Pub Date : 2025-11-10DOI: 10.1016/j.aam.2025.102995
Eri Matsudo , Kanako Oshiro , Gaishi Yamagishi
This is the first paper which discusses minimum numbers of “region” colors for knots, while minimum numbers of arc colors are well-studied. In this paper, we consider minimum numbers of colors of knots for Dehn colorings. In particular, we will show that for any odd prime number p and any Dehn p-colorable knot K, the minimum number of colors for K is at least . Moreover, we will define the -palette graph for a set of colors. The -palette graphs are quite useful to give candidates of sets of colors which might realize a nontrivially Dehn p-colored diagram. In Appendix, we also prove that for Dehn 5-colorable knot, the minimum number of colors is 4.
{"title":"Minimum numbers of Dehn colors of knots and R-palette graphs","authors":"Eri Matsudo , Kanako Oshiro , Gaishi Yamagishi","doi":"10.1016/j.aam.2025.102995","DOIUrl":"10.1016/j.aam.2025.102995","url":null,"abstract":"<div><div>This is the first paper which discusses minimum numbers of “region” colors for knots, while minimum numbers of arc colors are well-studied. In this paper, we consider minimum numbers of colors of knots for Dehn colorings. In particular, we will show that for any odd prime number <em>p</em> and any Dehn <em>p</em>-colorable knot <em>K</em>, the minimum number of colors for <em>K</em> is at least <span><math><mo>⌊</mo><msub><mrow><mi>log</mi></mrow><mrow><mn>2</mn></mrow></msub><mo></mo><mi>p</mi><mo>⌋</mo><mo>+</mo><mn>2</mn></math></span>. Moreover, we will define the <span><math><mi>R</mi></math></span>-palette graph for a set of colors. The <span><math><mi>R</mi></math></span>-palette graphs are quite useful to give candidates of sets of colors which might realize a nontrivially Dehn <em>p</em>-colored diagram. In Appendix, we also prove that for Dehn 5-colorable knot, the minimum number of colors is 4.</div></div>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"173 ","pages":"Article 102995"},"PeriodicalIF":1.3,"publicationDate":"2025-11-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145528744","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 : 2025-11-06DOI: 10.1016/j.aam.2025.102994
Wei Wang , Da Zhao
We provide a criterion to show that a graph is identified by its multivariate graph spectrum. Haemers conjectured that almost all graphs are identified by their spectra. Our approach suggests that almost all graphs are identified by their generalized block Laplacian spectra.
{"title":"Graph isomorphism and multivariate graph spectrum","authors":"Wei Wang , Da Zhao","doi":"10.1016/j.aam.2025.102994","DOIUrl":"10.1016/j.aam.2025.102994","url":null,"abstract":"<div><div>We provide a criterion to show that a graph is identified by its multivariate graph spectrum. Haemers conjectured that almost all graphs are identified by their spectra. Our approach suggests that almost all graphs are identified by their generalized block Laplacian spectra.</div></div>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"173 ","pages":"Article 102994"},"PeriodicalIF":1.3,"publicationDate":"2025-11-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145474011","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 : 2025-11-06DOI: 10.1016/j.aam.2025.102996
Mark Dukes, Andrew Mullins
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 big-choosers arrive with probability p and remove a match from the box with the most number remaining, and little-choosers arrive with probability and remove a match from the box with the least number remaining.
In this paper we consider Knuth's generalization for the case of k matchboxes in which there are big-choosers and little-choosers. We determine the generating function for the expected number of matches remaining in 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 p, which involves the study of an asymmetric random walk.
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.
{"title":"Knuth's big-chooser matchbox process: the case of many matchboxes","authors":"Mark Dukes, 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":"2025-11-06","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}
Pub Date : 2025-10-31DOI: 10.1016/j.aam.2025.102993
Runqiao Li , Andrew Y.Z. Wang , Zheng Xu
In this work, we introduce a new partition statistic, named block index, and explore its relationship with other well-known statistics, including Dyson's crank. We delve into the combinatorial significance of the block index, shedding light on its role in revealing the more intricate structure of certain recently discovered partition identities.
{"title":"Block index and integer partitions","authors":"Runqiao Li , Andrew Y.Z. Wang , Zheng Xu","doi":"10.1016/j.aam.2025.102993","DOIUrl":"10.1016/j.aam.2025.102993","url":null,"abstract":"<div><div>In this work, we introduce a new partition statistic, named block index, and explore its relationship with other well-known statistics, including Dyson's crank. We delve into the combinatorial significance of the block index, shedding light on its role in revealing the more intricate structure of certain recently discovered partition identities.</div></div>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"173 ","pages":"Article 102993"},"PeriodicalIF":1.3,"publicationDate":"2025-10-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145424662","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 : 2025-10-21DOI: 10.1016/j.aam.2025.102984
Kristina Ago, Bojan Bašić
The so-called MP-ratio 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 n-ary words is (trivially) 1, while the best possible upper bound is an open problem in the general case. It is solved for (where the optimal upper bound is 4) and for (where the optimal upper bound is 6). Also, it is known that in the n-ary case the optimal bound is between 2n and the order of growth . 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 2n, that is, we believe that for there are words whose MP-ratio is strictly larger than 2n.
