Structure Fault Tolerance of Fully Connected Cubic Networks

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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Bibliographic Details
Main Authors: Eminjan Sabir, Cheng-Kuan Lin
Format: Article
Language:English
Published: MDPI AG 2025-05-01
Series:Mathematics
Subjects:
connectivity
extra connectivity
structure connectivity
fully connected cubic networks
fault tolerance
Online Access:https://www.mdpi.com/2227-7390/13/9/1532
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Summary: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>.
ISSN:2227-7390

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