The Singularity of the <i>K</i><sub>4</sub> Homeomorphic Graph
Let <i>G</i> be a finite simple graph and let <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>A</mi><mo>(</mo><mi>G</mi><mo>)</mo></mrow&g...
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| Format: | Article |
| Language: | English |
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MDPI AG
2024-12-01
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| Series: | Axioms |
| Subjects: | |
| Online Access: | https://www.mdpi.com/2075-1680/14/1/17 |
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| Summary: | Let <i>G</i> be a finite simple graph and let <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>A</mi><mo>(</mo><mi>G</mi><mo>)</mo></mrow></semantics></math></inline-formula> be its adjacency matrix. Then, <i>G</i> is singular if <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>A</mi><mo>(</mo><mi>G</mi><mo>)</mo></mrow></semantics></math></inline-formula> is singular. The singularity of graphs is of certain interest in graph theory and algebraic combinatorics. For positive integers <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>a</mi><mi>i</mi></msub><mo>≥</mo><mn>2</mn></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>i</mi><mo>=</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mo>…</mo><mo>,</mo><mn>6</mn></mrow></semantics></math></inline-formula>. Insert <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>a</mi><mn>1</mn></msub><mo>−</mo><mn>2</mn></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>a</mi><mn>2</mn></msub><mo>−</mo><mn>2</mn></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>a</mi><mn>3</mn></msub><mo>−</mo><mn>2</mn></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>a</mi><mn>4</mn></msub><mo>−</mo><mn>2</mn></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>a</mi><mn>5</mn></msub><mo>−</mo><mn>2</mn></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>a</mi><mn>6</mn></msub><mo>−</mo><mn>2</mn></mrow></semantics></math></inline-formula> vertices in the six edges of the complete graph <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>K</mi><mn>4</mn></msub></semantics></math></inline-formula>, respectively, then the resulting graph is called the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>K</mi><mn>4</mn></msub></semantics></math></inline-formula> homeomorphic graph, denoted by <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>K</mi><mo>(</mo><msub><mi>a</mi><mn>1</mn></msub><mo>,</mo><msub><mi>a</mi><mn>2</mn></msub><mo>,</mo><msub><mi>a</mi><mn>3</mn></msub><mo>,</mo><msub><mi>a</mi><mn>4</mn></msub><mo>,</mo><msub><mi>a</mi><mn>5</mn></msub><mo>,</mo><msub><mi>a</mi><mn>6</mn></msub><mo>)</mo></mrow></semantics></math></inline-formula>. In this paper, we give the necessary and sufficient condition for the singularity of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>K</mi><mo>(</mo><msub><mi>a</mi><mn>1</mn></msub><mo>,</mo><msub><mi>a</mi><mn>2</mn></msub><mo>,</mo><msub><mi>a</mi><mn>3</mn></msub><mo>,</mo><msub><mi>a</mi><mn>4</mn></msub><mo>,</mo><msub><mi>a</mi><mn>5</mn></msub><mo>,</mo><msub><mi>a</mi><mn>6</mn></msub><mo>)</mo></mrow></semantics></math></inline-formula>, and we also show that the probability of a <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>K</mi><mn>4</mn></msub></semantics></math></inline-formula> homeomorphic graph <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>K</mi><mo>(</mo><msub><mi>a</mi><mn>1</mn></msub><mo>,</mo><msub><mi>a</mi><mn>2</mn></msub><mo>,</mo><msub><mi>a</mi><mn>3</mn></msub><mo>,</mo><msub><mi>a</mi><mn>4</mn></msub><mo>,</mo><msub><mi>a</mi><mn>5</mn></msub><mo>,</mo><msub><mi>a</mi><mn>6</mn></msub><mo>)</mo></mrow></semantics></math></inline-formula> being a singular graph is equal to <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mstyle scriptlevel="0" displaystyle="true"><mfrac><mn>193</mn><mn>512</mn></mfrac></mstyle></semantics></math></inline-formula>. |
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| ISSN: | 2075-1680 |