Teufl-Wagner公式的一种变体及其应用

IF 1 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":"Teufl-Wagner公式的一种变体及其应用","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":1.0000,"publicationDate":"2025-01-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"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\":1.0000,\"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}","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

摘要

设G = （V (G)）,E (G)) <math xmlns=“http://www.w3.org/1998/Math/MathML”altimg = " urn: x-wiley: 03649024:媒体:jgt23220: jgt23220 -数学- 0001”威利:位置= "方程/ jgt23220 -数学- 0001. png”祝辞& lt; mrow> & lt; mrow> & lt; mi> G< / mi> & lt; mo> = & lt; / mo> & lt; mrow> & lt; mo> (& lt; / mo> & lt; mrow> & lt; mi> V< / mi> & lt; mrow> & lt; mo> (& lt; / mo> & lt; mi&gt G< / mi> & lt; mo>) & lt; / mo> & lt; / mrow> & lt; mo>, & lt; / mo> & lt; mi> E< / mi> & lt; mrow> & lt; mo> (& lt; / mo> & lt; mi&gt G< / mi> & lt; mo>) & lt; / mo> & lt; / mrow> & lt; / mrow> & lt; mo>) & lt; / mo> & lt; / mrow> & lt; / mrow> & lt; / mrow> & lt; / math>和G * = (VG *)；E (g *))<math xmlns="http://www.w3.org/1998/Math/MathML" altimg=“urn:x-wiley:03649024:media:jgt23220:jgt23220-math-0002”威利:位置= "方程/ jgt23220 -数学- 0002. png”祝辞& lt; mrow> & lt; mrow> & lt; msup> & lt; mi> G< / mi> & lt; mo> \ unicode {x0002A} & lt; / mo> & lt; / msup> & lt; mo> = & lt; / mo> & lt; mrow> & lt; mo> (& lt; / mo> & lt; mrow> & lt; mi> V< / mi> & lt; mrow> & lt; mo> (& lt; / mo> & lt; msup> & lt; mi> G< / mi> & lt; mo> \ unicode {x0002A} & lt; / mo> & lt; / msup> & lt; mo>) & lt; / mo> & lt; / mrow> & lt; mo>, & lt; / mo> & lt; mi> E< / mi> & lt; mrow> & lt; mo> (& lt; / mo> & lt; msup> & lt;心肌梗死gt; G&lt / mi> & lt; mo> \ unicode {x0002A} & lt; / mo> & lt; / msup> & lt; mo>) & lt; / mo> & lt; / mrow> & lt; / mrow> & lt; mo>) & lt; / mo> & lt; / mrow> & lt; / mrow> & lt; / mrow> & lt; / math>是关于V (G)的两个电等效边权连通图&lt；math xmlns=“http://www.w3.org/1998/Math/MathML”altimg = " urn: x-wiley: 03649024:媒体:jgt23220: jgt23220 -数学- 0003“威利:位置= "方程/ jgt23220 -数学- 0003。 png”> < mrow&gt < mrow> &lt mi> V&lt / mi> < mrow&gt < mo&gt (< / mo&gt < mi> G&lt / mi> < mo&gt) < / mo> &lt / mrow> < / mrow&gt < / mrow> &lt / math>(因此V (G)⊆V (“urn:x-wiley:03649024:media:jgt23220:jgt23220-math-0004”魏:地方= "方程/ jgt23220-math-0004.png > < mrow&gt < mrow> &lt mi> V&lt / mi> &lt mrow> < mo&gt (< / mo&gt < mi> G&lt / mi> < mo&gt) < / mo> &lt / mrow> < mo&gt \ unicode {x02286 < / mo> &lt mi> V&lt / mi> < mrow&gt < mo> (&lt / mo> < msup&gt < mi> G&lt / mi> < mo&gt \ unicode {x0002A < / mo> &lt / msup> < mo&gt) < / mo> &lt / mrow> < / mrow&gt < / mrow> &lt / math&gt ; ).