Pub Date : 2025-12-23DOI: 10.1016/j.aam.2025.103023
Keunsu Kim , Jae-Hun Jung
We propose the Exact Multi-Parameter Persistent Homology (EMPH) method for the topological analysis of time-series data based on the Liouville torus. Assuming, as in Takens' embedding, that a time-series represents observations of an underlying dynamical system, we model the system as a Hamiltonian system of uncoupled one-dimensional harmonic oscillators. Under this setting, the Liouville torus arises naturally as a dynamical object, and the persistent homology of the Vietoris–Rips complex built on this torus can be interpreted through Fourier analysis. EMPH constructs a multi-parameter filtration framework using Fourier decomposition and provides a closed-form expression for the fibered barcode, an invariant obtained by restricting multi-parameter persistent homology along a specific ray. This formulation establishes a direct correspondence between the choice of a ray and the weighting of Fourier modes, enabling variable topological inferences by exploring different rays in the filtration space. Compared with conventional sliding window based analysis of time-series data, which is computationally expensive, EMPH yields exact barcode formulas with the symmetry of the Liouville torus, achieving much lower computational cost while maintaining comparable or superior accuracy. Thus, EMPH offers both computational efficiency and interpretive flexibility, bridging Fourier analysis and multi-parameter persistent homology in time-series data analysis.
{"title":"Exact multi-parameter persistent homology of time-series data: Fast and variable topological inferences","authors":"Keunsu Kim , Jae-Hun Jung","doi":"10.1016/j.aam.2025.103023","DOIUrl":"10.1016/j.aam.2025.103023","url":null,"abstract":"<div><div>We propose the Exact Multi-Parameter Persistent Homology (EMPH) method for the topological analysis of time-series data based on the Liouville torus. Assuming, as in Takens' embedding, that a time-series represents observations of an underlying dynamical system, we model the system as a Hamiltonian system of uncoupled one-dimensional harmonic oscillators. Under this setting, the Liouville torus arises naturally as a dynamical object, and the persistent homology of the Vietoris–Rips complex built on this torus can be interpreted through Fourier analysis. EMPH constructs a multi-parameter filtration framework using Fourier decomposition and provides a closed-form expression for the fibered barcode, an invariant obtained by restricting multi-parameter persistent homology along a specific ray. This formulation establishes a direct correspondence between the choice of a ray and the weighting of Fourier modes, enabling variable topological inferences by exploring different rays in the filtration space. Compared with conventional sliding window based analysis of time-series data, which is computationally expensive, EMPH yields exact barcode formulas with the symmetry of the Liouville torus, achieving much lower computational cost while maintaining comparable or superior accuracy. Thus, EMPH offers both computational efficiency and interpretive flexibility, bridging Fourier analysis and multi-parameter persistent homology in time-series data analysis.</div></div>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"174 ","pages":"Article 103023"},"PeriodicalIF":1.3,"publicationDate":"2025-12-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145840905","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 : 2025-12-15DOI: 10.1016/j.aam.2025.103020
Olya Mandelshtam, Jerónimo Valencia-Porras
Multiline queues are versatile combinatorial objects that play a key role in understanding the remarkable connection between the asymmetric simple exclusion process (ASEP) on a circle and Macdonald polynomials. Specializing the results of Corteel–Mandelshtam–Williams (2018) to the case yields a formula for the q-Whittaker polynomials through the Ferrari–Martin (2007) algorithm with a major index (maj) statistic. In this paper, we reinterpret the maj statistic as a charge statistic on reading words, thereby bypassing the Ferrari–Martin algorithm to obtain an elegant formula for the q-Whittaker polynomials. Our methods naturally extend to the case of bosonic multiline queues, with which we obtain analogous results for the modified Hall–Littlewood polynomials using a cocharge statistic on reading words.
Twisted multiline queues (TMLQs) are obtained from the action of the symmetric group on the rows of a multiline queue. The Ferrari–Martin algorithm was extended to TMLQs by Arita–Ayyer–Mallick–Prolhac (2011), and Aas–Grinberg–Scrimshaw (2020) showed it is preserved under this action. We extend these results by defining a maj statistic on TMLQs that is also preserved under this action. This yields a novel family of formulas, indexed by compositions, for the q-Whittaker polynomials. Additionally, we define a procedure on both TMLQs and bosonic multiline queues that we call collapsing, which can be realized via the Kashiwara (crystal) operators on type-A Kirillov–Reshetikhin crystals. As an application, we naturally recover the Lascoux–Schützenberger charge formula for the q-Whittaker and modified Hall–Littlewood polynomials, and the classical and dual Cauchy identities for Schur functions.
