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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2511.23190 |
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| _version_ | 1866911346322309120 |
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| author | Sorouhesh, Mohammad Reza Golriz, Mayam Panbehkar, Bozorg |
| author_facet | Sorouhesh, Mohammad Reza Golriz, Mayam Panbehkar, Bozorg |
| contents | We introduce the \emph{Generalized Latin Square Graph} $Γ(S)$ of a finite semigroup $S$. Since we record global factorization multiplicities and local alternative counts, we define three counting invariants $N_S,N_R,N_C$. This gives that we have a simple degree formula \[ \text{deg}(v)=2n-3+Q(v),\qquad Q(v)=N_S(s_k)-2N_R(v)-2N_C(v). \] We show that $Γ(S)$ is regular exactly when $Q$ is constant. We apply the framework to cancellative semigroups, bands, Brandt semigroups and null semigroups. For null semigroups, since we identify $Γ(S)\cong K_n\times K_n$, we compute the spectrum and energy. A concise computational appendix lists the \texttt{GAP} driver and representative outputs. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_23190 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Generalized Latin Square Graphs of Semigroups: A Counting Framework for Regularity and Spectra Sorouhesh, Mohammad Reza Golriz, Mayam Panbehkar, Bozorg Combinatorics 20M99, 05C25, 05B15, 20M17 G.2.1; G.2.2 We introduce the \emph{Generalized Latin Square Graph} $Γ(S)$ of a finite semigroup $S$. Since we record global factorization multiplicities and local alternative counts, we define three counting invariants $N_S,N_R,N_C$. This gives that we have a simple degree formula \[ \text{deg}(v)=2n-3+Q(v),\qquad Q(v)=N_S(s_k)-2N_R(v)-2N_C(v). \] We show that $Γ(S)$ is regular exactly when $Q$ is constant. We apply the framework to cancellative semigroups, bands, Brandt semigroups and null semigroups. For null semigroups, since we identify $Γ(S)\cong K_n\times K_n$, we compute the spectrum and energy. A concise computational appendix lists the \texttt{GAP} driver and representative outputs. |
| title | Generalized Latin Square Graphs of Semigroups: A Counting Framework for Regularity and Spectra |
| topic | Combinatorics 20M99, 05C25, 05B15, 20M17 G.2.1; G.2.2 |
| url | https://arxiv.org/abs/2511.23190 |