Generalized Latin Square Graphs of Semigroups: A Counting Framework for Regularity and Spectra

Abstract

We introduce the \emph{Generalized Latin Square Graph} $\Gamma(S)$ 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 $\Gamma(S)$ 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 $\Gamma(S)\cong K_n\times K_n$, we compute the spectrum and energy. A concise computational appendix lists the \texttt{GAP} driver and representative outputs.