设F = t1∪t2∪…∪T c <math xmlns="http://www.w3.org/1998/Math/MathML" altimg=“urn:x-wiley:03649024:media:jgt23220:jgt23220-math-0005”魏:地方= "方程/ jgt23220-math-0005.png > < mrow&gt < mrow> &lt mi> F< / mi&gt < mo> = &lt / mo> < msub&gt < mi> T&lt / mi> < mn&gt 1< / mn> &lt / msub> < mo&gt \ unicode {x0222A < / mo> < msub&gt < mi> T&lt / mi> < mn&gt 2< / mn> &lt / msub> < mo&gt \ unicode {x0222A < / mo> < mo&gt \ unicode {x022EF < / mo> &lt mspacewidth = " 0.25em / > < mo&gt \ unicode {x0222A < / mo> &lt msub> < mi&gt T< / mi> &lt mi> c< / mi&gt < / msub> &lt / mrow> < / mrow&gt &lt / math>在G&lt； 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">&lt；.xmlns="http://www.w3.org/1998/Math/MathML" altimg=“urn:x-wiley:03649024:media:jgt23220:jgt23220-math-0007”魏:地方= "方程/ jgt23220-math-0007.png > < mrow&gt < mrow> &lt mi> t&lt / mi> &lt mrow> < mo&gt (< / mo&gt < mi> G&lt / mi> < mo&gt) < / mo> &lt / mrow> < / mrow&gt < / mrow> &lt / math>点心》weights of spanning trees of G & lt;计算xmlns = " http://www.w3.org/1998/Math/MathML altimg = " urn: x-wiley 03649024:媒体:jgt23220: jgt23220-math-0008“魏:地方=方程/ jgt23220-math-0008.png > < mrow&gt < mrow&gt < mi> G&lt / mi> < / mrow&gt < / mrow> &lt / math>和t F (G) <math xmlns="http://www.w3。 org/1998/Math/MathML " altimg = " urn: x-wiley: 03649024:媒体:jgt23220: jgt23220 -数学- 0009“威利:位置= "方程/ jgt23220 -数学- 0009. png”祝辞& lt; mrow> & lt; mrow> & lt; msub> & lt; mi> t< / mi> & lt; mi> F< / mi> & lt; / msub> & lt; mrow> & lt; mo> (& lt; / mo> & lt; mi&gt G< / mi> & lt; mo>) & lt; / mo> & lt; / mrow> & lt; / mrow> & lt; / mrow> & lt; / math>F< 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>< /mrow></mrow></ mrow></mrow></ mrow></mrow></math&gt；,其中权重ω (H) <math xmlns="http://www.w3.org/1998/Math/MathML" altimg=“urn:x-wiley:03649024:media:jgt23220:jgt23220-math-0011”威利:位置= "方程/ jgt23220 -数学- 0011. png”祝辞& lt; mrow> & lt; mrow> & lt; mi> \ unicode {x003C9} & lt; / mi> & lt; mrow> & lt; mo> (& lt; / mo> & lt; mi&gt H< / mi> & lt; mo>) & lt; / mo> & lt; / mrow> & lt; / mrow> & lt; / mrow> & lt; / math>查看子图H&lt； 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></mrow></ mrow></mrow></ mrow></mrow></math&gt；> > /mrow></mrow></ mrow></mrow></ mrow></mrow></math>；是H&lt； math中边权的乘积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></mrow></ mrow></mrow></ mrow></math&gt；。假设G *⋅V (F)<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>< /jgt23220- jgt23220-math-0015.png"><mrow>< /msup>< /msup>< /msup>< /msup>< /msup>< /msup>< /msup>< /msup>< /msup>< /msup>< /msup>< /msup>< \unicode{x0002A}</mo></msup>< /msup>< /msup>< /msup>< /msup>&lt；为从G * &lt；math得到的边加权图xmlns="http://www.w3.org/1998/Math/MathML" altimg=“urn:x-wiley:03649024:media:jgt23220:jgt23220-math-0016”威利:位置= "方程/ jgt23220 -数学- 0016. png”祝辞& lt; mrow> & lt; mrow> & lt; msup> & lt; mi> G< / mi> & lt; mo> \ unicode {x0002A} & lt; / mo> & lt; / msup> & lt; / mrow> & lt; / mrow> & lt; / math>通过识别V （t1）中的所有顶点&lt；math xmlns="http://www.w3 "。 