{"title":"Macdonald polynomials at t = 0 through twisted multiline queues","authors":"Olya Mandelshtam, Jerónimo Valencia-Porras","doi":"10.1016/j.aam.2025.103020","DOIUrl":"10.1016/j.aam.2025.103020","url":null,"abstract":"<div><div>Multiline queues are versatile combinatorial objects that play a key role in understanding the remarkable connection between the asymmetric simple exclusion process (ASEP) on a circle and Macdonald polynomials. Specializing the results of Corteel–Mandelshtam–Williams (2018) to the <span><math><mi>t</mi><mo>=</mo><mn>0</mn></math></span> case yields a formula for the <em>q</em>-Whittaker polynomials through the Ferrari–Martin (2007) algorithm with a major index (<span>maj</span>) statistic. In this paper, we reinterpret the <span>maj</span> statistic as a <span>charge</span> statistic on reading words, thereby bypassing the Ferrari–Martin algorithm to obtain an elegant formula for the <em>q</em>-Whittaker polynomials. Our methods naturally extend to the case of <em>bosonic multiline queues</em>, with which we obtain analogous results for the modified Hall–Littlewood polynomials using a <span>cocharge</span> statistic on reading words.</div><div><em>Twisted multiline queues (TMLQs)</em> are obtained from the action of the symmetric group on the rows of a multiline queue. The Ferrari–Martin algorithm was extended to TMLQs by Arita–Ayyer–Mallick–Prolhac (2011), and Aas–Grinberg–Scrimshaw (2020) showed it is preserved under this action. We extend these results by defining a <span>maj</span> statistic on TMLQs that is also preserved under this action. This yields a novel family of formulas, indexed by compositions, for the <em>q</em>-Whittaker polynomials. Additionally, we define a procedure on both TMLQs and bosonic multiline queues that we call <em>collapsing</em>, which can be realized via the Kashiwara (crystal) operators on type-A Kirillov–Reshetikhin crystals. As an application, we naturally recover the Lascoux–Schützenberger <span>charge</span> formula for the <em>q</em>-Whittaker and modified Hall–Littlewood polynomials, and the classical and dual Cauchy identities for Schur functions.</div></div>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"174 ","pages":"Article 103020"},"PeriodicalIF":1.3,"publicationDate":"2025-12-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145791113","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 : 2025-12-11DOI: 10.1016/j.aam.2025.103019
Dandan Fan , Huiqiu Lin
Let G be a connected graph. If G contains a matching of size k, and every matching of size k is contained in a perfect matching of G, then G is said to be k-extendable. A k-regular spanning subgraph of G is called a k-factor. In this paper, we provide spectral conditions for a (balanced bipartite) graph with minimum degree δ to be k-extendable, and for the existence of a k-factor in a balanced bipartite graph, respectively. Our results generalize some previous results on perfect matchings of graphs, and extend the results in [12] and [27] to k-extendable graphs. Furthermore, our results generalize the result of Lu, Liu and Tian [24] to general regular factors. Additionally, using the equivalence of k edge-disjoint perfect matchings and k-factors in balanced bipartite graphs, our results can derive a spectral condition for the existence of k edge-disjoint perfect matchings in balanced bipartite graphs.
