A Variant of the Teufl-Wagner Formula and Applications

IF 0.9 3区数学 Q2 MATHEMATICS

Journal of Graph Theory Pub Date : 2025-01-06 DOI:10.1002/jgt.23220

Danyi Li, Weigen Yan

{"title":"A Variant of the Teufl-Wagner Formula and Applications","authors":"Danyi Li, Weigen Yan","doi":"10.1002/jgt.23220","DOIUrl":null,"url":null,"abstract":"<div>\n \n Let <math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>G</mi>\n \n <mo>=</mo>\n \n <mrow>\n <mo>(</mo>\n \n <mrow>\n <mi>V</mi>\n \n <mrow>\n <mo>(</mo>\n \n <mi>G</mi>\n \n <mo>)</mo>\n </mrow>\n \n <mo>,</mo>\n \n <mi>E</mi>\n \n <mrow>\n <mo>(</mo>\n \n <mi>G</mi>\n \n <mo>)</mo>\n </mrow>\n </mrow>\n \n <mo>)</mo>\n </mrow>\n </mrow>\n </mrow>\n <annotation> <math xmlns=\"http://www.w3.org/1998/Math/MathML\" altimg=\"urn:x-wiley:03649024:media:jgt23220:jgt23220-math-0001\" wiley:location=\"equation/jgt23220-math-0001.png\"><mrow><mrow><mi>G</mi><mo>=</mo><mrow><mo>(</mo><mrow><mi>V</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>,</mo><mi>E</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow><mo>)</mo></mrow></mrow></mrow></math></annotation>\n </semantics></math> and <math>\n <semantics>\n <mrow>\n \n <mrow>\n <msup>\n <mi>G</mi>\n \n <mo>*</mo>\n </msup>\n \n <mo>=</mo>\n \n <mrow>\n <mo>(</mo>\n \n <mrow>\n <mi>V</mi>\n \n <mrow>\n <mo>(</mo>\n \n <msup>\n <mi>G</mi>\n \n <mo>*</mo>\n </msup>\n \n <mo>)</mo>\n </mrow>\n \n <mo>,</mo>\n \n <mi>E</mi>\n \n <mrow>\n <mo>(</mo>\n \n <msup>\n <mi>G</mi>\n \n <mo>*</mo>\n </msup>\n \n <mo>)</mo>\n </mrow>\n </mrow>\n \n <mo>)</mo>\n </mrow>\n </mrow>\n </mrow>\n <annotation> <math xmlns=\"http://www.w3.org/1998/Math/MathML\" altimg=\"urn:x-wiley:03649024:media:jgt23220:jgt23220-math-0002\" wiley:location=\"equation/jgt23220-math-0002.png\"><mrow><mrow><msup><mi>G</mi><mo>\\unicode{x0002A}</mo></msup><mo>=</mo><mrow><mo>(</mo><mrow><mi>V</mi><mrow><mo>(</mo><msup><mi>G</mi><mo>\\unicode{x0002A}</mo></msup><mo>)</mo></mrow><mo>,</mo><mi>E</mi><mrow><mo>(</mo><msup><mi>G</mi><mo>\\unicode{x0002A}</mo></msup><mo>)</mo></mrow></mrow><mo>)</mo></mrow></mrow></mrow></math></annotation>\n </semantics></math> be two electrically equivalent edge-weighted connected graphs with respect to <math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>V</mi>\n \n <mrow>\n <mo>(</mo>\n \n <mi>G</mi>\n \n <mo>)</mo>\n </mrow>\n </mrow>\n </mrow>\n <annotation> <math xmlns=\"http://www.w3.org/1998/Math/MathML\" altimg=\"urn:x-wiley:03649024:media:jgt23220:jgt23220-math-0003\" wiley:location=\"equation/jgt23220-math-0003.png\"><mrow><mrow><mi>V</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></mrow></math></annotation>\n </semantics></math> (hence <math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>V</mi>\n \n <mrow>\n <mo>(</mo>\n \n <mi>G</mi>\n \n <mo>)</mo>\n </mrow>\n \n <mo>⊆</mo>\n \n <mi>V</mi>\n \n <mrow>\n <mo>(</mo>\n \n <msup>\n <mi>G</mi>\n \n <mo>*</mo>\n </msup>\n \n <mo>)</mo>\n </mrow>\n </mrow>\n </mrow>\n <annotation> <math xmlns=\"http://www.w3.org/1998/Math/MathML\" altimg=\"urn:x-wiley:03649024:media:jgt23220:jgt23220-math-0004\" wiley:location=\"equation/jgt23220-math-0004.png\"><mrow><mrow><mi>V</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>\\unicode{x02286}</mo><mi>V</mi><mrow><mo>(</mo><msup><mi>G</mi><mo>\\unicode{x0002A}</mo></msup><mo>)</mo></mrow></mrow></mrow></math></annotation>\n </semantics></math>). Let <math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>F</mi>\n \n <mo>=</mo>\n \n <msub>\n <mi>T</mi>\n \n <mn>1</mn>\n </msub>\n \n <mo>∪</mo>\n \n <msub>\n <mi>T</mi>\n \n <mn>2</mn>\n </msub>\n \n <mo>∪</mo>\n \n <mo>⋯</mo>\n <mspace></mspace>\n \n <mo>∪</mo>\n \n <msub>\n <mi>T</mi>\n \n <mi>c</mi>\n </msub>\n </mrow>\n </mrow>\n <annotation> <math xmlns=\"http://www.w3.org/1998/Math/MathML\" altimg=\"urn:x-wiley:03649024:media:jgt23220:jgt23220-math-0005\" wiley:location=\"equation/jgt23220-math-0005.png\"><mrow><mrow><mi>F</mi><mo>=</mo><msub><mi>T</mi><mn>1</mn></msub><mo>\\unicode{x0222A}</mo><msub><mi>T</mi><mn>2</mn></msub><mo>\\unicode{x0222A}</mo><mo>\\unicode{x022EF}</mo><mspace width=\"0.25em\"/><mo>\\unicode{x0222A}</mo><msub><mi>T</mi><mi>c</mi></msub></mrow></mrow></math></annotation>\n </semantics></math> be a forest in <math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>G</mi>\n </mrow>\n </mrow>\n <annotation> <math xmlns=\"http://www.w3.org/1998/Math/MathML\" altimg=\"urn:x-wiley:03649024:media:jgt23220:jgt23220-math-0006\" wiley:location=\"equation/jgt23220-math-0006.png\"><mrow><mrow><mi>G</mi></mrow></mrow></math></annotation>\n </semantics></math>. Denote by <math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>t</mi>\n \n <mrow>\n <mo>(</mo>\n \n <mi>G</mi>\n \n <mo>)</mo>\n </mrow>\n </mrow>\n </mrow>\n <annotation> <math xmlns=\"http://www.w3.org/1998/Math/MathML\" altimg=\"urn:x-wiley:03649024:media:jgt23220:jgt23220-math-0007\" wiley:location=\"equation/jgt23220-math-0007.png\"><mrow><mrow><mi>t</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></mrow></math></annotation>\n </semantics></math> the sum of weights of spanning trees of <math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>G</mi>\n </mrow>\n </mrow>\n <annotation> <math xmlns=\"http://www.w3.org/1998/Math/MathML\" altimg=\"urn:x-wiley:03649024:media:jgt23220:jgt23220-math-0008\" wiley:location=\"equation/jgt23220-math-0008.png\"><mrow><mrow><mi>G</mi></mrow></mrow></math></annotation>\n </semantics></math> and by <math>\n <semantics>\n <mrow>\n \n <mrow>\n <msub>\n <mi>t</mi>\n \n <mi>F</mi>\n </msub>\n \n <mrow>\n <mo>(</mo>\n \n <mi>G</mi>\n \n <mo>)</mo>\n </mrow>\n </mrow>\n </mrow>\n <annotation> <math xmlns=\"http://www.w3.org/1998/Math/MathML\" altimg=\"urn:x-wiley:03649024:media:jgt23220:jgt23220-math-0009\" wiley:location=\"equation/jgt23220-math-0009.png\"><mrow><mrow><msub><mi>t</mi><mi>F</mi></msub><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></mrow></math></annotation>\n </semantics></math> the sum of weights of spanning trees each of which containing all edges in <math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>F</mi>\n </mrow>\n </mrow>\n <annotation> <math xmlns=\"http://www.w3.org/1998/Math/MathML\" altimg=\"urn:x-wiley:03649024:media:jgt23220:jgt23220-math-0010\" wiley:location=\"equation/jgt23220-math-0010.png\"><mrow><mrow><mi>F</mi></mrow></mrow></math></annotation>\n </semantics></math>, where the weight <math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>ω</mi>\n \n <mrow>\n <mo>(</mo>\n \n <mi>H</mi>\n \n <mo>)</mo>\n </mrow>\n </mrow>\n </mrow>\n <annotation> <math