arXiv - MATH - Combinatorics最新文献_第5页

Counting Flows of $b$-compatible Graphs 计算兼容 $b$ 图形的流量

arXiv - MATH - Combinatorics Pub Date : 2024-09-15 DOI: arxiv-2409.09634

Houshan Fu, Xiangyu Ren, Suijie Wang

{"title":"Counting Flows of $b$-compatible Graphs","authors":"Houshan Fu, Xiangyu Ren, Suijie Wang","doi":"arxiv-2409.09634","DOIUrl":"https://doi.org/arxiv-2409.09634","url":null,"abstract":"Kochol introduced the assigning polynomial $F(G,alpha;k)$ to count\u0000nowhere-zero $(A,b)$-flows of a graph $G$, where $A$ is a finite Abelian group\u0000and $alpha$ is a ${0,1}$-assigning from a family $Lambda(G)$ of certain\u0000nonempty vertex subsets of $G$ to ${0,1}$. We introduce the concepts of\u0000$b$-compatible graph and $b$-compatible broken bond to give an explicit formula\u0000for the assigning polynomials and to examine their coefficients. More\u0000specifically, for a function $b:V(G)to A$, let $alpha_{G,b}$ be a\u0000${0,1}$-assigning of $G$ such that for each $XinLambda(G)$,\u0000$alpha_{G,b}(X)=0$ if and only if $sum_{vin X}b(v)=0$. We show that for any\u0000${0,1}$-assigning $alpha$ of $G$, if there exists a function $b:V(G)to A$\u0000such that $G$ is $b$-compatible and $alpha=alpha_{G,b}$, then the assigning\u0000polynomial $F(G,alpha;k)$ has the $b$-compatible spanning subgraph expansion\u0000[ F(G,alpha;k)=sum_{substack{Ssubseteq E(G),G-Smbox{ is\u0000$b$-compatible}}}(-1)^{|S|}k^{m(G-S)}, ] and is the following form\u0000$F(G,alpha;k)=sum_{i=0}^{m(G)}(-1)^ia_i(G,alpha)k^{m(G)-i}$, where each\u0000$a_i(G,alpha)$ is the number of subsets $S$ of $E(G)$ having $i$ edges such\u0000that $G-S$ is $b$-compatible and $S$ contains no $b$-compatible broken bonds\u0000with respect to a total order on $E(G)$. Applying the counting interpretation,\u0000we also obtain unified comparison relations for the signless coefficients of\u0000assigning polynomials. Namely, for any ${0,1}$-assignings $alpha,alpha'$ of\u0000$G$, if there exist functions $b:V(G)to A$ and $b':V(G)to A'$ such that $G$\u0000is both $b$-compatible and $b'$-compatible, $alpha=alpha_{G,b}$,\u0000$alpha'=alpha_{G,b'}$ and $alpha(X)lealpha'(X)$ for all $XinLambda(G)$,\u0000then [ a_i(G,alpha)le a_i(G,alpha') quad mbox{ for }quad i=0,1,ldots,\u0000m(G). ]","PeriodicalId":501407,"journal":{"name":"arXiv - MATH - Combinatorics","volume":"12 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142255076","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

Invariants of Bipartite Kneser B type-MakeLowercase{k} graphs 双方 Kneser B 型-/MakeLowercase{k}图的不变式

