Pub Date : 2025-12-01Epub Date: 2025-07-15DOI: 10.1016/j.aam.2025.102936
Chunyang Dou , Bo Ning , Xing Peng
<div><div>Let <span><math><mi>H</mi></math></span> be a family of graphs. The Turán number <span><math><mrow><mi>ex</mi></mrow><mo>(</mo><mi>n</mi><mo>,</mo><mi>H</mi><mo>)</mo></math></span> is the maximum possible number of edges in an <em>n</em>-vertex graph which does not contain any member of <span><math><mi>H</mi></math></span> as a subgraph. As a common generalization of Turán's theorem and Erdős-Gallai theorem on the Turán number of matchings, Alon and Frankl determined <span><math><mrow><mi>ex</mi></mrow><mo>(</mo><mi>n</mi><mo>,</mo><mi>H</mi><mo>)</mo></math></span> for <span><math><mi>H</mi><mo>=</mo><mo>{</mo><msub><mrow><mi>K</mi></mrow><mrow><mi>r</mi></mrow></msub><mo>,</mo><msub><mrow><mi>M</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>}</mo></math></span>, where <span><math><msub><mrow><mi>M</mi></mrow><mrow><mi>k</mi></mrow></msub></math></span> is a matching of size <em>k</em>. Replacing <span><math><msub><mrow><mi>M</mi></mrow><mrow><mi>k</mi></mrow></msub></math></span> by <span><math><msub><mrow><mi>P</mi></mrow><mrow><mi>k</mi></mrow></msub></math></span>, Katona and Xiao obtained the Turán number of <span><math><mi>H</mi><mo>=</mo><mo>{</mo><msub><mrow><mi>K</mi></mrow><mrow><mi>r</mi></mrow></msub><mo>,</mo><msub><mrow><mi>P</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>}</mo></math></span> for <span><math><mi>r</mi><mo>≤</mo><mo>⌊</mo><mi>k</mi><mo>/</mo><mn>2</mn><mo>⌋</mo></math></span> and sufficiently large <em>n</em>. In addition, they proposed a conjecture for the case where <span><math><mi>r</mi><mo>≥</mo><mo>⌊</mo><mi>k</mi><mo>/</mo><mn>2</mn><mo>⌋</mo><mo>+</mo><mn>1</mn></math></span> and <em>n</em> is sufficiently large. Motivated by the fact that the result for <span><math><mrow><mi>ex</mi></mrow><mo>(</mo><mi>n</mi><mo>,</mo><msub><mrow><mi>P</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>)</mo></math></span> can be deduced from the one for <span><math><mrow><mi>ex</mi></mrow><mo>(</mo><mi>n</mi><mo>,</mo><msub><mrow><mi>C</mi></mrow><mrow><mo>≥</mo><mi>k</mi></mrow></msub><mo>)</mo></math></span>, we investigate the Turán number of <span><math><mo>{</mo><msub><mrow><mi>K</mi></mrow><mrow><mi>r</mi></mrow></msub><mo>,</mo><msub><mrow><mi>C</mi></mrow><mrow><mo>≥</mo><mi>k</mi></mrow></msub><mo>}</mo></math></span> in this paper, where <span><math><msub><mrow><mi>C</mi></mrow><mrow><mo>≥</mo><mi>k</mi></mrow></msub></math></span> denotes the set of cycles of length at least <em>k</em>. In other words, we aim to determine the maximum number of edges in graphs with clique number at most <span><math><mi>r</mi><mo>−</mo><mn>1</mn></math></span> and circumference at most <span><math><mi>k</mi><mo>−</mo><mn>1</mn></math></span>. For <span><math><mi>H</mi><mo>=</mo><mo>{</mo><msub><mrow><mi>K</mi></mrow><mrow><mi>r</mi></mrow></msub><mo>,</mo><msub><mrow><mi>C</mi></mrow><mrow><mo>≥</mo><mi>k</mi></mrow></msub><mo>}</mo></math></span>, we are able to show the value of <span><math><mrow><mi>ex</mi></mrow><mo>(</mo><mi>
