Structure Fault Tolerance of Fully Connected Cubic Networks
An interconnection network is usually modeled by a graph, and fault tolerance of the interconnection network is often measured by connectivity of the graph. Given a connected subgraph <i>L</i> of a graph <i>G</i> and non-negative integer <i>t</i>, the <i>t&l...
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2025-05-01
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| author | Eminjan Sabir Cheng-Kuan Lin |
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| description | An interconnection network is usually modeled by a graph, and fault tolerance of the interconnection network is often measured by connectivity of the graph. Given a connected subgraph <i>L</i> of a graph <i>G</i> and non-negative integer <i>t</i>, the <i>t</i>-extra connectivity <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>κ</mi><mi>t</mi></msub><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula>, the <i>L</i>-structure connectivity <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>κ</mi><mo>(</mo><mi>G</mi><mo>;</mo><mi>L</mi><mo>)</mo></mrow></semantics></math></inline-formula> and the <i>t</i>-extra <i>L</i>-structure connectivity <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>κ</mi><mi>g</mi></msub><mrow><mo>(</mo><mi>G</mi><mo>;</mo><mi>L</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> of <i>G</i> can provide new metrics to measure the fault tolerance of a network represented by <i>G</i>. Fully connected cubic networks <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi mathvariant="script">FC</mi><mi>n</mi></msub></semantics></math></inline-formula> are a class of hierarchical networks which enjoy the strengths of a constant vertex degree and good expansibility. In this paper, we determine <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>κ</mi><mi>t</mi></msub><mrow><mo>(</mo><msub><mi mathvariant="script">FC</mi><mi>n</mi></msub><mo>)</mo></mrow></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>κ</mi><mo>(</mo><msub><mi mathvariant="script">FC</mi><mi>n</mi></msub><mo>;</mo><mi>L</mi><mo>)</mo></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>κ</mi><mi>t</mi></msub><mrow><mo>(</mo><msub><mi mathvariant="script">FC</mi><mi>n</mi></msub><mo>;</mo><mi>L</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>t</mi><mo>=</mo><mn>1</mn></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>L</mi><mo>∈</mo><mo>{</mo><msub><mi>K</mi><mrow><mn>1</mn><mo>,</mo><mn>1</mn></mrow></msub><mo>,</mo><msub><mi>K</mi><mrow><mn>1</mn><mo>,</mo><mn>2</mn></mrow></msub><mo>,</mo><msub><mi>K</mi><mrow><mn>1</mn><mo>,</mo><mn>3</mn></mrow></msub><mo>}</mo></mrow></semantics></math></inline-formula>. We also establish the edge versions <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>λ</mi><mi>t</mi></msub><mrow><mo>(</mo><msub><mi mathvariant="script">FC</mi><mi>n</mi></msub><mo>)</mo></mrow></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>λ</mi><mo>(</mo><msub><mi mathvariant="script">FC</mi><mi>n</mi></msub><mo>;</mo><mi>L</mi><mo>)</mo></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>λ</mi><mi>t</mi></msub><mrow><mo>(</mo><msub><mi mathvariant="script">FC</mi><mi>n</mi></msub><mo>;</mo><mi>L</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>t</mi><mo>=</mo><mn>1</mn></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>L</mi><mo>∈</mo><mo>{</mo><msub><mi>K</mi><mrow><mn>1</mn><mo>,</mo><mn>1</mn></mrow></msub><mo>,</mo><msub><mi>K</mi><mrow><mn>1</mn><mo>,</mo><mn>2</mn></mrow></msub><mo>}</mo></mrow></semantics></math></inline-formula>. |
| format | Article |
| id | doaj-art-2a09d1f1a5254629afbeffacc4b7edc6 |
| institution | OA Journals |
| issn | 2227-7390 |
| language | English |