{"title":"Spectral conditions for k-extendability and k-factors of bipartite graphs","authors":"Dandan Fan , Huiqiu Lin","doi":"10.1016/j.aam.2025.103019","DOIUrl":"10.1016/j.aam.2025.103019","url":null,"abstract":"<div><div>Let <em>G</em> be a connected graph. If <em>G</em> contains a matching of size <em>k</em>, and every matching of size <em>k</em> is contained in a perfect matching of <em>G</em>, then <em>G</em> is said to be <em>k-extendable</em>. A <em>k</em>-regular spanning subgraph of <em>G</em> is called a <em>k-factor</em>. In this paper, we provide spectral conditions for a (balanced bipartite) graph with minimum degree <em>δ</em> to be <em>k</em>-extendable, and for the existence of a <em>k</em>-factor in a balanced bipartite graph, respectively. Our results generalize some previous results on perfect matchings of graphs, and extend the results in <span><span>[12]</span></span> and <span><span>[27]</span></span> to <em>k</em>-extendable graphs. Furthermore, our results generalize the result of Lu, Liu and Tian <span><span>[24]</span></span> to general regular factors. Additionally, using the equivalence of <em>k</em> edge-disjoint perfect matchings and <em>k</em>-factors in balanced bipartite graphs, our results can derive a spectral condition for the existence of <em>k</em> edge-disjoint perfect matchings in balanced bipartite graphs.</div></div>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"174 ","pages":"Article 103019"},"PeriodicalIF":1.3,"publicationDate":"2025-12-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145738743","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 : 2025-12-05DOI: 10.1016/j.aam.2025.103004
Nursel Erey , Sara Faridi , Tài Huy Hà , Takayuki Hibi , Selvi Kara , Susan Morey
Let be the edge ideal of a gapfree graph G. An open conjecture of Nevo and Peeva states that has a linear resolution for . We present a promising approach to this challenging conjecture by investigating the stronger property of linear quotients. Specifically, we make the conjecture that if has linear quotients for some integer , then has linear quotients for all . We give a partial solution to this conjecture by considering a special order of the generators of . It is known that if G does not contain a cricket, a diamond, or a cycle of length 4, then has a linear resolution for . We construct a family of gapfree graphs G containing a cricket, a diamond, a together with a cycle of length 5 as induced subgraphs of G for which has linear quotients for .
{"title":"Gapfree graphs and powers of edge ideals with linear quotients","authors":"Nursel Erey , Sara Faridi , Tài Huy Hà , Takayuki Hibi , Selvi Kara , Susan Morey","doi":"10.1016/j.aam.2025.103004","DOIUrl":"10.1016/j.aam.2025.103004","url":null,"abstract":"<div><div>Let <span><math><mi>I</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span> be the edge ideal of a gapfree graph <em>G</em>. An open conjecture of Nevo and Peeva states that <span><math><mi>I</mi><msup><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mrow><mi>q</mi></mrow></msup></math></span> has a linear resolution for <span><math><mi>q</mi><mo>≫</mo><mn>0</mn></math></span>. We present a promising approach to this challenging conjecture by investigating the stronger property of linear quotients. Specifically, we make the conjecture that if <span><math><mi>I</mi><msup><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mrow><mi>q</mi></mrow></msup></math></span> has linear quotients for some integer <span><math><mi>q</mi><mo>≥</mo><mn>1</mn></math></span>, then <span><math><mi>I</mi><msup><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mrow><mi>s</mi></mrow></msup></math></span> has linear quotients for all <span><math><mi>s</mi><mo>≥</mo><mi>q</mi></math></span>. We give a partial solution to this conjecture by considering a special order of the generators of <span><math><mi>I</mi><msup><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mrow><mi>q</mi></mrow></msup></math></span>. It is known that if <em>G</em> does not contain a cricket, a diamond, or a cycle <span><math><msub><mrow><mi>C</mi></mrow><mrow><mn>4</mn></mrow></msub></math></span> of length 4, then <span><math><mi>I</mi><msup><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mrow><mi>q</mi></mrow></msup></math></span> has a linear resolution for <span><math><mi>q</mi><mo>≥</mo><mn>2</mn></math></span>. We construct a family of gapfree graphs <em>G</em> containing a cricket, a diamond, a <span><math><msub><mrow><mi>C</mi></mrow><mrow><mn>4</mn></mrow></msub></math></span> together with a cycle <span><math><msub><mrow><mi>C</mi></mrow><mrow><mn>5</mn></mrow></msub></math></span> of length 5 as induced subgraphs of <em>G</em> for which <span><math><mi>I</mi><msup><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mrow><mi>q</mi></mrow></msup></math></span> has linear quotients for <span><math><mi>q</mi><mo>≥</mo><mn>2</mn></math></span>.</div></div>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"174 ","pages":"Article 103004"},"PeriodicalIF":1.3,"publicationDate":"2025-12-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145685686","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 : 2025-12-04DOI: 10.1016/j.aam.2025.103003
Yulia Alexandr , Kristen Dawson , Hannah Friedman , Fatemeh Mohammadi , Pardis Semnani , Teresa Yu
We study a family of determinantal ideals whose decompositions encode the structural zeros in conditional independence models with hidden variables. We provide explicit decompositions of these ideals and, for certain subclasses of models, we show that this is a decomposition into radical ideals by displaying Gröbner bases for the components. We identify conditions under which the components are prime, and establish formulas for the dimensions of these prime ideals. We show that the components in the decomposition can be grouped into equivalence classes defined by their combinatorial structure, and we derive a closed formula for the number of such classes.