{"title":"The number of edges in graphs with bounded clique number and circumference","authors":"Chunyang Dou , Bo Ning , Xing Peng","doi":"10.1016/j.aam.2025.102936","DOIUrl":"10.1016/j.aam.2025.102936","url":null,"abstract":"<div><div>Let <span><math><mi>H</mi></math></span> be a family of graphs. The Turán number <span><math><mrow><mi>ex</mi></mrow><mo>(</mo><mi>n</mi><mo>,</mo><mi>H</mi><mo>)</mo></math></span> is the maximum possible number of edges in an <em>n</em>-vertex graph which does not contain any member of <span><math><mi>H</mi></math></span> as a subgraph. As a common generalization of Turán's theorem and Erdős-Gallai theorem on the Turán number of matchings, Alon and Frankl determined <span><math><mrow><mi>ex</mi></mrow><mo>(</mo><mi>n</mi><mo>,</mo><mi>H</mi><mo>)</mo></math></span> for <span><math><mi>H</mi><mo>=</mo><mo>{</mo><msub><mrow><mi>K</mi></mrow><mrow><mi>r</mi></mrow></msub><mo>,</mo><msub><mrow><mi>M</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>}</mo></math></span>, where <span><math><msub><mrow><mi>M</mi></mrow><mrow><mi>k</mi></mrow></msub></math></span> is a matching of size <em>k</em>. Replacing <span><math><msub><mrow><mi>M</mi></mrow><mrow><mi>k</mi></mrow></msub></math></span> by <span><math><msub><mrow><mi>P</mi></mrow><mrow><mi>k</mi></mrow></msub></math></span>, Katona and Xiao obtained the Turán number of <span><math><mi>H</mi><mo>=</mo><mo>{</mo><msub><mrow><mi>K</mi></mrow><mrow><mi>r</mi></mrow></msub><mo>,</mo><msub><mrow><mi>P</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>}</mo></math></span> for <span><math><mi>r</mi><mo>≤</mo><mo>⌊</mo><mi>k</mi><mo>/</mo><mn>2</mn><mo>⌋</mo></math></span> and sufficiently large <em>n</em>. In addition, they proposed a conjecture for the case where <span><math><mi>r</mi><mo>≥</mo><mo>⌊</mo><mi>k</mi><mo>/</mo><mn>2</mn><mo>⌋</mo><mo>+</mo><mn>1</mn></math></span> and <em>n</em> is sufficiently large. Motivated by the fact that the result for <span><math><mrow><mi>ex</mi></mrow><mo>(</mo><mi>n</mi><mo>,</mo><msub><mrow><mi>P</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>)</mo></math></span> can be deduced from the one for <span><math><mrow><mi>ex</mi></mrow><mo>(</mo><mi>n</mi><mo>,</mo><msub><mrow><mi>C</mi></mrow><mrow><mo>≥</mo><mi>k</mi></mrow></msub><mo>)</mo></math></span>, we investigate the Turán number of <span><math><mo>{</mo><msub><mrow><mi>K</mi></mrow><mrow><mi>r</mi></mrow></msub><mo>,</mo><msub><mrow><mi>C</mi></mrow><mrow><mo>≥</mo><mi>k</mi></mrow></msub><mo>}</mo></math></span> in this paper, where <span><math><msub><mrow><mi>C</mi></mrow><mrow><mo>≥</mo><mi>k</mi></mrow></msub></math></span> denotes the set of cycles of length at least <em>k</em>. In other words, we aim to determine the maximum number of edges in graphs with clique number at most <span><math><mi>r</mi><mo>−</mo><mn>1</mn></math></span> and circumference at most <span><math><mi>k</mi><mo>−</mo><mn>1</mn></math></span>. For <span><math><mi>H</mi><mo>=</mo><mo>{</mo><msub><mrow><mi>K</mi></mrow><mrow><mi>r</mi></mrow></msub><mo>,</mo><msub><mrow><mi>C</mi></mrow><mrow><mo>≥</mo><mi>k</mi></mrow></msub><mo>}</mo></math></span>, we are able to show the value of <span><math><mrow><mi>ex</mi></mrow><mo>(</mo><mi>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"171 ","pages":"Article 102936"},"PeriodicalIF":1.0,"publicationDate":"2025-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144632923","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 : 2025-12-01Epub Date: 2025-07-23DOI: 10.1016/j.aam.2025.102938
Giuseppe Cotardo , Alberto Ravagnani , Ferdinando Zullo
We investigate the Whitney numbers of the first kind of rank-metric lattices, which are closely linked to the open problem of enumerating rank-metric codes having prescribed parameters. We apply methods from the theory of hyperovals and linear sets to compute these Whitney numbers for infinite families of rank-metric lattices. As an application of our results, we prove asymptotic estimates on the density function of certain rank-metric codes that have been conjectured in previous work.
