Pub Date : 2026-05-01Epub Date: 2026-01-27DOI: 10.1016/j.aam.2026.103046
Gianira N. Alfarano , Eimear Byrne
In this paper, we describe properties of the characteristic polynomial of a weighted lattice and show that it has a recursive description, which we use to obtain results on the critical exponent of q-polymatroids. We give a Critical Theorem for representable q-polymatroids and we provide a lower bound on the critical exponent. We show that q-polymatroids arising from certain families of rank-metric codes attain this lower bound.
{"title":"Recursive properties of the characteristic polynomial of weighted lattices","authors":"Gianira N. Alfarano , Eimear Byrne","doi":"10.1016/j.aam.2026.103046","DOIUrl":"10.1016/j.aam.2026.103046","url":null,"abstract":"<div><div>In this paper, we describe properties of the characteristic polynomial of a weighted lattice and show that it has a recursive description, which we use to obtain results on the critical exponent of <em>q</em>-polymatroids. We give a Critical Theorem for representable <em>q</em>-polymatroids and we provide a lower bound on the critical exponent. We show that <em>q</em>-polymatroids arising from certain families of rank-metric codes attain this lower bound.</div></div>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"176 ","pages":"Article 103046"},"PeriodicalIF":1.3,"publicationDate":"2026-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146080813","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 : 2026-05-01Epub Date: 2026-02-03DOI: 10.1016/j.aam.2026.103052
Changjiang Bu , Lixiang Chen , Yongtang Shi
It is well known that the algebraic multiplicity of an eigenvalue of a graph (or real symmetric matrix) is equal to the dimension of its corresponding linear eigen-subspace, also known as the geometric multiplicity. However, for hypergraphs, the relationship between these two multiplicities remains an open problem. For a graph and , the k-power hypergraph is a k-uniform hypergraph obtained by adding new vertices to each edge of G, who always has non-real eigenvalues. In this paper, we determine the second-largest modulus Λ among the eigenvalues of , which is indeed an eigenvalue of . The projective eigenvariety associated with Λ is the set of the eigenvectors of corresponding to Λ considered in the complex projective space. We show that the dimension of is zero, i.e., there are finitely many eigenvectors corresponding to Λ up to a scalar. We give both the algebraic multiplicity of Λ and the total multiplicity of the eigenvector in in terms of the number of the weakest edges of G. Our results show that these two multiplicities are equal.
{"title":"On the second-largest modulus among the eigenvalues of a power hypergraph","authors":"Changjiang Bu , Lixiang Chen , Yongtang Shi","doi":"10.1016/j.aam.2026.103052","DOIUrl":"10.1016/j.aam.2026.103052","url":null,"abstract":"<div><div>It is well known that the algebraic multiplicity of an eigenvalue of a graph (or real symmetric matrix) is equal to the dimension of its corresponding linear eigen-subspace, also known as the geometric multiplicity. However, for hypergraphs, the relationship between these two multiplicities remains an open problem. For a graph <span><math><mi>G</mi><mo>=</mo><mo>(</mo><mi>V</mi><mo>,</mo><mi>E</mi><mo>)</mo></math></span> and <span><math><mi>k</mi><mo>≥</mo><mn>3</mn></math></span>, the <em>k</em>-power hypergraph <span><math><msup><mrow><mi>G</mi></mrow><mrow><mo>(</mo><mi>k</mi><mo>)</mo></mrow></msup></math></span> is a <em>k</em>-uniform hypergraph obtained by adding <span><math><mi>k</mi><mo>−</mo><mn>2</mn></math></span> new vertices to each edge of <em>G</em>, who always has non-real eigenvalues. In this paper, we determine the second-largest modulus Λ among the eigenvalues of <span><math><msup><mrow><mi>G</mi></mrow><mrow><mo>(</mo><mi>k</mi><mo>)</mo></mrow></msup></math></span>, which is indeed an eigenvalue of <span><math><msup><mrow><mi>G</mi></mrow><mrow><mo>(</mo><mi>k</mi><mo>)</mo></mrow></msup></math></span>. The projective eigenvariety <span><math><msub><mrow><mi>V</mi></mrow><mrow><mi>Λ</mi></mrow></msub></math></span> associated with Λ is the set of the eigenvectors of <span><math><msup><mrow><mi>G</mi></mrow><mrow><mo>(</mo><mi>k</mi><mo>)</mo></mrow></msup></math></span> corresponding to Λ considered in the complex projective space. We show that the dimension of <span><math><msub><mrow><mi>V</mi></mrow><mrow><mi>Λ</mi></mrow></msub></math></span> is zero, i.e., there are finitely many eigenvectors corresponding to Λ up to a scalar. We give both the algebraic multiplicity of Λ and the total multiplicity of the eigenvector in <span><math><msub><mrow><mi>V</mi></mrow><mrow><mi>Λ</mi></mrow></msub></math></span> in terms of the number of the weakest edges of <em>G</em>. Our results show that these two multiplicities are equal.</div></div>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"176 ","pages":"Article 103052"},"PeriodicalIF":1.3,"publicationDate":"2026-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146190640","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 : 2026-05-01Epub Date: 2026-02-10DOI: 10.1016/j.aam.2026.103056
Marco Longinetti , Simone Naldi , Adriana Venturi
The R-hulloid, in the Euclidean space , of the set of vertices V of a tetrahedron T is the minimal closed set containing V such that its complement is the union of open balls of radius R. When R is greater than the circumradius of T, the boundary of the R-hulloid consists of V and possibly of four spherical subsets of well defined spheres of radius R through the vertices of T. The existence of a value such that these subsets collapse into a point , in the interior of T, is investigated; in such a case belongs to four spheres of radius , each one through three vertices of T and not containing the fourth one. As a consequence, the range of ρ such that V is a ρ-body is described completely. This work generalizes to dimension three previous results, proved in the planar case and related to the three circles Johnson's Theorem.
{"title":"R-hulloid of the vertices of a tetrahedron","authors":"Marco Longinetti , Simone Naldi , Adriana Venturi","doi":"10.1016/j.aam.2026.103056","DOIUrl":"10.1016/j.aam.2026.103056","url":null,"abstract":"<div><div>The <em>R</em>-hulloid, in the Euclidean space <span><math><msup><mrow><mi>R</mi></mrow><mrow><mn>3</mn></mrow></msup></math></span>, of the set of vertices <em>V</em> of a tetrahedron <em>T</em> is the minimal closed set containing <em>V</em> such that its complement is the union of open balls of radius <em>R</em>. When <em>R</em> is greater than the circumradius of <em>T</em>, the boundary of the <em>R</em>-hulloid consists of <em>V</em> and possibly of four spherical subsets of well defined spheres of radius <em>R</em> through the vertices of <em>T</em>. The existence of a value <span><math><msup><mrow><mi>R</mi></mrow><mrow><mo>⁎</mo></mrow></msup></math></span> such that these subsets collapse into a point <span><math><msup><mrow><mi>O</mi></mrow><mrow><mo>⁎</mo></mrow></msup></math></span>, in the interior of <em>T</em>, is investigated; in such a case <span><math><msup><mrow><mi>O</mi></mrow><mrow><mo>⁎</mo></mrow></msup></math></span> belongs to four spheres of radius <span><math><msup><mrow><mi>R</mi></mrow><mrow><mo>⁎</mo></mrow></msup></math></span>, each one through three vertices of <em>T</em> and not containing the fourth one. As a consequence, the range of <em>ρ</em> such that <em>V</em> is a <em>ρ</em>-body is described completely. This work generalizes to dimension three previous results, proved in the planar case and related to the three circles Johnson's Theorem.</div></div>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"176 ","pages":"Article 103056"},"PeriodicalIF":1.3,"publicationDate":"2026-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146190620","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 : 2026-05-01Epub Date: 2026-02-09DOI: 10.1016/j.aam.2026.103057
Fernanda Moreira Baêta
Extending classical results on polytopal approximation of convex bodies, we derive asymptotic formulas for the weighted approximation of smooth convex functions by piecewise affine convex functions as the number of their facets tends to infinity. These asymptotic expressions are formulated in terms of a functional that extends the notion of affine surface area to the functional setting.