xmlns=\"http://www.w3.org/1998/Math/MathML\" altimg=\"urn:x-wiley:03649024:media:jgt23220:jgt23220-math-0011\" wiley:location=\"equation/jgt23220-math-0011.png\"><mrow><mrow><mi>\\unicode{x003C9}</mi><mrow><mo>(</mo><mi>H</mi><mo>)</mo></mrow></mrow></mrow></math></annotation>\n </semantics></math> of a subgraph <math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>H</mi>\n </mrow>\n </mrow>\n <annotation> <math xmlns=\"http://www.w3.org/1998/Math/MathML\" altimg=\"urn:x-wiley:03649024:media:jgt23220:jgt23220-math-0012\" wiley:location=\"equation/jgt23220-math-0012.png\"><mrow><mrow><mi>H</mi></mrow></mrow></math></annotation>\n </semantics></math> of <math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>G</mi>\n </mrow>\n </mrow>\n <annotation> <math xmlns=\"http://www.w3.org/1998/Math/MathML\" altimg=\"urn:x-wiley:03649024:media:jgt23220:jgt23220-math-0013\" wiley:location=\"equation/jgt23220-math-0013.png\"><mrow><mrow><mi>G</mi></mrow></mrow></math></annotation>\n </semantics></math> is the product of weights of edges in <math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>H</mi>\n </mrow>\n </mrow>\n <annotation> <math xmlns=\"http://www.w3.org/1998/Math/MathML\" altimg=\"urn:x-wiley:03649024:media:jgt23220:jgt23220-math-0014\" wiley:location=\"equation/jgt23220-math-0014.png\"><mrow><mrow><mi>H</mi></mrow></mrow></math></annotation>\n </semantics></math>. Suppose that <math>\n <semantics>\n <mrow>\n \n <mrow>\n <msup>\n <mi>G</mi>\n \n <mo>*</mo>\n </msup>\n \n <mo>⋅</mo>\n \n <mi>V</mi>\n \n <mrow>\n <mo>(</mo>\n \n <mi>F</mi>\n \n <mo>)</mo>\n </mrow>\n </mrow>\n </mrow>\n <annotation> <math xmlns=\"http://www.w3.org/1998/Math/MathML\" altimg=\"urn:x-wiley:03649024:media:jgt23220:jgt23220-math-0015\" wiley:location=\"equation/jgt23220-math-0015.png\"><mrow><mrow><msup><mi>G</mi><mo>\\unicode{x0002A}</mo></msup><mo>\\unicode{x022C5}</mo><mi>V</mi><mrow><mo>(</mo><mi>F</mi><mo>)</mo></mrow></mrow></mrow></math></annotation>\n </semantics></math> is the edge-weighted graph obtained from <math>\n <semantics>\n <mrow>\n \n <mrow>\n <msup>\n <mi>G</mi>\n \n <mo>*</mo>\n </msup>\n </mrow>\n </mrow>\n <annotation> <math xmlns=\"http://www.w3.org/1998/Math/MathML\" altimg=\"urn:x-wiley:03649024:media:jgt23220:jgt23220-math-0016\" wiley:location=\"equation/jgt23220-math-0016.png\"><mrow><mrow><msup><mi>G</mi><mo>\\unicode{x0002A}</mo></msup></mrow></mrow></math></annotation>\n </semantics></math> by identifying all vertices in <math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>V</mi>\n \n <mrow>\n <mo>(</mo>\n \n <msub>\n <mi>T</mi>\n \n <mi>i</mi>\n </msub>\n \n <mo>)</mo>\n </mrow>\n </mrow>\n </mrow>\n <annotation> <math xmlns=\"http://www.w3.org/1998/Math/MathML\" altimg=\"urn:x-wiley:03649024:media:jgt23220:jgt23220-math-0017\" wiley:location=\"equation/jgt23220-math-0017.png\"><mrow><mrow><mi>V</mi><mrow><mo>(</mo><msub><mi>T</mi><mi>i</mi></msub><mo>)</mo></mrow></mrow></mrow></math></annotation>\n </semantics></math> of <math>\n <semantics>\n <mrow>\n \n <mrow>\n <msup>\n <mi>G</mi>\n \n <mo>*</mo>\n </msup>\n </mrow>\n </mrow>\n <annotation> <math xmlns=\"http://www.w3.org/1998/Math/MathML\" altimg=\"urn:x-wiley:03649024:media:jgt23220:jgt23220-math-0018\" wiley:location=\"equation/jgt23220-math-0018.png\"><mrow><mrow><msup><mi>G</mi><mo>\\unicode{x0002A}</mo></msup></mrow></mrow></math></annotation>\n </semantics></math> into a new vertex <math>\n <semantics>\n <mrow>\n \n <mrow>\n <msub>\n <mi>u</mi>\n \n <mi>i</mi>\n </msub>\n </mrow>\n </mrow>\n <annotation> <math xmlns=\"http://www.w3.org/1998/Math/MathML\" altimg=\"urn:x-wiley:03649024:media:jgt23220:jgt23220-math-0019\" wiley:location=\"equation/jgt23220-math-0019.png\"><mrow><mrow><msub><mi>u</mi><mi>i</mi></msub></mrow></mrow></math></annotation>\n </semantics></math> for <math>\n <semantics>\n <mrow>\n \n <mrow>\n <mn>1</mn>\n \n <mo>≤</mo>\n \n <mi>i</mi>\n \n <mo>≤</mo>\n \n <mi>c</mi>\n </mrow>\n </mrow>\n <annotation> <math xmlns=\"http://www.w3.org/1998/Math/MathML\" altimg=\"urn:x-wiley:03649024:media:jgt23220:jgt23220-math-0020\" wiley:location=\"equation/jgt23220-math-0020.png\"><mrow><mrow><mn>1</mn><mo>\\unicode{x02264}</mo><mi>i</mi><mo>\\unicode{x02264}</mo><mi>c</mi></mrow></mrow></math></annotation>\n </semantics></math>. In this paper, we obtain a variant of the Teufl-Wagner formula (Linear Alg Appl, 432 (2010), 441–457) and prove that <math>\n <semantics>\n <mrow>\n \n <mrow>\n <mfrac>\n <mrow>\n <mi>t</mi>\n \n <mrow>\n <mo>(</mo>\n \n <mi>G</mi>\n \n <mo>)</mo>\n </mrow>\n </mrow>\n \n <mrow>\n <mi>t</mi>\n \n <mrow>\n <mo>(</mo>\n \n <msup>\n <mi>G</mi>\n \n <mo>*</mo>\n </msup>\n \n <mo>)</mo>\n </mrow>\n </mrow>\n </mfrac>\n \n <mo>=</mo>\n \n <mfrac>\n <mrow>\n <msub>\n <mi>t</mi>\n \n <mi>F</mi>\n </msub>\n \n <mrow>\n <mo>(</mo>\n \n <mi>G</mi>\n \n <mo>)</mo>\n </mrow>\n \n <mo>∕</mo>\n \n <mi>ω</mi>\n \n <mrow>\n <mo>(</mo>\n \n <mi>F</mi>\n \n <mo>)</mo>\n </mrow>\n </mrow>\n \n <mrow>\n <mi>t</mi>\n \n <mrow>\n <mo>(</mo>\n \n <mrow>\n <msup>\n <mi>G</mi>\n \n <mo>*</mo>\n </msup>\n \n <mo>⋅</mo>\n \n <mi>V</mi>\n \n <mrow>\n <mo>(</mo>\n \n <mi>F</mi>\n \n <mo>)</mo>\n </mrow>\n </mrow>\n \n <mo>)</mo>\n </mrow>\n </mrow>\n </mfrac>\n </mrow>\n </mrow>\n <annotation> <math xmlns=\"http://www.w3.org/1998/Math/MathML\" altimg=\"urn:x-wiley:03649024:media:jgt23220:jgt23220-math-0021\" wiley:location=\"equation/jgt23220-math-0021.png\"><mrow><mrow><mfrac><mrow><mi>t</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow><mrow><mi>t</mi><mrow><mo>(</mo><msup><mi>G</mi><mo>\\unicode{x0002A}</mo></msup><mo>)</mo></mrow></mrow></mfrac><mo>=</mo><mfrac><mrow><msub><mi>t</mi><mi>F</mi></msub><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>\\unicode{x02215}</mo><mi>\\unicode{x003C9}</mi><mrow><mo>(</mo><mi>F</mi><mo>)</mo></mrow></mrow><mrow><mi>t</mi><mrow><mo>(</mo><mrow><msup><mi>G</mi><mo>\\unicode{x0002A}</mo></msup><mo>\\unicode{x022C5}</mo><mi>V</mi><mrow><mo>(</mo><mi>F</mi><mo>)</mo></mrow></mrow><mo>)</mo></mrow></mrow></mfrac></mrow></mrow></math></annotation>\n </semantics></math>. As applications, we enumerate spanning trees of some graphs containing all edges in a given forest and give a simple proof of Moon's formula (Mathematika, 11 (1964), 95–98) and the Dong-Ge formula (J Graph Theory, 101 (2022), 79–94). In particular, we count spanning trees with a perfect matching in some graphs.\n </div>","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":"109 1","pages":"68-75"},"PeriodicalIF":0.9000,"publicationDate":"2025-01-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Graph Theory","FirstCategoryId":"100","ListUrlMain":"https://onlinelibrary.wiley.com/doi/10.1002/jgt.23220","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}