{"title":"Whitney numbers of rank-metric lattices and code enumeration","authors":"Giuseppe Cotardo , Alberto Ravagnani , Ferdinando Zullo","doi":"10.1016/j.aam.2025.102938","DOIUrl":"10.1016/j.aam.2025.102938","url":null,"abstract":"<div><div>We investigate the Whitney numbers of the first kind of rank-metric lattices, which are closely linked to the open problem of enumerating rank-metric codes having prescribed parameters. We apply methods from the theory of hyperovals and linear sets to compute these Whitney numbers for infinite families of rank-metric lattices. As an application of our results, we prove asymptotic estimates on the density function of certain rank-metric codes that have been conjectured in previous work.</div></div>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"171 ","pages":"Article 102938"},"PeriodicalIF":1.0,"publicationDate":"2025-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144686919","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 : 2025-10-01Epub Date: 2025-07-03DOI: 10.1016/j.aam.2025.102931
Guoce Xin , Chen Zhang
Sylvester's denumerant is a quantity that counts the number of nonnegative integer solutions to the equation , where is a sequence of positive integers with . We present a polynomial time algorithm in N for computing when a is bounded and t is a parameter. The proposed algorithm is rooted in the use of cyclotomic polynomials and builds upon recent results by Xin-Zhang-Zhang on the efficient computation of generalized Todd polynomials. The algorithm has been implemented in Maple under the name Cyc-Denum and demonstrates superior performance when compared to Sills-Zeilberger's Maple package PARTITIONS.
{"title":"A polynomial time algorithm for Sylvester waves when entries are bounded","authors":"Guoce Xin , Chen Zhang","doi":"10.1016/j.aam.2025.102931","DOIUrl":"10.1016/j.aam.2025.102931","url":null,"abstract":"<div><div>Sylvester's denumerant <span><math><mi>d</mi><mo>(</mo><mi>t</mi><mo>;</mo><mi>a</mi><mo>)</mo></math></span> is a quantity that counts the number of nonnegative integer solutions to the equation <span><math><msubsup><mrow><mo>∑</mo></mrow><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>N</mi></mrow></msubsup><msub><mrow><mi>a</mi></mrow><mrow><mi>i</mi></mrow></msub><msub><mrow><mi>x</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>=</mo><mi>t</mi></math></span>, where <span><math><mi>a</mi><mo>=</mo><mo>(</mo><msub><mrow><mi>a</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>a</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>a</mi></mrow><mrow><mi>N</mi></mrow></msub><mo>)</mo></math></span> is a sequence of positive integers with <span><math><mi>gcd</mi><mo></mo><mo>(</mo><mi>a</mi><mo>)</mo><mo>=</mo><mn>1</mn></math></span>. We present a polynomial time algorithm in <em>N</em> for computing <span><math><mi>d</mi><mo>(</mo><mi>t</mi><mo>;</mo><mi>a</mi><mo>)</mo></math></span> when <strong><em>a</em></strong> is bounded and <em>t</em> is a parameter. The proposed algorithm is rooted in the use of cyclotomic polynomials and builds upon recent results by Xin-Zhang-Zhang on the efficient computation of generalized Todd polynomials. The algorithm has been implemented in <span>Maple</span> under the name <span>Cyc-Denum</span> and demonstrates superior performance when <span><math><msub><mrow><mi>a</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>≤</mo><mn>500</mn></math></span> compared to Sills-Zeilberger's <span>Maple</span> package <span>PARTITIONS</span>.</div></div>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"170 ","pages":"Article 102931"},"PeriodicalIF":1.0,"publicationDate":"2025-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144535520","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 : 2025-10-01Epub Date: 2025-07-09DOI: 10.1016/j.aam.2025.102932
V. Berthé , S. Puzynina
An infinite word generated by a substitution is rigid if all the substitutions which fix this word are powers of the same substitution. Sturmian words as well as characteristic Arnoux-Rauzy words that are generated by iterating a substitution are known to be rigid. In the present paper, we prove that all Arnoux-Rauzy words generated by iterating a substitution are rigid. The proof relies on two main ingredients: first, the fact that the primitive substitutions that fix an Arnoux-Rauzy word share a common power, and secondly, the notion of normal form of an episturmian substitution (i.e., a substitution that fixes an Arnoux-Rauzy word). The main difficulty is then of a combinatorial nature and relies on the normalization process when taking powers of episturmian substitutions: the normal form of a square is not necessarily equal to the square of the normal forms.