{"title":"Asymptotic weighted approximation of convex functions","authors":"Fernanda Moreira Baêta","doi":"10.1016/j.aam.2026.103057","DOIUrl":"10.1016/j.aam.2026.103057","url":null,"abstract":"<div><div>Extending classical results on polytopal approximation of convex bodies, we derive asymptotic formulas for the weighted approximation of smooth convex functions by piecewise affine convex functions as the number of their facets tends to infinity. These asymptotic expressions are formulated in terms of a functional that extends the notion of affine surface area to the functional setting.</div></div>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"176 ","pages":"Article 103057"},"PeriodicalIF":1.3,"publicationDate":"2026-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146190621","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 : 2026-05-01Epub Date: 2026-01-27DOI: 10.1016/j.aam.2026.103051
Shane Chern , Lin Jiu , Shuhan Li , Liuquan Wang
It is a standard result that the Hankel determinants for a sequence stay invariant after performing the binomial transform on this sequence. In this work, we extend the scenario to q-binomial transforms and study the behavior of the leading coefficient in such Hankel determinants. We also investigate the leading coefficient in the Hankel determinants for even-indexed Bernoulli polynomials with recourse to a curious binomial transform. In particular, the degrees of these Hankel determinants share the same nature as those in one of the q-binomial cases.
{"title":"Leading coefficient in the Hankel determinants related to binomial and q-binomial transforms","authors":"Shane Chern , Lin Jiu , Shuhan Li , Liuquan Wang","doi":"10.1016/j.aam.2026.103051","DOIUrl":"10.1016/j.aam.2026.103051","url":null,"abstract":"<div><div>It is a standard result that the Hankel determinants for a sequence stay invariant after performing the binomial transform on this sequence. In this work, we extend the scenario to <em>q</em>-binomial transforms and study the behavior of the leading coefficient in such Hankel determinants. We also investigate the leading coefficient in the Hankel determinants for even-indexed Bernoulli polynomials with recourse to a curious binomial transform. In particular, the degrees of these Hankel determinants share the same nature as those in one of the <em>q</em>-binomial cases.</div></div>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"176 ","pages":"Article 103051"},"PeriodicalIF":1.3,"publicationDate":"2026-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146080818","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 : 2026-05-01Epub Date: 2026-01-23DOI: 10.1016/j.aam.2026.103048
Deke Zhao , Zhankui Xiao
The paper aims to provide a categorification of the Brenti–Welker identity involving Eulerian numbers in (Adv. Appl. Math. 42 (2009): 545–556) by lifting it from an enumerative equality to an isomorphism of symmetric group representations. To do so, we study the decomposition of the tensor product of and modules affording Foulkes characters as modules of the symmetric group. The main ingredient of the proof is a combinatorial identity which may be of independent interest.
{"title":"A categorification of the Brenti–Welker identity","authors":"Deke Zhao , Zhankui Xiao","doi":"10.1016/j.aam.2026.103048","DOIUrl":"10.1016/j.aam.2026.103048","url":null,"abstract":"<div><div>The paper aims to provide a categorification of the Brenti–Welker identity involving Eulerian numbers in (Adv. Appl. Math. 42 (2009): 545–556) by lifting it from an enumerative equality to an isomorphism of symmetric group representations. To do so, we study the decomposition of the tensor product of <span><math><msup><mrow><mo>(</mo><msup><mrow><mi>C</mi></mrow><mrow><mi>r</mi></mrow></msup><mo>)</mo></mrow><mrow><mo>⊗</mo><mi>n</mi></mrow></msup></math></span> and modules affording Foulkes characters as modules of the symmetric group. The main ingredient of the proof is a combinatorial identity which may be of independent interest.</div></div>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"176 ","pages":"Article 103048"},"PeriodicalIF":1.3,"publicationDate":"2026-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146039693","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 : 2026-05-01Epub Date: 2026-01-28DOI: 10.1016/j.aam.2026.103054
Jian Cao
Polynomial expansions of analytic solutions of the heat equation occupy important positions in disciplines such as mathematics and physics [68]. In this paper, we introduce q-3D hypergeometric polynomials and find their corresponding q-heat equations, which were motivated by Ismail and Zhang (2016) [32] and (2017) [33]. We deduce several types of generating functions for q-3D hypergeometric polynomials and Askey–Wilson type integral involving q-3D hypergeometric polynomials by the method of heat equation type q-partial differential equations. In addition, we generalize some results of Ismail and Zhang (2017) [33], Milne (1997) [49] and Jia (2021) [38].