| publishDate | 2025-05-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj-art-2a09d1f1a5254629afbeffacc4b7edc62025-08-20T01:49:28ZengMDPI AGMathematics2227-73902025-05-01139153210.3390/math13091532Structure Fault Tolerance of Fully Connected Cubic NetworksEminjan Sabir0Cheng-Kuan Lin1College of Mathematics and System Sciences, Xinjiang University, Urumqi 830046, ChinaDepartment of Computer Science, National Yang Ming Chiao Tung University, Hsinchu 30010, TaiwanAn interconnection network is usually modeled by a graph, and fault tolerance of the interconnection network is often measured by connectivity of the graph. Given a connected subgraph <i>L</i> of a graph <i>G</i> and non-negative integer <i>t</i>, the <i>t</i>-extra connectivity <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>κ</mi><mi>t</mi></msub><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula>, the <i>L</i>-structure connectivity <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>κ</mi><mo>(</mo><mi>G</mi><mo>;</mo><mi>L</mi><mo>)</mo></mrow></semantics></math></inline-formula> and the <i>t</i>-extra <i>L</i>-structure connectivity <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>κ</mi><mi>g</mi></msub><mrow><mo>(</mo><mi>G</mi><mo>;</mo><mi>L</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> of <i>G</i> can provide new metrics to measure the fault tolerance of a network represented by <i>G</i>. Fully connected cubic networks <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi mathvariant="script">FC</mi><mi>n</mi></msub></semantics></math></inline-formula> are a class of hierarchical networks which enjoy the strengths of a constant vertex degree and good expansibility. In this paper, we determine <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>κ</mi><mi>t</mi></msub><mrow><mo>(</mo><msub><mi mathvariant="script">FC</mi><mi>n</mi></msub><mo>)</mo></mrow></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>κ</mi><mo>(</mo><msub><mi mathvariant="script">FC</mi><mi>n</mi></msub><mo>;</mo><mi>L</mi><mo>)</mo></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>κ</mi><mi>t</mi></msub><mrow><mo>(</mo><msub><mi mathvariant="script">FC</mi><mi>n</mi></msub><mo>;</mo><mi>L</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>t</mi><mo>=</mo><mn>1</mn></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>L</mi><mo>∈</mo><mo>{</mo><msub><mi>K</mi><mrow><mn>1</mn><mo>,</mo><mn>1</mn></mrow></msub><mo>,</mo><msub><mi>K</mi><mrow><mn>1</mn><mo>,</mo><mn>2</mn></mrow></msub><mo>,</mo><msub><mi>K</mi><mrow><mn>1</mn><mo>,</mo><mn>3</mn></mrow></msub><mo>}</mo></mrow></semantics></math></inline-formula>. We also establish the edge versions <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>λ</mi><mi>t</mi></msub><mrow><mo>(</mo><msub><mi mathvariant="script">FC</mi><mi>n</mi></msub><mo>)</mo></mrow></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>λ</mi><mo>(</mo><msub><mi mathvariant="script">FC</mi><mi>n</mi></msub><mo>;</mo><mi>L</mi><mo>)</mo></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>λ</mi><mi>t</mi></msub><mrow><mo>(</mo><msub><mi mathvariant="script">FC</mi><mi>n</mi></msub><mo>;</mo><mi>L</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>t</mi><mo>=</mo><mn>1</mn></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>L</mi><mo>∈</mo><mo>{</mo><msub><mi>K</mi><mrow><mn>1</mn><mo>,</mo><mn>1</mn></mrow></msub><mo>,</mo><msub><mi>K</mi><mrow><mn>1</mn><mo>,</mo><mn>2</mn></mrow></msub><mo>}</mo></mrow></semantics></math></inline-formula>.https://www.mdpi.com/2227-7390/13/9/1532connectivityextra connectivitystructure connectivityfully connected cubic networksfault tolerance |
| spellingShingle | Eminjan Sabir Cheng-Kuan Lin Structure Fault Tolerance of Fully Connected Cubic Networks Mathematics connectivity extra connectivity structure connectivity fully connected cubic networks fault tolerance |
| title | Structure Fault Tolerance of Fully Connected Cubic Networks |
| title_full | Structure Fault Tolerance of Fully Connected Cubic Networks |
| title_fullStr | Structure Fault Tolerance of Fully Connected Cubic Networks |
| title_full_unstemmed | Structure Fault Tolerance of Fully Connected Cubic Networks |
| title_short | Structure Fault Tolerance of Fully Connected Cubic Networks |
| title_sort | structure fault tolerance of fully connected cubic networks |
| topic | connectivity extra connectivity structure connectivity fully connected cubic networks fault tolerance |
| url | https://www.mdpi.com/2227-7390/13/9/1532 |
| work_keys_str_mv | AT eminjansabir structurefaulttoleranceoffullyconnectedcubicnetworks AT chengkuanlin structurefaulttoleranceoffullyconnectedcubicnetworks |