{"title":"The optimal upper bound on the MP-ratio for quaternary words","authors":"Kristina Ago, 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":"2025-10-21","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}
Pub Date : 2025-10-20DOI: 10.1016/j.aam.2025.102985
Fengming Dong , Meiqiao Zhang
In his article [J. Comb. Theory Ser. B16 (1974), 168–174], Tutte called two graphs T-equivalent (i.e., codichromatic) if they have the same Tutte polynomial and showed that graphs G and are T-equivalent if is obtained from G by flipping a rotor (i.e., replacing it by its mirror) of order at most 5, where a rotor of order k in G is an induced subgraph R having an automorphism ψ with a vertex orbit of size k such that every vertex of R is only adjacent to vertices in R unless it is in this vertex orbit. In this article, we show the above result due to Tutte can be extended to a rotor R of order if the subgraph of G induced by all those edges of G which are not in R satisfies certain conditions. Also, we provide a new method for generating infinitely many non-isomorphic T-equivalent pairs of graphs.
在他的文章中[J]。合成杆。Ser的理论。B 16 (1974), 168 - 174], Tutte叫两个图形T-equivalent(即codichromatic)如果他们有相同的Tutte多项式和显示,图G, G T-equivalent如果G是来自G翻转一个转子(即取代它的镜像)的订单最多5 k阶转子在G是一种诱导子图R有自同构与一个顶点ψ轨道{ψ(u):我≥0}的k大小的每个顶点只相邻顶点在R,除非它是在这个顶点轨道。在本文中，我们证明了由于Tutte的上述结果可以推广到k≥6阶的转子R，如果G的所有不在R中的边所诱导的G的子图满足一定的条件。此外，我们还提供了一种生成无限多个非同构t等价图对的新方法。
{"title":"A study on T-equivalent graphs","authors":"Fengming Dong , Meiqiao Zhang","doi":"10.1016/j.aam.2025.102985","DOIUrl":"10.1016/j.aam.2025.102985","url":null,"abstract":"<div><div>In his article [<em>J. Comb. Theory Ser. B</em> <strong>16</strong> (1974), 168–174], Tutte called two graphs <em>T</em>-equivalent (i.e., codichromatic) if they have the same Tutte polynomial and showed that graphs <em>G</em> and <span><math><msup><mrow><mi>G</mi></mrow><mrow><mo>′</mo></mrow></msup></math></span> are <em>T</em>-equivalent if <span><math><msup><mrow><mi>G</mi></mrow><mrow><mo>′</mo></mrow></msup></math></span> is obtained from <em>G</em> by flipping a rotor (i.e., replacing it by its mirror) of order at most 5, where a rotor of order <em>k</em> in <em>G</em> is an induced subgraph <em>R</em> having an automorphism <em>ψ</em> with a vertex orbit <span><math><mo>{</mo><msup><mrow><mi>ψ</mi></mrow><mrow><mi>i</mi></mrow></msup><mo>(</mo><mi>u</mi><mo>)</mo><mo>:</mo><mi>i</mi><mo>≥</mo><mn>0</mn><mo>}</mo></math></span> of size <em>k</em> such that every vertex of <em>R</em> is only adjacent to vertices in <em>R</em> unless it is in this vertex orbit. In this article, we show the above result due to Tutte can be extended to a rotor <em>R</em> of order <span><math><mi>k</mi><mo>≥</mo><mn>6</mn></math></span> if the subgraph of <em>G</em> induced by all those edges of <em>G</em> which are not in <em>R</em> satisfies certain conditions. Also, we provide a new method for generating infinitely many non-isomorphic <em>T</em>-equivalent pairs of graphs.</div></div>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"173 ","pages":"Article 102985"},"PeriodicalIF":1.3,"publicationDate":"2025-10-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145363042","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 : 2025-10-13DOI: 10.1016/j.aam.2025.102982
Ludovic Schwob
Double cosets appear in many contexts in combinatorics, for example in the enumeration of certain objects up to symmetries. Double cosets in a quotient of the form have an inverse, and can be their own inverse. In this paper we present various formulas enumerating double cosets, and in particular self-inverse double cosets. We study double cosets in classical groups, especially the symmetric groups and the general linear groups, explaining how to obtain the information on their conjugacy classes required to apply our formulas. We also consider double cosets of parabolic subgroups of Coxeter groups of type B.