{"title":"On the rigidity of Arnoux-Rauzy words","authors":"V. Berthé , S. Puzynina","doi":"10.1016/j.aam.2025.102932","DOIUrl":"10.1016/j.aam.2025.102932","url":null,"abstract":"<div><div>An infinite word generated by a substitution is rigid if all the substitutions which fix this word are powers of the same substitution. Sturmian words as well as characteristic Arnoux-Rauzy words that are generated by iterating a substitution are known to be rigid. In the present paper, we prove that all Arnoux-Rauzy words generated by iterating a substitution are rigid. The proof relies on two main ingredients: first, the fact that the primitive substitutions that fix an Arnoux-Rauzy word share a common power, and secondly, the notion of normal form of an episturmian substitution (i.e., a substitution that fixes an Arnoux-Rauzy word). The main difficulty is then of a combinatorial nature and relies on the normalization process when taking powers of episturmian substitutions: the normal form of a square is not necessarily equal to the square of the normal forms.</div></div>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"170 ","pages":"Article 102932"},"PeriodicalIF":1.0,"publicationDate":"2025-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144579417","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 : 2025-10-01Epub Date: 2025-06-09DOI: 10.1016/j.aam.2025.102916
Yaroslav Shitov
The positive part of the field consists of Puiseux series with positive real leading terms. Answering a question of Yu, we show that, if M is an matrix with entries in and rank two, then there are an matrix A and matrix B with entries in such that . We discuss the problem in larger ranks and answer a further question arisen in a work of Brandenburg, Loho, and Sinn.
{"title":"On the nonnegative ranks of matrices in Puiseux series fields","authors":"Yaroslav Shitov","doi":"10.1016/j.aam.2025.102916","DOIUrl":"10.1016/j.aam.2025.102916","url":null,"abstract":"<div><div>The <em>positive part</em> <span><math><msup><mrow><mi>C</mi></mrow><mrow><mo>+</mo></mrow></msup></math></span> of the field <span><math><mi>C</mi><mo>{</mo><mo>{</mo><mi>t</mi><mo>}</mo><mo>}</mo></math></span> consists of Puiseux series with positive real leading terms. Answering a question of Yu, we show that, if <em>M</em> is an <span><math><mi>m</mi><mo>×</mo><mi>n</mi></math></span> matrix with entries in <span><math><msup><mrow><mi>C</mi></mrow><mrow><mo>+</mo></mrow></msup></math></span> and rank two, then there are an <span><math><mi>m</mi><mo>×</mo><mn>2</mn></math></span> matrix <em>A</em> and <span><math><mn>2</mn><mo>×</mo><mi>n</mi></math></span> matrix <em>B</em> with entries in <span><math><msup><mrow><mi>C</mi></mrow><mrow><mo>+</mo></mrow></msup></math></span> such that <span><math><mi>M</mi><mo>=</mo><mi>A</mi><mi>B</mi></math></span>. We discuss the problem in larger ranks and answer a further question arisen in a work of Brandenburg, Loho, and Sinn.</div></div>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"170 ","pages":"Article 102916"},"PeriodicalIF":1.0,"publicationDate":"2025-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144239672","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 : 2025-10-01Epub Date: 2025-06-30DOI: 10.1016/j.aam.2025.102930
Eliana Duarte , Dmitrii Pavlov , Maximilian Wiesmann
Algebro-geometric methods have proven to be very successful in the study of graphical models in statistics. In this paper we introduce the foundations to carry out a similar study of their quantum counterparts. These quantum graphical models are families of quantum states satisfying certain locality or correlation conditions encoded by a graph. We lay out several ways to associate an algebraic variety to a quantum graphical model. The classical graphical models can be recovered from most of these varieties by restricting to quantum states represented by diagonal matrices. We study fundamental properties of these varieties and provide algorithms to compute their defining equations. Moreover, we study quantum information projections to quantum exponential families defined by graphs and prove a quantum analogue of Birch's Theorem.