{"title":"Decomposing conditional independence ideals with hidden variables","authors":"Yulia Alexandr , Kristen Dawson , Hannah Friedman , Fatemeh Mohammadi , Pardis Semnani , Teresa Yu","doi":"10.1016/j.aam.2025.103003","DOIUrl":"10.1016/j.aam.2025.103003","url":null,"abstract":"<div><div>We study a family of determinantal ideals whose decompositions encode the structural zeros in conditional independence models with hidden variables. We provide explicit decompositions of these ideals and, for certain subclasses of models, we show that this is a decomposition into radical ideals by displaying Gröbner bases for the components. We identify conditions under which the components are prime, and establish formulas for the dimensions of these prime ideals. We show that the components in the decomposition can be grouped into equivalence classes defined by their combinatorial structure, and we derive a closed formula for the number of such classes.</div></div>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"174 ","pages":"Article 103003"},"PeriodicalIF":1.3,"publicationDate":"2025-12-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145665380","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 : 2025-11-24DOI: 10.1016/j.aam.2025.103001
Takuya Saito
Discriminantal arrangements are hyperplane arrangements that are generalization of braid arrangements. They are constructed from given hyperplane arrangements, but their combinatorics are not invariant under combinatorial equivalence. However, it is known that the combinatorics of the discriminantal arrangements are constant on a Zariski open set of the space of hyperplane arrangements. In the present paper, we introduce -singularity varieties in the space of hyperplane arrangements to classify discriminantal arrangements and show that the Zariski open set is the complement of -singularity varieties. We study their basic properties and operations and provide examples, including infinite families of -singularity varieties. In particular, the operation that we call degeneration is a powerful tool for constructing -singularity varieties. As an application, we provide a list of -singularity varieties for spaces of small line arrangements.
{"title":"Degeneration in discriminantal arrangements","authors":"Takuya Saito","doi":"10.1016/j.aam.2025.103001","DOIUrl":"10.1016/j.aam.2025.103001","url":null,"abstract":"<div><div>Discriminantal arrangements are hyperplane arrangements that are generalization of braid arrangements. They are constructed from given hyperplane arrangements, but their combinatorics are not invariant under combinatorial equivalence. However, it is known that the combinatorics of the discriminantal arrangements are constant on a Zariski open set of the space of hyperplane arrangements. In the present paper, we introduce <span><math><mo>(</mo><mi>T</mi><mo>,</mo><mi>r</mi><mo>)</mo></math></span>-singularity varieties in the space of hyperplane arrangements to classify discriminantal arrangements and show that the Zariski open set is the complement of <span><math><mo>(</mo><mi>T</mi><mo>,</mo><mi>r</mi><mo>)</mo></math></span>-singularity varieties. We study their basic properties and operations and provide examples, including infinite families of <span><math><mo>(</mo><mi>T</mi><mo>,</mo><mi>r</mi><mo>)</mo></math></span>-singularity varieties. In particular, the operation that we call degeneration is a powerful tool for constructing <span><math><mo>(</mo><mi>T</mi><mo>,</mo><mi>r</mi><mo>)</mo></math></span>-singularity varieties. As an application, we provide a list of <span><math><mo>(</mo><mi>T</mi><mo>,</mo><mi>r</mi><mo>)</mo></math></span>-singularity varieties for spaces of small line arrangements.</div></div>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"173 ","pages":"Article 103001"},"PeriodicalIF":1.3,"publicationDate":"2025-11-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145623588","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 : 2025-11-18DOI: 10.1016/j.aam.2025.103000
Chang Liu , Jianping Li , Shuchao Li , Yuantian Yu