{"title":"On the enumeration of double cosets and self-inverse double cosets","authors":"Ludovic Schwob","doi":"10.1016/j.aam.2025.102982","DOIUrl":"10.1016/j.aam.2025.102982","url":null,"abstract":"<div><div>Double cosets appear in many contexts in combinatorics, for example in the enumeration of certain objects up to symmetries. Double cosets in a quotient of the form <span><math><mi>H</mi><mo>﹨</mo><mi>G</mi><mo>/</mo><mi>H</mi></math></span> have an inverse, and can be their own inverse. In this paper we present various formulas enumerating double cosets, and in particular self-inverse double cosets. We study double cosets in classical groups, especially the symmetric groups and the general linear groups, explaining how to obtain the information on their conjugacy classes required to apply our formulas. We also consider double cosets of parabolic subgroups of Coxeter groups of type B.</div></div>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"173 ","pages":"Article 102982"},"PeriodicalIF":1.3,"publicationDate":"2025-10-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145320793","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 : 2025-10-10DOI: 10.1016/j.aam.2025.102983
Christine Cho , James Oxley , Suijie Wang
If and are circuits in a matroid M with in and e in , then M has a circuit such that . This strong circuit elimination axiom is inherently asymmetric. A matroid M has the symmetric strong circuit elimination property (SSCE) if, when the above conditions hold and , there is a circuit with . 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.
{"title":"The symmetric strong circuit elimination property","authors":"Christine Cho , James Oxley , 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":"2025-10-10","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}
Pub Date : 2025-10-09DOI: 10.1016/j.aam.2025.102977
Xiangzi Meng , Hao Pan
The Eulerian number counts all permutations on having exactly k ascents. In this paper, we give an enumerative proof of the following congruence: where p is prime, and .
欧拉数< nk >计算{0,1，…，n−1}上的所有恰好有k个上升的排列。本文给出了下列同余的一个枚举证明：< ap−1bp+l >≡(−1)b(l+1)a−1(a−1b)(modp)，其中p为素数，0≤b<；a且0≤l≤p−1。
{"title":"Enumerative proof of a curious congruence for Eulerian numbers","authors":"Xiangzi Meng , Hao Pan","doi":"10.1016/j.aam.2025.102977","DOIUrl":"10.1016/j.aam.2025.102977","url":null,"abstract":"<div><div>The Eulerian number <span><math><mo>〈</mo><mtable><mtr><mtd><mi>n</mi></mtd></mtr><mtr><mtd><mi>k</mi></mtd></mtr></mtable><mo>〉</mo></math></span> counts all permutations on <span><math><mo>{</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>,</mo><mo>…</mo><mo>,</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo>}</mo></math></span> having exactly <em>k</em> ascents. In this paper, we give an enumerative proof of the following congruence:<span><span><span><math><mrow><mo>〈</mo><mtable><mtr><mtd><mrow><mi>a</mi><mi>p</mi><mo>−</mo><mn>1</mn></mrow></mtd></mtr><mtr><mtd><mrow><mi>b</mi><mi>p</mi><mo>+</mo><mi>l</mi></mrow></mtd></mtr></mtable><mo>〉</mo></mrow><mo>≡</mo><msup><mrow><mo>(</mo><mo>−</mo><mn>1</mn><mo>)</mo></mrow><mrow><mi>b</mi></mrow></msup><msup><mrow><mo>(</mo><mi>l</mi><mo>+</mo><mn>1</mn><mo>)</mo></mrow><mrow><mi>a</mi><mo>−</mo><mn>1</mn></mrow></msup><mrow><mo>(</mo><mtable><mtr><mtd><mrow><mi>a</mi><mo>−</mo><mn>1</mn></mrow></mtd></mtr><mtr><mtd><mi>b</mi></mtd></mtr></mtable><mo>)</mo></mrow><mspace></mspace><mo>(</mo><mrow><mi>mod</mi></mrow><mspace></mspace><mi>p</mi><mo>)</mo><mo>,</mo></math></span></span></span> where <em>p</em> is prime, <span><math><mn>0</mn><mo>≤</mo><mi>b</mi><mo><</mo><mi>a</mi></math></span> and <span><math><mn>0</mn><mo>≤</mo><mi>l</mi><mo>≤</mo><mi>p</mi><mo>−</mo><mn>1</mn></math></span>.</div></div>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"173 ","pages":"Article 102977"},"PeriodicalIF":1.3,"publicationDate":"2025-10-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145268467","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 : 2025-10-09DOI: 10.1016/j.aam.2025.102980
Hongying Lin , Bo Zhou
Let m 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 m. 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 m. Given m, there is exactly one positive integer n such that . We establish some structural properties of the extremal graphs for all m and solve the problem for . 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.
{"title":"Extremal distance spectral radius of graphs with fixed size","authors":"Hongying Lin , 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><</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":"2025-10-09","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}
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