Exploring the Structural and Traversal Properties of Total Graphs over Finite Rings
This paper extends the concept of the total graph <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>T</mi><mi mathvariant="sans-serif">Γ</mi></msub><...
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2025-05-01
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| author | Ali Al Khabyah Nazim Ikram Ali |
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| description | This paper extends the concept of the total graph <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>T</mi><mi mathvariant="sans-serif">Γ</mi></msub><mrow><mo>(</mo><mi>R</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> associated with a commutative ring to the three-fold Cartesian product <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>R</mi><mo>=</mo><msub><mi mathvariant="double-struck">Z</mi><mi>n</mi></msub><mo>×</mo><msub><mi mathvariant="double-struck">Z</mi><mi>m</mi></msub><mo>×</mo><msub><mi mathvariant="double-struck">Z</mi><mi>p</mi></msub></mrow></semantics></math></inline-formula>, where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>n</mi><mo>,</mo><mi>m</mi><mo>,</mo><mi>p</mi><mo>></mo><mn>1</mn></mrow></semantics></math></inline-formula>. We present complete and self-contained proofs for a wide range of graph-theoretic properties of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>T</mi><mi mathvariant="sans-serif">Γ</mi></msub><mrow><mo>(</mo><mi>R</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula>, including connectivity, diameter, regularity conditions, clique and independence numbers, and exact criteria for Hamiltonicity and Eulericity. We also derive improved lower bounds for the genus and characterize the automorphism group in both general and symmetric cases. Each result is illustrated through concrete numerical examples for clarity. Beyond theoretical contributions, we discuss potential applications in cryptographic key-exchange systems, fault-tolerant network architectures, and algebraic code design. This work generalizes and deepens prior studies on two-factor total graphs, and establishes a foundational framework for future exploration of higher-dimensional total graphs over finite commutative rings. |
| format | Article |
| id | doaj-art-0a4f968f4e4440f19a183e225bbde4cc |
| institution | OA Journals |
| issn | 2075-1680 |
| language | English |
| publishDate | 2025-05-01 |
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| spelling | doaj-art-0a4f968f4e4440f19a183e225bbde4cc2025-08-20T02:33:39ZengMDPI AGAxioms2075-16802025-05-0114538610.3390/axioms14050386Exploring the Structural and Traversal Properties of Total Graphs over Finite RingsAli Al Khabyah0Nazim1Ikram Ali2Department of Mathematics, College of Science, King Khalid University, Abha 61413, Saudi ArabiaDepartment of Applied Sciences, Meerut Institute of Engineering and Technology (MIET), Meerut 250005, IndiaDepartment of Computer Science, School of Engineering & Technology, Shri Venkateshwara University, Gajraula, Amroha 244236, IndiaThis paper extends the concept of the total graph <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>T</mi><mi mathvariant="sans-serif">Γ</mi></msub><mrow><mo>(</mo><mi>R</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> associated with a commutative ring to the three-fold Cartesian product <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>R</mi><mo>=</mo><msub><mi mathvariant="double-struck">Z</mi><mi>n</mi></msub><mo>×</mo><msub><mi mathvariant="double-struck">Z</mi><mi>m</mi></msub><mo>×</mo><msub><mi mathvariant="double-struck">Z</mi><mi>p</mi></msub></mrow></semantics></math></inline-formula>, where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>n</mi><mo>,</mo><mi>m</mi><mo>,</mo><mi>p</mi><mo>></mo><mn>1</mn></mrow></semantics></math></inline-formula>. We present complete and self-contained proofs for a wide range of graph-theoretic properties of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>T</mi><mi mathvariant="sans-serif">Γ</mi></msub><mrow><mo>(</mo><mi>R</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula>, including connectivity, diameter, regularity conditions, clique and independence numbers, and exact criteria for Hamiltonicity and Eulericity. We also derive improved lower bounds for the genus and characterize the automorphism group in both general and symmetric cases. Each result is illustrated through concrete numerical examples for clarity. Beyond theoretical contributions, we discuss potential applications in cryptographic key-exchange systems, fault-tolerant network architectures, and algebraic code design. This work generalizes and deepens prior studies on two-factor total graphs, and establishes a foundational framework for future exploration of higher-dimensional total graphs over finite commutative rings.https://www.mdpi.com/2075-1680/14/5/386total graphfinite commutative ringsautomorphism groupgraph traversal properties |
| spellingShingle | Ali Al Khabyah Nazim Ikram Ali Exploring the Structural and Traversal Properties of Total Graphs over Finite Rings Axioms total graph finite commutative rings automorphism group graph traversal properties |
| title | Exploring the Structural and Traversal Properties of Total Graphs over Finite Rings |
| title_full | Exploring the Structural and Traversal Properties of Total Graphs over Finite Rings |
| title_fullStr | Exploring the Structural and Traversal Properties of Total Graphs over Finite Rings |
| title_full_unstemmed | Exploring the Structural and Traversal Properties of Total Graphs over Finite Rings |
| title_short | Exploring the Structural and Traversal Properties of Total Graphs over Finite Rings |
| title_sort | exploring the structural and traversal properties of total graphs over finite rings |
| topic | total graph finite commutative rings automorphism group graph traversal properties |
| url | https://www.mdpi.com/2075-1680/14/5/386 |
| work_keys_str_mv | AT alialkhabyah exploringthestructuralandtraversalpropertiesoftotalgraphsoverfiniterings AT nazim exploringthestructuralandtraversalpropertiesoftotalgraphsoverfiniterings AT ikramali exploringthestructuralandtraversalpropertiesoftotalgraphsoverfiniterings |