arXiv - MATH - Combinatorics Pub Date : 2024-09-14 DOI: arxiv-2409.09317

Jayakumar C, Sreekumar K. G., Manilal K., Ismail Naci Cangul

引用次数: 0

Subdivisions and near-linear stable sets 细分和近线性稳定集合

arXiv - MATH - Combinatorics Pub Date : 2024-09-14 DOI: arxiv-2409.09400

Tung Nguyen, Alex Scott, Paul Seymour

引用次数: 0

Trees and near-linear stable sets 树和近线稳定集

arXiv - MATH - Combinatorics Pub Date : 2024-09-14 DOI: arxiv-2409.09397

Tung Nguyen, Alex Scott, Paul Seymour

引用次数: 0

A Riordan array family for some integrable lattice models 一些可积分晶格模型的瑞尔丹阵列族

arXiv - MATH - Combinatorics Pub Date : 2024-09-14 DOI: arxiv-2409.09547

Paul Barry

引用次数: 0

A degenerate version of Brion's formula 布里昂公式的退化版本

arXiv - MATH - Combinatorics Pub Date : 2024-09-14 DOI: arxiv-2409.09544

Carsten Peterson

{"title":"A degenerate version of Brion's formula","authors":"Carsten Peterson","doi":"arxiv-2409.09544","DOIUrl":"https://doi.org/arxiv-2409.09544","url":null,"abstract":"Let $mathfrak{p} subset V$ be a polytope and $xi in V_{mathbb{C}}^*$. We\u0000obtain an expression for $I(mathfrak{p}; alpha) := int_{mathfrak{p}}\u0000e^{langle alpha, x rangle} dx$ as a sum of meromorphic functions in $alpha\u0000in V^*_{mathbb{C}}$ parametrized by the faces $mathfrak{f}$ of\u0000$mathfrak{p}$ on which $langle xi, x rangle$ is constant. Each term only\u0000depends on the local geometry of $mathfrak{p}$ near $mathfrak{f}$ (and on\u0000$xi$) and is holomorphic at $alpha = xi$. When $langle xi, cdot rangle$\u0000is only constant on the vertices of $mathfrak{p}$ our formula reduces to\u0000Brion's formula. Suppose $mathfrak{p}$ is a rational polytope with respect to a lattice\u0000$Lambda$. We obtain an expression for $S(mathfrak{p}; alpha) :=\u0000sum_{lambda in mathfrak{p} cap Lambda} e^{langle alpha, lambda\u0000rangle}$ as a sum of meromorphic functions parametrized by the faces\u0000$mathfrak{f}$ on which $e^{langle xi, x rangle} = 1$ on a finite index\u0000sublattice of $text{lin}(mathfrak{f}) cap Lambda$. Each term only depends\u0000on the local geometry of $mathfrak{p}$ near $mathfrak{f}$ (and on $xi$ and\u0000$Lambda$) and is holomorphic at $alpha = xi$. When $e^{langle xi, cdot\u0000rangle} neq 1$ at any non-zero lattice point on a line through the origin\u0000parallel to an edge of $mathfrak{p}$, our formula reduces to Brion's formula,\u0000and when $xi = 0$, it reduces to the Ehrhart quasi-polynomial. Our formulas are particularly useful for understanding how\u0000$I(mathfrak{p}(h); xi)$ and $S(mathfrak{p}(h); xi)$ vary in a family of\u0000polytopes $mathfrak{p}(h)$ with the same normal fan. When considering dilates\u0000of a fixed polytope, our formulas may be viewed as polytopal analogues of\u0000Laplace's method and the method of stationary phase. Such expressions naturally\u0000show up in analysis on symmetric spaces and affine buildings.","PeriodicalId":501407,"journal":{"name":"arXiv - MATH - Combinatorics","volume":"70 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142255078","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

Subsquares in random Latin squares 随机拉丁方格中的子方格

arXiv - MATH - Combinatorics Pub Date : 2024-09-13 DOI: arxiv-2409.08446

Jack Allsop, Ian M. Wanless

引用次数: 0

Critical Thresholds for Maximum Cardinality Matching on General Hypergraphs 一般超图上最大卡方匹配的临界阈值

arXiv - MATH - Combinatorics Pub Date : 2024-09-13 DOI: arxiv-2409.09155

Christopher Sumnicht, Jamison W. Weber, Dhanush R. Giriyan, Arunabha Sen

引用次数: 0

On a generalization of a result of Kleitman 关于克莱特曼一个结果的推广

arXiv - MATH - Combinatorics Pub Date : 2024-09-13 DOI: arxiv-2409.08694

Ryan R. Martin, Balázs Patkós