Non-Isomorphic Cayley Graphs of Metacyclic Groups of Order 8<i>p</i> with the Same Spectrum
The spectrum of a graph <inline-formula><math display="inline"><semantics><mi mathvariant="normal">Γ</mi></semantics></math></inline-formula>, denoted by <inline-formula><math display="inline"><semantics>...
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| Main Authors: | , , |
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| Format: | Article |
| Language: | English |
| Published: |
MDPI AG
2025-06-01
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| Series: | Mathematics |
| Subjects: | |
| Online Access: | https://www.mdpi.com/2227-7390/13/12/1903 |
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| Summary: | The spectrum of a graph <inline-formula><math display="inline"><semantics><mi mathvariant="normal">Γ</mi></semantics></math></inline-formula>, denoted by <inline-formula><math display="inline"><semantics><mrow><mi>S</mi><mi>p</mi><mi>e</mi><mi>c</mi><mo>(</mo><mi mathvariant="normal">Γ</mi><mo>)</mo></mrow></semantics></math></inline-formula>, is the multiset of eigenvalues of its adjacency matrix. A Cayley graph <inline-formula><math display="inline"><semantics><mrow><mi>C</mi><mi>a</mi><mi>y</mi><mo>(</mo><mi>G</mi><mo>,</mo><mi>S</mi><mo>)</mo></mrow></semantics></math></inline-formula> of a finite group <i>G</i> is called Cay-DS (Cayley graph determined by its spectrum) if, for any other Cayley graph <inline-formula><math display="inline"><semantics><mrow><mi>C</mi><mi>a</mi><mi>y</mi><mo>(</mo><mi>G</mi><mo>,</mo><mi>T</mi><mo>)</mo></mrow></semantics></math></inline-formula>, <inline-formula><math display="inline"><semantics><mrow><mi>S</mi><mi>p</mi><mi>e</mi><mi>c</mi><mo>(</mo><mi>C</mi><mi>a</mi><mi>y</mi><mo>(</mo><mi>G</mi><mo>,</mo><mi>S</mi><mo>)</mo><mo>)</mo><mo>=</mo><mi>S</mi><mi>p</mi><mi>e</mi><mi>c</mi><mo>(</mo><mi>C</mi><mi>a</mi><mi>y</mi><mo>(</mo><mi>G</mi><mo>,</mo><mi>T</mi><mo>)</mo><mo>)</mo></mrow></semantics></math></inline-formula> implies <inline-formula><math display="inline"><semantics><mrow><mi>C</mi><mi>a</mi><mi>y</mi><mo>(</mo><mi>G</mi><mo>,</mo><mi>S</mi><mo>)</mo><mo>≅</mo><mi>C</mi><mi>a</mi><mi>y</mi><mo>(</mo><mi>G</mi><mo>,</mo><mi>T</mi><mo>)</mo></mrow></semantics></math></inline-formula>. A group <i>G</i> is said to be Cay-DS if all Cayley graphs of <i>G</i> are Cay-DS. An interesting open problem in the area of algebraic graph theory involves characterizing finite Cay-DS groups or constructing non-isomorphic Cayley graphs of a non-Cay-DS group that share the same spectrum. The present paper contributes to parts of this problem of metacyclic groups <inline-formula><math display="inline"><semantics><msub><mi>M</mi><mrow><mn>8</mn><mi>p</mi></mrow></msub></semantics></math></inline-formula> of order <inline-formula><math display="inline"><semantics><mrow><mn>8</mn><mi>p</mi></mrow></semantics></math></inline-formula> (with center of order 4), where <i>p</i> is an odd prime, in terms of irreducible characters, which are first presented. Then some new families of pairwise non-isomorphic Cayley graph pairs of <inline-formula><math display="inline"><semantics><msub><mi>M</mi><mrow><mn>8</mn><mi>p</mi></mrow></msub></semantics></math></inline-formula> (<inline-formula><math display="inline"><semantics><mrow><mi>p</mi><mo>≥</mo><mn>5</mn></mrow></semantics></math></inline-formula>) with the same spectrum are found. As a conclusion, this paper concludes that <inline-formula><math display="inline"><semantics><msub><mi>M</mi><mrow><mn>8</mn><mi>p</mi></mrow></msub></semantics></math></inline-formula> is Cay-DS if and only if <inline-formula><math display="inline"><semantics><mrow><mi>p</mi><mo>=</mo><mn>3</mn></mrow></semantics></math></inline-formula>. |
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| ISSN: | 2227-7390 |