{"title":"Algebraic geometry of quantum graphical models","authors":"Eliana Duarte , Dmitrii Pavlov , Maximilian Wiesmann","doi":"10.1016/j.aam.2025.102930","DOIUrl":"10.1016/j.aam.2025.102930","url":null,"abstract":"<div><div>Algebro-geometric methods have proven to be very successful in the study of graphical models in statistics. In this paper we introduce the foundations to carry out a similar study of their quantum counterparts. These quantum graphical models are families of quantum states satisfying certain locality or correlation conditions encoded by a graph. We lay out several ways to associate an algebraic variety to a quantum graphical model. The classical graphical models can be recovered from most of these varieties by restricting to quantum states represented by diagonal matrices. We study fundamental properties of these varieties and provide algorithms to compute their defining equations. Moreover, we study quantum information projections to quantum exponential families defined by graphs and prove a quantum analogue of Birch's Theorem.</div></div>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"170 ","pages":"Article 102930"},"PeriodicalIF":1.0,"publicationDate":"2025-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144513870","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 : 2025-10-01Epub Date: 2025-06-20DOI: 10.1016/j.aam.2025.102928
Arthur Bik , Orlando Marigliano
We propose a classification of all one-dimensional discrete statistical models with maximum likelihood degree one based on their rational parametrization. We show how all such models can be constructed from members of a smaller class of ‘fundamental models’ using a finite number of simple operations. We introduce ‘chipsplitting games’, a class of combinatorial games on a grid which we use to represent fundamental models. This combinatorial perspective enables us to show that there are only finitely many fundamental models in the probability simplex for .
{"title":"Classifying one-dimensional discrete models with maximum likelihood degree one","authors":"Arthur Bik , Orlando Marigliano","doi":"10.1016/j.aam.2025.102928","DOIUrl":"10.1016/j.aam.2025.102928","url":null,"abstract":"<div><div>We propose a classification of all one-dimensional discrete statistical models with maximum likelihood degree one based on their rational parametrization. We show how all such models can be constructed from members of a smaller class of ‘fundamental models’ using a finite number of simple operations. We introduce ‘chipsplitting games’, a class of combinatorial games on a grid which we use to represent fundamental models. This combinatorial perspective enables us to show that there are only finitely many fundamental models in the probability simplex <span><math><msub><mrow><mi>Δ</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> for <span><math><mi>n</mi><mo>≤</mo><mn>4</mn></math></span>.</div></div>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"170 ","pages":"Article 102928"},"PeriodicalIF":1.0,"publicationDate":"2025-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144330513","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 : 2025-10-01Epub Date: 2025-06-20DOI: 10.1016/j.aam.2025.102927
Songlin Guo , Wei Wang , Wei Wang
Suppose G is a controllable graph of order n with adjacency matrix A. Let (e is the all-ones vector) and ('s are eigenvalues of A) be the walk matrix and the discriminant of G, respectively. Wang and Yu (arXiv:1608.01144) [21] showed that if is odd and squarefree, then G is determined by its generalized spectrum (DGS). Using the primary decomposition theorem, we obtain a new criterion for a graph G to be DGS without the squarefreeness assumption on . Examples are further given to illustrate the effectiveness of the proposed criterion, compared with the two existing methods to deal with the difficulty of non-squarefreeness.
