recover the original <em>Q</em>-polynomial structure. We interpret the new <em>Q</em>-polynomial structure using the quantum group <span><math><msub><mrow><mi>U</mi></mrow><mrow><msup><mrow><mi>q</mi></mrow><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup></mrow></msub><mo>(</mo><msub><mrow><mi>sl</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></math></span> in the equitable presentation. We use the new <em>Q</em>-polynomial structure to obtain analogs of the four split decompositions that appear in the theory of <em>Q</em>-polynomial distance-regular graphs.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 2","pages":"Article 114321"},"PeriodicalIF":0.7,"publicationDate":"2024-11-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142655719","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-11-12DOI: 10.1016/j.disc.2024.114323
Amal Alofi, Mark Dukes
The lacking polynomial is a graph polynomial introduced by Chan, Marckert, and Selig in 2013 that is closely related to the Tutte polynomial of a graph. It arose by way of a generalization of the Abelian sandpile model and is essentially the generating function of the level statistic on the set of recurrent configurations, called stochastically recurrent states, for that model. In this note we consider the lacking polynomial of the complete bipartite graph. We classify the stochastically recurrent states of the stochastic sandpile model on the complete bipartite graphs and where the sink is always an element of the set counted by the first index. We use these characterizations to give explicit formulae for the lacking polynomials of these graphs. Log-concavity of the sequence of coefficients of these two lacking polynomials is proven, and we conjecture log-concavity holds for this general class of graphs.
{"title":"A note on the lacking polynomial of the complete bipartite graph","authors":"Amal Alofi, Mark Dukes","doi":"10.1016/j.disc.2024.114323","DOIUrl":"10.1016/j.disc.2024.114323","url":null,"abstract":"<div><div>The lacking polynomial is a graph polynomial introduced by Chan, Marckert, and Selig in 2013 that is closely related to the Tutte polynomial of a graph. It arose by way of a generalization of the Abelian sandpile model and is essentially the generating function of the level statistic on the set of recurrent configurations, called stochastically recurrent states, for that model. In this note we consider the lacking polynomial of the complete bipartite graph. We classify the stochastically recurrent states of the stochastic sandpile model on the complete bipartite graphs <span><math><msub><mrow><mi>K</mi></mrow><mrow><mn>2</mn><mo>,</mo><mi>n</mi></mrow></msub></math></span> and <span><math><msub><mrow><mi>K</mi></mrow><mrow><mi>m</mi><mo>,</mo><mn>2</mn></mrow></msub></math></span> where the sink is always an element of the set counted by the first index. We use these characterizations to give explicit formulae for the lacking polynomials of these graphs. Log-concavity of the sequence of coefficients of these two lacking polynomials is proven, and we conjecture log-concavity holds for this general class of graphs.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 2","pages":"Article 114323"},"PeriodicalIF":0.7,"publicationDate":"2024-11-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142655716","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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