Counting flows of b-compatible graphs

IF 1.3 3区数学 Q3 MATHEMATICS, APPLIED Advances in Applied Mathematics Pub Date : 2025-04-14 DOI:10.1016/j.aam.2025.102901

Houshan Fu , Xiangyu Ren , Suijie Wang

引用次数: 0

Abstract

Kochol introduced the assigning polynomial

F (G, α; k)

to count nowhere-zero

(A, b)

-flows of a graph G, where A is a finite Abelian group and α is a

{0, 1}

-assigning from a family

Λ (G)

of certain nonempty vertex subsets of G to

{0, 1}

. We introduce the concepts of b-compatible graph and b-compatible broken bond to give an explicit formula for the assigning polynomials and to examine their coefficients. More specifically, for a function

b : V (G) \to A

, let

α_{G, b}

be a

{0, 1}

-assigning of G such that for each

X \in Λ (G)

α_{G, b} (X) = 0

if and only if

\sum_{v \in X} b (v) = 0

. We show that for any

{0, 1}

-assigning α of G, if there exists a function

b : V (G) \to A

such that G is b-compatible and

α = α_{G, b}

, then the assigning polynomial

F (G, α; k)

has the b-compatible spanning subgraph expansion

F (G, α; k) = \sum_{\begin{matrix} S \subseteq E (G), \\ G - S is b -compatible \end{matrix}} {(- 1)}^{| S |} k^{m (G - S)},

and is the following form

F (G, α; k) = \sum_{i = 0}^{m (G)} {(- 1)}^{i} a_{i} (G, α) k^{m (G) - i},

where each

a_{i} (G, α)

is the number of subsets S of

E (G)

having i edges such that

G - S

is b-compatible and S contains no b-compatible broken bonds with respect to a total order on

E (G)

. Applying the counting interpretation, we also obtain unified comparison relations for the signless coefficients of assigning polynomials. Namely, for any

{0, 1}

-assignings

α, α^{'}

of G, if there exist functions

b : V (G) \to A

and

b^{'} : V (G) \to A^{'}

such that G is both b-compatible and

b^{'}

-compatible,

α = α_{G, b}

α^{'} = α_{G, b^{'}}

and

α (X) \leq α^{'} (X)

for all

X \in Λ (G)

, then

a_{i} (G, α) \leq a_{i} (G, α^{'}) for i = 0, 1, \dots, m (G) .

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计算 b 兼容图的流量

Kochol引入赋值多项式F（G,α;k）来计算图G的无零(A,b)-流，其中A是有限阿贝尔群，α是A{0,1}-从G的某些非空顶点子集的族Λ(G)中赋值到{0,1}。我们引入了b相容图和b相容断键的概念，给出了赋值多项式的显式公式，并检验了它们的系数。更具体地说，对于函数b:V(G)→a，设αG，b为a {0,1}- G的赋值使得对于每个X∈Λ(G), αG,b(X)=0当且仅当∑V∈Xb(V)=0。证明了对于G的任意{0,1}-赋值α，若存在函数b:V(G)→a，使得G是b相容且α=αG,b，则赋值多项式F（G,α;k）具有b相容的生成子图展开式F(G,α;k)=∑S≥≥E(G),G−S≥≥|S≥|km(G−S)，其形式为F(G,α;k)=∑i=0m(G)(−1)iai(G,α)km(G)−i。其中每个ai（G,α）是E(G)的子集S有i条边使得G−S与b相容且S不包含与b相容的断键对于E(G)的总阶的个数。应用计数解释，我们还得到了赋值多项式的无符号系数的统一比较关系。即对任何{0,1}-assigningsα,α’G的,如果存在函数b: V (G)→A和b的:V (G)→这样G b-compatible和b的兼容,α=αG b,α=αG b和α(X)≤α的所有X (X)∈Λ(G), thenai (G,α)≤ai (G,α')= 0,1,…,m (G)。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Advances in Applied Mathematics 数学-应用数学

CiteScore

2.00

自引率

9.10%

发文量

审稿时长

85 days

期刊介绍： Interdisciplinary in its coverage, Advances in Applied Mathematics is dedicated to the publication of original and survey articles on rigorous methods and results in applied mathematics. The journal features articles on discrete mathematics, discrete probability theory, theoretical statistics, mathematical biology and bioinformatics, applied commutative algebra and algebraic geometry, convexity theory, experimental mathematics, theoretical computer science, and other areas. Emphasizing papers that represent a substantial mathematical advance in their field, the journal is an excellent source of current information for mathematicians, computer scientists, applied mathematicians, physicists, statisticians, and biologists. Over the past ten years, Advances in Applied Mathematics has published research papers written by many of the foremost mathematicians of our time.

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