{"title":"Primary decomposition theorem and generalized spectral characterization of graphs","authors":"Songlin Guo , Wei Wang , Wei Wang","doi":"10.1016/j.aam.2025.102927","DOIUrl":"10.1016/j.aam.2025.102927","url":null,"abstract":"<div><div>Suppose <em>G</em> is a controllable graph of order <em>n</em> with adjacency matrix <em>A</em>. Let <span><math><mi>W</mi><mo>=</mo><mo>[</mo><mi>e</mi><mo>,</mo><mi>A</mi><mi>e</mi><mo>,</mo><mo>…</mo><mo>,</mo><msup><mrow><mi>A</mi></mrow><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup><mi>e</mi><mo>]</mo></math></span> (<em>e</em> is the all-ones vector) and <span><math><mi>Δ</mi><mo>=</mo><msub><mrow><mo>∏</mo></mrow><mrow><mi>i</mi><mo>></mo><mi>j</mi></mrow></msub><msup><mrow><mo>(</mo><msub><mrow><mi>α</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>−</mo><msub><mrow><mi>α</mi></mrow><mrow><mi>j</mi></mrow></msub><mo>)</mo></mrow><mrow><mn>2</mn></mrow></msup></math></span> (<span><math><msub><mrow><mi>α</mi></mrow><mrow><mi>i</mi></mrow></msub></math></span>'s are eigenvalues of <em>A</em>) be the walk matrix and the discriminant of <em>G</em>, respectively. Wang and Yu (<span><span>arXiv:1608.01144</span><svg><path></path></svg></span>) <span><span>[21]</span></span> showed that if<span><span><span><math><mi>θ</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>:</mo><mo>=</mo><mi>gcd</mi><mo></mo><mo>{</mo><msup><mrow><mn>2</mn></mrow><mrow><mo>−</mo><mo>⌊</mo><mfrac><mrow><mi>n</mi></mrow><mrow><mn>2</mn></mrow></mfrac><mo>⌋</mo></mrow></msup><mi>det</mi><mo></mo><mi>W</mi><mo>,</mo><mi>Δ</mi><mo>}</mo></math></span></span></span> is odd and squarefree, then <em>G</em> is determined by its generalized spectrum (DGS). Using the primary decomposition theorem, we obtain a new criterion for a graph <em>G</em> to be DGS without the squarefreeness assumption on <span><math><mi>θ</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span>. Examples are further given to illustrate the effectiveness of the proposed criterion, compared with the two existing methods to deal with the difficulty of non-squarefreeness.</div></div>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"170 ","pages":"Article 102927"},"PeriodicalIF":1.0,"publicationDate":"2025-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144321228","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 : 2025-10-01Epub Date: 2025-07-08DOI: 10.1016/j.aam.2025.102933
Christine Cho, James Oxley
Let M and N be matroids such that N is the image of M under a rank-preserving weak map. Generalizing results of Lucas, we prove that, for x and y positive, if and only if or . We give a number of consequences of this result.
{"title":"Weak maps and the Tutte polynomial","authors":"Christine Cho, James Oxley","doi":"10.1016/j.aam.2025.102933","DOIUrl":"10.1016/j.aam.2025.102933","url":null,"abstract":"<div><div>Let <em>M</em> and <em>N</em> be matroids such that <em>N</em> is the image of <em>M</em> under a rank-preserving weak map. Generalizing results of Lucas, we prove that, for <em>x</em> and <em>y</em> positive, <span><math><mi>T</mi><mo>(</mo><mi>M</mi><mo>;</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>)</mo><mo>≥</mo><mi>T</mi><mo>(</mo><mi>N</mi><mo>;</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>)</mo></math></span> if and only if <span><math><mi>x</mi><mo>+</mo><mi>y</mi><mo>≥</mo><mi>x</mi><mi>y</mi></math></span> or <span><math><mi>M</mi><mo>≅</mo><mi>N</mi></math></span>. We give a number of consequences of this result.</div></div>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"170 ","pages":"Article 102933"},"PeriodicalIF":1.0,"publicationDate":"2025-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144572076","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 : 2025-10-01Epub Date: 2025-06-20DOI: 10.1016/j.aam.2025.102929
Giulio Cerbai , Anders Claesson , Bruce E. Sagan
Ascent sequences play a key role in the combinatorics of Fishburn structures. Difference ascent sequences are a natural generalization obtained by replacing ascents with d-ascents. We have recently extended the so-called hat map to difference ascent sequences, and self-modified difference ascent sequences are the fixed points under this map. We characterize self-modified difference ascent sequences and enumerate them in terms of certain generalized Fibonacci polynomials. Furthermore, we describe the corresponding subset of d-Fishburn permutations.
{"title":"Self-modified difference ascent sequences","authors":"Giulio Cerbai , Anders Claesson , Bruce E. Sagan","doi":"10.1016/j.aam.2025.102929","DOIUrl":"10.1016/j.aam.2025.102929","url":null,"abstract":"<div><div>Ascent sequences play a key role in the combinatorics of Fishburn structures. Difference ascent sequences are a natural generalization obtained by replacing ascents with <em>d</em>-ascents. We have recently extended the so-called hat map to difference ascent sequences, and self-modified difference ascent sequences are the fixed points under this map. We characterize self-modified difference ascent sequences and enumerate them in terms of certain generalized Fibonacci polynomials. Furthermore, we describe the corresponding subset of <em>d</em>-Fishburn permutations.</div></div>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"170 ","pages":"Article 102929"},"PeriodicalIF":1.0,"publicationDate":"2025-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144321229","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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