引用次数: 0

Abstract

Let $G = (V (G), E (G))$ and $G^{*} = (V (G^{*}), E (G^{*}))$ be two electrically equivalent edge-weighted connected graphs with respect to $V (G)$ (hence $V (G) \subseteq V (G^{*})$ ). Let $F = T_{1} \cup T_{2} \cup \dots \cup T_{c}$ be a forest in $G$ . Denote by $t (G)$ the sum of weights of spanning trees of $G$ and by $t_{F} (G)$ the sum of weights of spanning trees each of which containing all edges in $F$ , where the weight $ω (H)$ of a subgraph $H$ of $G$ is the product of weights of edges in $H$ . Suppose that $G^{*} \cdot V (F)$ is the edge-weighted graph obtained from $G^{*}$ by identifying all vertices in $V (T_{i})$ of $G^{*}$ into a new vertex $u_{i}$ for $1 \leq i \leq c$ . In this paper, we obtain a variant of the Teufl-Wagner formula (Linear Alg Appl, 432 (2010), 441–457) and prove that $\frac{t (G)}{t (G^{*})} = \frac{t_{F} (G) ∕ ω (F)}{t (G^{*} \cdot V (F))}$ . As applications, we enumerate spanning trees of some graphs containing all edges in a given forest and give a simple proof of Moon's formula (Mathematika, 11 (1964), 95–98) and the Dong-Ge formula (J Graph Theory, 101 (2022), 79–94). In particular, we count spanning trees with a perfect matching in some graphs.

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来源期刊

Journal of Graph Theory 数学-数学

CiteScore

1.60

自引率

22.20%

发文量

130

审稿时长

6-12 weeks

期刊介绍： The Journal of Graph Theory is devoted to a variety of topics in graph theory, such as structural results about graphs, graph algorithms with theoretical emphasis, and discrete optimization on graphs. The scope of the journal also includes related areas in combinatorics and the interaction of graph theory with other mathematical sciences. A subscription to the Journal of Graph Theory includes a subscription to the Journal of Combinatorial Designs .