热方程解析解的多项式展开式在数学、物理等学科中占有重要地位[68]。本文引入了由Ismail and Zhang(2016)[32]和（2017）[33]提出的q-3D超几何多项式，并找到了其对应的q-heat方程。利用热方程型q-偏微分方程的方法推导了q-3D超几何多项式的几种生成函数和涉及q-3D超几何多项式的Askey-Wilson型积分。此外，我们还推广了Ismail and Zhang (2017) b[33]、Milne(1997)[49]和Jia(2021)[38]的一些结果。
{"title":"The generalized q-heat equations for q-3D hypergeometric polynomials with applications to generating functions and Askey–Wilson integrals","authors":"Jian Cao","doi":"10.1016/j.aam.2026.103054","DOIUrl":"10.1016/j.aam.2026.103054","url":null,"abstract":"<div><div>Polynomial expansions of analytic solutions of the heat equation occupy important positions in disciplines such as mathematics and physics <span><span>[68]</span></span>. In this paper, we introduce <em>q</em>-3D hypergeometric polynomials and find their corresponding <em>q</em>-heat equations, which were motivated by Ismail and Zhang (2016) <span><span>[32]</span></span> and (2017) <span><span>[33]</span></span>. We deduce several types of generating functions for <em>q</em>-3D hypergeometric polynomials and Askey–Wilson type integral involving <em>q</em>-3D hypergeometric polynomials by the method of heat equation type <em>q</em>-partial differential equations. In addition, we generalize some results of Ismail and Zhang (2017) <span><span>[33]</span></span>, Milne (1997) <span><span>[49]</span></span> and Jia (2021) <span><span>[38]</span></span>.</div></div>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"176 ","pages":"Article 103054"},"PeriodicalIF":1.3,"publicationDate":"2026-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146080819","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 : 2026-05-01Epub Date: 2026-01-28DOI: 10.1016/j.aam.2026.103050
Aaron Robertson
We extend Deuber's theorem on -sets to hold over the multidimensional positive integer lattices. This leads to a multidimensional Rado theorem where we are guaranteed monochromatic multidimensional points in all finite colorings of where the set of coordinates satisfies the given linear Rado system.
{"title":"A multidimensional Rado Theorem","authors":"Aaron Robertson","doi":"10.1016/j.aam.2026.103050","DOIUrl":"10.1016/j.aam.2026.103050","url":null,"abstract":"<div><div>We extend Deuber's theorem on <span><math><mo>(</mo><mi>m</mi><mo>,</mo><mi>p</mi><mo>,</mo><mi>c</mi><mo>)</mo></math></span>-sets to hold over the multidimensional positive integer lattices. This leads to a multidimensional Rado theorem where we are guaranteed monochromatic multidimensional points in all finite colorings of <span><math><msup><mrow><mo>(</mo><msup><mrow><mi>Z</mi></mrow><mrow><mo>+</mo></mrow></msup><mo>)</mo></mrow><mrow><mi>d</mi></mrow></msup></math></span> where the <span><math><msup><mrow><mi>i</mi></mrow><mrow><mi>th</mi></mrow></msup></math></span> set of coordinates satisfies the <span><math><msup><mrow><mi>i</mi></mrow><mrow><mi>th</mi></mrow></msup></math></span> given linear Rado system.</div></div>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"176 ","pages":"Article 103050"},"PeriodicalIF":1.3,"publicationDate":"2026-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146080817","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 : 2026-05-01Epub Date: 2026-02-12DOI: 10.1016/j.aam.2026.103055
Gunnar Fløystad
We get new Hopf algebras (HA): 1. A wealth of quotient HA's of the Malvenuto-Reutenauer HA (the Loday-Ronco HA being a special case). They consist of the permutations avoiding an arbitrary set of permutations without global descents, 2. A HA of pairs of parking filtrations, and 3. Four HA of pairs of preorders.