<div><div>The Brualdi-Hoffman-Turán problem, a central topic in spectral graph theory, seeks to determine maximum spectral radius <span><math><mi>ρ</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span> of an <em>F</em>-free graph <em>G</em> with <em>m</em> edges. This problem has attracted significant attention in recent years. Let <span><math><msubsup><mrow><mi>C</mi></mrow><mrow><mi>l</mi></mrow><mrow><mo>+</mo></mrow></msubsup></math></span> denote the theta graph obtained by adding a chord between two vertices at distance 2 on cycle <span><math><msub><mrow><mi>C</mi></mrow><mrow><mi>l</mi></mrow></msub></math></span>. Zhai, Lin, and Shu <span><span>[32]</span></span> conjectured that, for <span><math><mi>k</mi><mo>⩾</mo><mn>2</mn></math></span> and sufficiently large <em>m</em>, if <em>G</em> is <span><math><msub><mrow><mi>C</mi></mrow><mrow><mn>2</mn><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msub></math></span>-free or <span><math><msub><mrow><mi>C</mi></mrow><mrow><mn>2</mn><mi>k</mi><mo>+</mo><mn>2</mn></mrow></msub></math></span>-free, then <span><math><mi>ρ</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>⩽</mo><mfrac><mrow><mi>k</mi><mo>−</mo><mn>1</mn><mo>+</mo><msqrt><mrow><mn>4</mn><mi>m</mi><mo>−</mo><msup><mrow><mi>k</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>+</mo><mn>1</mn></mrow></msqrt></mrow><mrow><mn>2</mn></mrow></mfrac></math></span>, with equality if and only if <span><math><mi>G</mi><mo>≅</mo><msub><mrow><mi>K</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>∨</mo><mo>(</mo><mfrac><mrow><mi>m</mi></mrow><mrow><mi>k</mi></mrow></mfrac><mo>−</mo><mfrac><mrow><mi>k</mi><mo>−</mo><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac><mo>)</mo><msub><mrow><mi>K</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span>. This conjecture was highlighted in Liu and Ning's survey <span><span>[18]</span></span> as one of the twenty unsolved problems in spectral graph theory. Subsequently, Y.T. Li proposed an even stronger conjecture, which claims that the upper bound on <span><math><mi>ρ</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span> and the corresponding extremal graph hold for <span><math><msubsup><mrow><mi>C</mi></mrow><mrow><mn>2</mn><mi>k</mi><mo>+</mo><mn>1</mn></mrow><mrow><mo>+</mo></mrow></msubsup></math></span>-free, or <span><math><msubsup><mrow><mi>C</mi></mrow><mrow><mn>2</mn><mi>k</mi><mo>+</mo><mn>2</mn></mrow><mrow><mo>+</mo></mrow></msubsup></math></span>-free graphs. Recently, Li, Zhai, and Shu <span><span>[14]</span></span> resolved both conjectures completely. Note that the above extremal graph <span><math><msub><mrow><mi>K</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>∨</mo><mo>(</mo><mfrac><mrow><mi>m</mi></mrow><mrow><mi>k</mi></mrow></mfrac><mo>−</mo><mfrac><mrow><mi>k</mi><mo>−</mo><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac><mo>)</mo><msub><mrow><mi>K</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span> is well-defined only if <span><math><mi>m</mi><mo>+</mo><mfrac><mrow><mi>k</mi><mo>(</mo><mi>k</mi><mo>+</mo><mn>1</mn><mo>)</
{"title":"A Brualdi-Hoffman-Turán problem on theta graph","authors":"Chang Liu , Jianping Li , Shuchao Li , Yuantian Yu","doi":"10.1016/j.aam.2025.103000","DOIUrl":"10.1016/j.aam.2025.103000","url":null,"abstract":"<div><div>The Brualdi-Hoffman-Turán problem, a central topic in spectral graph theory, seeks to determine maximum spectral radius <span><math><mi>ρ</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span> of an <em>F</em>-free graph <em>G</em> with <em>m</em> edges. This problem has attracted significant attention in recent years. Let <span><math><msubsup><mrow><mi>C</mi></mrow><mrow><mi>l</mi></mrow><mrow><mo>+</mo></mrow></msubsup></math></span> denote the theta graph obtained by adding a chord between two vertices at distance 2 on cycle <span><math><msub><mrow><mi>C</mi></mrow><mrow><mi>l</mi></mrow></msub></math></span>. Zhai, Lin, and Shu <span><span>[32]</span></span> conjectured that, for <span><math><mi>k</mi><mo>⩾</mo><mn>2</mn></math></span> and sufficiently large <em>m</em>, if <em>G</em> is <span><math><msub><mrow><mi>C</mi></mrow><mrow><mn>2</mn><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msub></math></span>-free or <span><math><msub><mrow><mi>C</mi></mrow><mrow><mn>2</mn><mi>k</mi><mo>+</mo><mn>2</mn></mrow></msub></math></span>-free, then <span><math><mi>ρ</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>⩽</mo><mfrac><mrow><mi>k</mi><mo>−</mo><mn>1</mn><mo>+</mo><msqrt><mrow><mn>4</mn><mi>m</mi><mo>−</mo><msup><mrow><mi>k</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>+</mo><mn>1</mn></mrow></msqrt></mrow><mrow><mn>2</mn></mrow></mfrac></math></span>, with equality if and only if <span><math><mi>G</mi><mo>≅</mo><msub><mrow><mi>K</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>∨</mo><mo>(</mo><mfrac><mrow><mi>m</mi></mrow><mrow><mi>k</mi></mrow></mfrac><mo>−</mo><mfrac><mrow><mi>k</mi><mo>−</mo><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac><mo>)</mo><msub><mrow><mi>K</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span>. This conjecture was highlighted in Liu and Ning's survey <span><span>[18]</span></span> as one of the twenty unsolved problems in spectral graph theory. Subsequently, Y.T. Li proposed an even stronger conjecture, which claims that the upper bound on <span><math><mi>ρ</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span> and the corresponding extremal graph hold for <span><math><msubsup><mrow><mi>C</mi></mrow><mrow><mn>2</mn><mi>k</mi><mo>+</mo><mn>1</mn></mrow><mrow><mo>+</mo></mrow></msubsup></math></span>-free, or <span><math><msubsup><mrow><mi>C</mi></mrow><mrow><mn>2</mn><mi>k</mi><mo>+</mo><mn>2</mn></mrow><mrow><mo>+</mo></mrow></msubsup></math></span>-free graphs. Recently, Li, Zhai, and Shu <span><span>[14]</span></span> resolved both conjectures completely. Note that the above extremal graph <span><math><msub><mrow><mi>K</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>∨</mo><mo>(</mo><mfrac><mrow><mi>m</mi></mrow><mrow><mi>k</mi></mrow></mfrac><mo>−</mo><mfrac><mrow><mi>k</mi><mo>−</mo><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac><mo>)</mo><msub><mrow><mi>K</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span> is well-defined only if <span><math><mi>m</mi><mo>+</mo><mfrac><mrow><mi>k</mi><mo>(</mo><mi>k</mi><mo>+</mo><mn>1</mn><mo>)</","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"173 ","pages":"Article 103000"},"PeriodicalIF":1.3,"publicationDate":"2025-11-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145578779","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 : 2025-11-15DOI: 10.1016/j.aam.2025.102999
Moussa Ahmia , José L. Ramírez , Diego Villamizar
Arslan, Altoum, and Zaarour introduced an inversion statistic for generalized symmetric groups [5]. In this work, we study the distribution of this statistic over colored permutations, including derangements and involutions. By establishing a bijective correspondence between colored permutations and colored Lehmer codes, we develop a unified framework for enumerating colored Mahonian numbers and analyzing their combinatorial properties. We derive explicit formulas, recurrence relations, and generating functions for the number of inversions in these families, extending classical results to the colored setting. We conclude with explicit expressions for inversions in colored derangements and involutions.
{"title":"Inversions in colored permutations, derangements, and involutions","authors":"Moussa Ahmia , José L. Ramírez , Diego Villamizar","doi":"10.1016/j.aam.2025.102999","DOIUrl":"10.1016/j.aam.2025.102999","url":null,"abstract":"<div><div>Arslan, Altoum, and Zaarour introduced an inversion statistic for generalized symmetric groups <span><span>[5]</span></span>. In this work, we study the distribution of this statistic over colored permutations, including derangements and involutions. By establishing a bijective correspondence between colored permutations and colored Lehmer codes, we develop a unified framework for enumerating colored Mahonian numbers and analyzing their combinatorial properties. We derive explicit formulas, recurrence relations, and generating functions for the number of inversions in these families, extending classical results to the colored setting. We conclude with explicit expressions for inversions in colored derangements and involutions.</div></div>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"173 ","pages":"Article 102999"},"PeriodicalIF":1.3,"publicationDate":"2025-11-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145528747","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 : 2025-11-15DOI: 10.1016/j.aam.2025.102997
William Gustafson
We introduce posets of simple vertex labeled minors of graphs and a generalization to the level of polymatroids, collectively termed minor posets. We show that any minor poset is isomorphic to the face poset of a regular CW sphere, and in particular, is Eulerian. We establish cd-index inequalities induced by strong maps, a tight upper bound for cd-indices of minor posets and a tight lower bound for cd-indices of minor posets arising from lattices of maximal length.
