Advances in Applied Mathematics最新文献

{"title":"Extremal distance spectral radius of graphs with fixed size","authors":"Hongying Lin ,&nbsp;Bo Zhou","doi":"10.1016/j.aam.2025.102980","DOIUrl":"10.1016/j.aam.2025.102980","url":null,"abstract":"<div><div>Let <em>m</em> be a positive integer. Brualdi and Hoffman proposed the problem to determine the (connected) graphs with maximum adjacency spectral radius in a given graph class and they posed a conjecture for the class of graphs with given size <em>m</em>. After partial results due to Friedland and Stanley, Rowlinson completely confirmed the conjecture. The distance spectral radius of a connected graph is the largest eigenvalue of its distance matrix. We investigate the problem to determine the connected graphs with minimum distance spectral radius in the class of graphs with size <em>m</em>. Given <em>m</em>, there is exactly one positive integer <em>n</em> such that <span><math><mrow><mo>(</mo><mtable><mtr><mtd><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></mtd></mtr><mtr><mtd><mn>2</mn></mtd></mtr></mtable><mo>)</mo></mrow><mo>&lt;</mo><mi>m</mi><mo>≤</mo><mrow><mo>(</mo><mtable><mtr><mtd><mi>n</mi></mtd></mtr><mtr><mtd><mn>2</mn></mtd></mtr></mtable><mo>)</mo></mrow></math></span>. We establish some structural properties of the extremal graphs for all <em>m</em> and solve the problem for <span><math><mrow><mo>(</mo><mtable><mtr><mtd><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></mtd></mtr><mtr><mtd><mn>2</mn></mtd></mtr></mtable><mo>)</mo></mrow><mo>+</mo><mi>max</mi><mo>⁡</mo><mo>{</mo><mfrac><mrow><mi>n</mi><mo>−</mo><mn>6</mn></mrow><mrow><mn>2</mn></mrow></mfrac><mo>,</mo><mn>1</mn><mo>}</mo><mo>≤</mo><mi>m</mi><mo>≤</mo><mrow><mo>(</mo><mtable><mtr><mtd><mi>n</mi></mtd></mtr><mtr><mtd><mn>2</mn></mtd></mtr></mtable><mo>)</mo></mrow></math></span>. We give a conjecture for the remaining case. To prove the main results, we also determine the complements of forests of fixed order with large and small distance spectral radius.</div></div>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"173 ","pages":"Article 102980"},"PeriodicalIF":1.3,"publicationDate":"2026-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145268468","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}

{"title":"The symmetric strong circuit elimination property","authors":"Christine Cho ,&nbsp;James Oxley ,&nbsp;Suijie Wang","doi":"10.1016/j.aam.2025.102983","DOIUrl":"10.1016/j.aam.2025.102983","url":null,"abstract":"<div><div>If <span><math><msub><mrow><mi>C</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span> and <span><math><msub><mrow><mi>C</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span> are circuits in a matroid <em>M</em> with <span><math><msub><mrow><mi>e</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span> in <span><math><msub><mrow><mi>C</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>−</mo><msub><mrow><mi>C</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span> and <em>e</em> in <span><math><msub><mrow><mi>C</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>∩</mo><msub><mrow><mi>C</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span>, then <em>M</em> has a circuit <span><math><msub><mrow><mi>C</mi></mrow><mrow><mn>3</mn></mrow></msub></math></span> such that <span><math><mi>e</mi><mo>∈</mo><msub><mrow><mi>C</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>⊆</mo><mo>(</mo><msub><mrow><mi>C</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>∪</mo><msub><mrow><mi>C</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo><mo>−</mo><mi>e</mi></math></span>. This strong circuit elimination axiom is inherently asymmetric. A matroid <em>M</em> has the symmetric strong circuit elimination property (SSCE) if, when the above conditions hold and <span><math><msub><mrow><mi>e</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>∈</mo><msub><mrow><mi>C</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>−</mo><msub><mrow><mi>C</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span>, there is a circuit <span><math><msubsup><mrow><mi>C</mi></mrow><mrow><mn>3</mn></mrow><mrow><mo>′</mo></mrow></msubsup></math></span> with <span><math><mo>{</mo><msub><mrow><mi>e</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>e</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>}</mo><mo>⊆</mo><msubsup><mrow><mi>C</mi></mrow><mrow><mn>3</mn></mrow><mrow><mo>′</mo></mrow></msubsup><mo>⊆</mo><mo>(</mo><msub><mrow><mi>C</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>∪</mo><msub><mrow><mi>C</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo><mo>−</mo><mi>e</mi></math></span>. We prove that a connected matroid has this property if and only if it has no two skew circuits. We also characterize such matroids in terms of forbidden series minors, and we give a new matroid axiom system that is built around a modification of SSCE.</div></div>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"173 ","pages":"Article 102983"},"PeriodicalIF":1.3,"publicationDate":"2026-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145268469","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}