org/1998/Math/MathML altimg = " urn: x-wiley 03649024:媒体:jgt23220: jgt23220-math-0017“魏:地方=方程/ jgt23220-math-0017.png > &lt mrow> < mrow&gt < mi> V&lt / mi> < mrow&gt < mo> (&lt / mo> < msub&gt < mi> T&lt / mi> < mi&gt i< / mi> &lt / msub> < mo&gt) < / mo> &lt / mrow> < / mrow&gt < / mrow> &lt / math>xmlns="http://www.w3.org/1998/Math/MathML" altimg=“urn:x-wiley:03649024:media:jgt23220:jgt23220- math0018 ”魏:地方= "方程/ jgt23220-math-0018.png > < mrow&gt < mrow> &lt msup> &lt mi> G&lt / mi> < mo&gt \ unicode {x0002A < / mo> &lt / msup> < / mrow&gt < / mrow> &lt / math>数学xmlns="http://www.w3.org/1998/Math/MathML" altimg=“urn:x-wiley:03649024:media:jgt23220:jgt23220-math-0019”魏:地方= "方程/ jgt23220-math-0019.png > < mrow&gt < mrow> &lt msub> < mi&gt u< / mi> &lt mi> i< / mi&gt < / msub> &lt / mrow> < / mrow&gt &lt / math>math xmlns="http://www.w3.org/1998/Math/MathML" altimg=“urn:x-wiley:03649024:media:jgt23220:jgt23220-math-0020”魏:地方= "方程/ jgt23220-math-0020.png > < mrow&gt < mrow> &lt mn> 1&lt / mn> < mo&gt \ unicode {x02264 < / mo> &lt mi> i&lt / mi> < mo&gt \ unicode {x02264 < / mo> &lt mi> c< / mi&gt < / mrow> &lt / mrow> &lt / math>．在本文中，我们得到了Teufl-Wagner公式的一个变体（线性数学应用，432(2010)）。441-457)证明t (G) t（g *） =F (G)∕ω (F) t (G *⋅v (f)) <math xmlns="http://www.w3.org/1998/Math/MathML" altimg=“urn:x-wiley:03649024:media:jgt23220:jgt23220-math-0021”威利:位置= "方程/ jgt23220 -数学- 0021. png”祝辞& lt; mrow> & lt; mrow> & lt; mfrac> & lt; mrow> & lt; mi> t< / mi> & lt; mrow> & lt; mo> (& lt; / mo> & lt; mi&gt G< / mi> & lt; mo>) & lt; / mo> & lt; / mrow> & lt; / mrow> & lt; mrow> & lt; mi> t< / mi> & lt; mrow> & lt; mo> (& lt; / mo> & lt; msup> & lt; mi> G< / mi> & lt; mo> \ unicode {x0002A} & lt; / mo> & lt; / msup> & lt; mo>) & lt; / mo> & lt; / mrow> & lt; / mrow> & lt; / mfrac> & lt; mo> = & lt; / mo> & lt; mfrac> & lt; mrow> & lt; msub>& lt; mi&gt t< / mi> & lt; mi> F< / mi> & lt; / msub> & lt; mrow> & lt; mo> (& lt; / mo> & lt; mi&gt G< / mi> & lt; mo>) & lt; / mo> & lt; / mrow> & lt; mo> \ unicode {x02215} & lt; / mo> & lt; mi> \ unicode {x003C9} & lt; / mi> & lt; mrow> & lt; mo> (& lt; / mo> & lt; mi&gt F< / mi> & lt; mo>) & lt; / mo> & lt; / mrow> & lt; / mrow> & lt; mrow> & lt; mi> t< / mi> & lt; mrow> & lt; mo> (& lt; / mo> & lt; mrow> & lt; msup> & lt; mi> G< / mi> & lt; mo> \ unicode {x0002A} & lt; / mo> & lt; / msup> & lt; mo>unicode {x022C5} \ & lt; / mo> & lt; mi> V< / mi> & lt; mrow> & lt; mo> (& lt; / mo> & lt; mi&gt F< / mi> & lt; mo>) & lt; / mo> & lt; / mrow> & lt; / mrow> & lt; mo>) & lt; / mo> & lt; / mrow> & lt; / mrow> & lt; / mfrac> & lt; / mrow> & lt; / mrow> & lt; / math>。作为应用，我们列举了给定森林中包含所有边的一些图的生成树，并给出了Moon公式（Mathematika, 11(1964), 95-98）和Dong-Ge公式（J Graph Theory, 101(2022), 79-94）的简单证明。特别地，我们对一些图中具有完美匹配的生成树进行计数。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

查看原文本刊更多论文

A Variant of the Teufl-Wagner Formula and Applications

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.

求助全文

通过发布文献求助，成功后即可免费获取论文全文。去求助

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 .