New concepts in this setting are: 1. a category whose objects are sets, but morphisms are represented by matrices of natural numbers, and 2. restriction species on sets coming with pairs of natural transformations to the species of preorders. These induce two coproducts and . Dualizing gives product and coproduct , giving bimonoid species.
{"title":"Combinatorial Hopf algebras from restriction species with preorder cuts","authors":"Gunnar Fløystad","doi":"10.1016/j.aam.2026.103055","DOIUrl":"10.1016/j.aam.2026.103055","url":null,"abstract":"<div><div>We get new Hopf algebras (HA): 1. A wealth of quotient HA's of the Malvenuto-Reutenauer HA (the Loday-Ronco HA being a special case). They consist of the permutations avoiding an <em>arbitrary</em> set of permutations without global descents, 2. A HA of pairs of parking filtrations, and 3. Four HA of pairs of preorders.</div><div>New concepts in this setting are: 1. a category <span><math><mi>se</mi><msub><mrow><mi>t</mi></mrow><mrow><mi>N</mi></mrow></msub></math></span> whose objects are sets, but morphisms are represented by matrices of natural numbers, and 2. restriction species <span><math><mi>S</mi></math></span> on sets coming with pairs of natural transformations <span><math><msub><mrow><mi>π</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>π</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>:</mo><mi>S</mi><mo>→</mo><mrow><mi>Pre</mi></mrow></math></span> to the species of preorders. These induce two coproducts <span><math><msub><mrow><mi>Δ</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span> and <span><math><msub><mrow><mi>Δ</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span>. Dualizing <span><math><msub><mrow><mi>Δ</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span> gives product <span><math><msub><mrow><mi>μ</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span> and coproduct <span><math><msub><mrow><mi>Δ</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span>, giving bimonoid species.</div></div>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"176 ","pages":"Article 103055"},"PeriodicalIF":1.3,"publicationDate":"2026-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146190619","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 : 2026-05-01Epub Date: 2026-02-03DOI: 10.1016/j.aam.2026.103053
Yichao Chen
The permutation-partition pair was initially introduced by Stahl in 1980 and independently by Archdeacon in a geometric form in 1986. This pair is a generalization of graphs and oriented embeddings of graphs. This paper introduces a concept of a pre-signed graph, which serves as an extension of both signed graphs and signed graph embeddings. Additionally, we extend a theorem for counting the number of faces of oriented embeddings of graphs that pass through a given cut-edge set to an embedding on any surface. Finally, we extend certain theorems proposed by Stahl and Lee regarding the average genus of graphs to include signed graphs.
{"title":"Pre-signed graphs: A reformulation of signed graphs and their embeddings","authors":"Yichao Chen","doi":"10.1016/j.aam.2026.103053","DOIUrl":"10.1016/j.aam.2026.103053","url":null,"abstract":"<div><div>The permutation-partition pair was initially introduced by Stahl in 1980 and independently by Archdeacon in a geometric form in 1986. This pair is a generalization of graphs and oriented embeddings of graphs. This paper introduces a concept of a pre-signed graph, which serves as an extension of both signed graphs and signed graph embeddings. Additionally, we extend a theorem for counting the number of faces of oriented embeddings of graphs that pass through a given cut-edge set to an embedding on any surface. Finally, we extend certain theorems proposed by Stahl and Lee regarding the average genus of graphs to include signed graphs.</div></div>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"176 ","pages":"Article 103053"},"PeriodicalIF":1.3,"publicationDate":"2026-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146190641","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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