{"title":"Hybrid pipe dreams for key polynomials","authors":"Yihan Xiao ,&nbsp;Rui Xiong ,&nbsp;Haofeng Zhang","doi":"10.1016/j.aam.2025.102979","DOIUrl":"10.1016/j.aam.2025.102979","url":null,"abstract":"<div><div>We develop a family of new combinatorial models for key polynomials. It is similar to the hybrid pipe dream model for Schubert polynomials defined recently by Knutson and Udell.</div></div>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"173 ","pages":"Article 102979"},"PeriodicalIF":1.3,"publicationDate":"2026-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145268470","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}

{"title":"d-Separated permutations and q-Stirling numbers of the first kind","authors":"Rosena R.X. Du,&nbsp;Yun Li","doi":"10.1016/j.aam.2025.102976","DOIUrl":"10.1016/j.aam.2025.102976","url":null,"abstract":"&lt;div&gt;&lt;div&gt;Let &lt;em&gt;d&lt;/em&gt; be a nonnegative integer, a &lt;em&gt;d&lt;/em&gt;-separated permutation is a permutation in which every two left-to-right minima are at distance greater than &lt;em&gt;d&lt;/em&gt;. More precisely, for &lt;span&gt;&lt;math&gt;&lt;mi&gt;π&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;π&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;π&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;⋯&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;π&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;∈&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;S&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt;, suppose that &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;π&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;π&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mo&gt;…&lt;/mo&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;π&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt; are the left-to-right minima of &lt;em&gt;π&lt;/em&gt; with &lt;span&gt;&lt;math&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;mo&gt;≥&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/math&gt;&lt;/span&gt; and &lt;span&gt;&lt;math&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;&lt;&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;&lt;&lt;/mo&gt;&lt;mo&gt;⋯&lt;/mo&gt;&lt;mo&gt;&lt;&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;≤&lt;/mo&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt;, then &lt;em&gt;π&lt;/em&gt; is &lt;em&gt;d&lt;/em&gt;-separated if &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;j&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;j&lt;/mi&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;&gt;&lt;/mo&gt;&lt;mi&gt;d&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt; for each &lt;em&gt;j&lt;/em&gt;, &lt;span&gt;&lt;math&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;&lt;&lt;/mo&gt;&lt;mi&gt;j&lt;/mi&gt;&lt;mo&gt;≤&lt;/mo&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt;. In this paper we study different enumerative properties on &lt;em&gt;d&lt;/em&gt;-separated permutations. We first give a recurrence formula of the numbers &lt;span&gt;&lt;math&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;c&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;d&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; of &lt;em&gt;d&lt;/em&gt;-separated permutations in &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;S&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt; with exactly &lt;em&gt;k&lt;/em&gt; left-to-right minima. Then we study the inversion and co-inversion polynomials of &lt;em&gt;d&lt;/em&gt;-separated permutations, and give &lt;em&gt;q&lt;/em&gt;-analogue &lt;span&gt;&lt;math&gt;&lt;msubsup&gt;&lt;mrow&gt;&lt;mi&gt;c&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;q&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;d&lt;/mi&gt;&lt;/mrow&gt;&lt;/msubsup&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; and &lt;span&gt;&lt;math&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;q&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt;-analogue &lt;span&gt;&lt;math&gt;&lt;msubsup&gt;&lt;mrow&gt;&lt;mi&gt;c&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;q&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;d&lt;/mi&gt;&lt;/mrow&gt;&lt;/msubsup&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; of &lt;span&gt;&lt;math&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;c&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;d&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; for any &lt;em&gt;d&lt;/em&gt;. Note that when &lt;span&gt;&lt;math&gt;&lt;mi&gt;d&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/math&gt;&lt;/span&gt;, 0-separated permutations are just all permutations in &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;S&lt;/mi&gt;&lt;/mrow","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"173 ","pages":"Article 102976"},"PeriodicalIF":1.3,"publicationDate":"2026-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145220933","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}