{"title":"Lattice minors and Eulerian posets","authors":"William Gustafson","doi":"10.1016/j.aam.2025.102997","DOIUrl":"10.1016/j.aam.2025.102997","url":null,"abstract":"<div><div>We introduce posets of simple vertex labeled minors of graphs and a generalization to the level of polymatroids, collectively termed minor posets. We show that any minor poset is isomorphic to the face poset of a regular CW sphere, and in particular, is Eulerian. We establish <strong>cd</strong>-index inequalities induced by strong maps, a tight upper bound for <strong>cd</strong>-indices of minor posets and a tight lower bound for <strong>cd</strong>-indices of minor posets arising from lattices of maximal length.</div></div>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"173 ","pages":"Article 102997"},"PeriodicalIF":1.3,"publicationDate":"2025-11-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145528746","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 : 2025-11-12DOI: 10.1016/j.aam.2025.102998
Tomasz Jędrzejak
We give a description of the structure of factor rings for the where n is an integer (which is not a square). For example, we prove that is isomorphic to the ring of integers modulo for relatively prime . We also characterize the structure of for arbitrary integers . Finally, we describe for non-principal ideals I. We also present many corollaries regarding irreducible and prime elements in and give numerous examples. We only use methods from elementary number theory and basic ring theory.
{"title":"The structure of factor rings of Z[n]","authors":"Tomasz Jędrzejak","doi":"10.1016/j.aam.2025.102998","DOIUrl":"10.1016/j.aam.2025.102998","url":null,"abstract":"<div><div>We give a description of the structure of factor rings for the <span><math><mi>Z</mi><mrow><mo>[</mo><msqrt><mrow><mi>n</mi></mrow></msqrt><mo>]</mo></mrow></math></span> where <em>n</em> is an integer (which is not a square). For example, we prove that <span><math><mi>Z</mi><mrow><mo>[</mo><msqrt><mrow><mi>n</mi></mrow></msqrt><mo>]</mo></mrow><mo>/</mo><mrow><mo>(</mo><mi>a</mi><mo>+</mo><mi>b</mi><msqrt><mrow><mi>n</mi></mrow></msqrt><mo>)</mo></mrow></math></span> is isomorphic to the ring of integers modulo <span><math><mo>|</mo><msup><mrow><mi>a</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>−</mo><mi>n</mi><msup><mrow><mi>b</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>|</mo></math></span> for relatively prime <span><math><mi>a</mi><mo>,</mo><mi>b</mi></math></span>. We also characterize the structure of <span><math><mi>Z</mi><mrow><mo>[</mo><msqrt><mrow><mi>n</mi></mrow></msqrt><mo>]</mo></mrow><mo>/</mo><mrow><mo>(</mo><mi>a</mi><mo>+</mo><mi>b</mi><msqrt><mrow><mi>n</mi></mrow></msqrt><mo>)</mo></mrow></math></span> for arbitrary integers <span><math><mi>a</mi><mo>,</mo><mi>b</mi></math></span>. Finally, we describe <span><math><mi>Z</mi><mrow><mo>[</mo><msqrt><mrow><mi>n</mi></mrow></msqrt><mo>]</mo></mrow><mo>/</mo><mi>I</mi></math></span> for non-principal ideals <em>I</em>. We also present many corollaries regarding irreducible and prime elements in <span><math><mi>Z</mi><mrow><mo>[</mo><msqrt><mrow><mi>n</mi></mrow></msqrt><mo>]</mo></mrow></math></span> and give numerous examples. We only use methods from elementary number theory and basic ring theory.</div></div>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"173 ","pages":"Article 102998"},"PeriodicalIF":1.3,"publicationDate":"2025-11-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145528745","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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