{"title":"A class of generalized chord Minkowski problems","authors":"Lu Zhang","doi":"10.1016/j.aam.2025.102978","DOIUrl":"10.1016/j.aam.2025.102978","url":null,"abstract":"<div><div>In this paper, we consider a class of generalized chord integrals in integral geometry, where the integrand is a generalized kernel that replaces the Riesz kernel. The generalized chord measure arises from the study of the generalized chord integral of convex bodies. We pose the Minkowski problem for the generalized chord measure and obtain the existence of solutions to the related Minkowski problem.</div></div>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"173 ","pages":"Article 102978"},"PeriodicalIF":1.3,"publicationDate":"2026-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145268595","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}

{"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":"2026-02-01","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}

{"title":"A Brualdi-Hoffman-Turán problem on theta graph","authors":"Chang Liu ,&nbsp;Jianping Li ,&nbsp;Shuchao Li ,&nbsp;Yuantian Yu","doi":"10.1016/j.aam.2025.103000","DOIUrl":"10.1016/j.aam.2025.103000","url":null,"abstract":"&lt;div&gt;&lt;div&gt;The Brualdi-Hoffman-Turán problem, a central topic in spectral graph theory, seeks to determine maximum spectral radius &lt;span&gt;&lt;math&gt;&lt;mi&gt;ρ&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; of an &lt;em&gt;F&lt;/em&gt;-free graph &lt;em&gt;G&lt;/em&gt; with &lt;em&gt;m&lt;/em&gt; edges. This problem has attracted significant attention in recent years. Let &lt;span&gt;&lt;math&gt;&lt;msubsup&gt;&lt;mrow&gt;&lt;mi&gt;C&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;l&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;/mrow&gt;&lt;/msubsup&gt;&lt;/math&gt;&lt;/span&gt; denote the theta graph obtained by adding a chord between two vertices at distance 2 on cycle &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;C&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;l&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt;. Zhai, Lin, and Shu &lt;span&gt;&lt;span&gt;[32]&lt;/span&gt;&lt;/span&gt; conjectured that, for &lt;span&gt;&lt;math&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;mo&gt;⩾&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/math&gt;&lt;/span&gt; and sufficiently large &lt;em&gt;m&lt;/em&gt;, if &lt;em&gt;G&lt;/em&gt; is &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;C&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt;-free or &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;C&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt;-free, then &lt;span&gt;&lt;math&gt;&lt;mi&gt;ρ&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;⩽&lt;/mo&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;msqrt&gt;&lt;mrow&gt;&lt;mn&gt;4&lt;/mn&gt;&lt;mi&gt;m&lt;/mi&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msqrt&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;/math&gt;&lt;/span&gt;, with equality if and only if &lt;span&gt;&lt;math&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;mo&gt;≅&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;K&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;∨&lt;/mo&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;mi&gt;m&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;K&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt;. This conjecture was highlighted in Liu and Ning's survey &lt;span&gt;&lt;span&gt;[18]&lt;/span&gt;&lt;/span&gt; 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 &lt;span&gt;&lt;math&gt;&lt;mi&gt;ρ&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; and the corresponding extremal graph hold for &lt;span&gt;&lt;math&gt;&lt;msubsup&gt;&lt;mrow&gt;&lt;mi&gt;C&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;/mrow&gt;&lt;/msubsup&gt;&lt;/math&gt;&lt;/span&gt;-free, or &lt;span&gt;&lt;math&gt;&lt;msubsup&gt;&lt;mrow&gt;&lt;mi&gt;C&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;/mrow&gt;&lt;/msubsup&gt;&lt;/math&gt;&lt;/span&gt;-free graphs. Recently, Li, Zhai, and Shu &lt;span&gt;&lt;span&gt;[14]&lt;/span&gt;&lt;/span&gt; resolved both conjectures completely. Note that the above extremal graph &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;K&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;∨&lt;/mo&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;mi&gt;m&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;K&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt; is well-defined only if &lt;span&gt;&lt;math&gt;&lt;mi&gt;m&lt;/mi&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;)&lt;/","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"173 ","pages":"Article 103000"},"PeriodicalIF":1.3,"publicationDate":"2026-02-01","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}

{"title":"Knuth's big-chooser matchbox process: the case of many matchboxes","authors":"Mark Dukes,&nbsp;Andrew Mullins","doi":"10.1016/j.aam.2025.102996","DOIUrl":"10.1016/j.aam.2025.102996","url":null,"abstract":"<div><div>Banach's matchbox problem considers the setting of two matchboxes that each initially contain the same number of matches. Boxes are chosen with equal probability and a match removed each time. The problem concerns the law of the number of matches remaining in one box once the other box empties. Knuth considered a generalization of this problem whereby <em>big-choosers</em> arrive with probability <em>p</em> and remove a match from the box with the most number remaining, and <em>little-choosers</em> arrive with probability <span><math><mn>1</mn><mo>−</mo><mi>p</mi></math></span> and remove a match from the box with the least number remaining.</div><div>In this paper we consider Knuth's generalization for the case of <em>k</em> matchboxes in which there are <em>big-choosers</em> and <em>little-choosers</em>. We determine the generating function for the expected number of matches remaining in <span><math><mi>k</mi><mo>−</mo><mn>1</mn></math></span> matchboxes once a box first empties, a quantity we refer to as the ‘residue’. Interestingly, this generating function is a quotient whose denominator contains a generating function for a special case of the Raney numbers. The form for this generating function allows us to give an expression for the expected residue in terms of a sum that involves diagonal state return probabilities, where a diagonal state is a configuration in which all matchboxes each contain the same number of matches. We use analytic techniques to determine the asymptotic behavior of this expected value for all values of <em>p</em>, which involves the study of an asymmetric random walk.</div><div>In addition to this we consider the expected value of the order of the first return to a diagonal state and determine the asymptotic behavior of this quantity. The coefficients of the diagonal state probability generating function are shown to be related to ‘manila folder configurations in a filing cabinet’, and we make this connection precise. This allows us to use known results for the enumeration of such manila folder configurations to give a closed form expression for the diagonal state return probabilities.</div></div>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"173 ","pages":"Article 102996"},"PeriodicalIF":1.3,"publicationDate":"2026-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145474010","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}

{"title":"The optimal upper bound on the MP-ratio for quaternary words","authors":"Kristina Ago,&nbsp;Bojan Bašić","doi":"10.1016/j.aam.2025.102984","DOIUrl":"10.1016/j.aam.2025.102984","url":null,"abstract":"<div><div>The so-called <em>MP-ratio</em> is a kind of measure of how “packed with palindromes” a given word is. The lower bound on the MP-ratio for the set of all <em>n</em>-ary words is (trivially) 1, while the best possible upper bound is an open problem in the general case. It is solved for <span><math><mi>n</mi><mo>=</mo><mn>2</mn></math></span> (where the optimal upper bound is 4) and for <span><math><mi>n</mi><mo>=</mo><mn>3</mn></math></span> (where the optimal upper bound is 6). Also, it is known that in the <em>n</em>-ary case the optimal bound is between 2<em>n</em> and the order of growth <span><math><mi>n</mi><msup><mrow><mn>2</mn></mrow><mrow><mfrac><mrow><mi>n</mi></mrow><mrow><mn>2</mn></mrow></mfrac></mrow></msup></math></span>. In this article we solve this problem for quaternary words, for which we show that the best possible upper bound on the MP-ratio equals 8. We believe that this is the last case in which the result is 2<em>n</em>, that is, we believe that for <span><math><mi>n</mi><mo>⩾</mo><mn>5</mn></math></span> there are words whose MP-ratio is strictly larger than 2<em>n</em>.</div></div>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"173 ","pages":"Article 102984"},"PeriodicalIF":1.3,"publicationDate":"2026-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145333347","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}

{"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":"2026-02-01","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}