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

TL;DR

This paper introduces a new class of graphs derived from finite semigroups, providing a counting framework to analyze their regularity and spectral properties, with applications to various semigroup types.

Contribution

It defines generalized Latin square graphs of semigroups and establishes a counting invariants framework to determine regularity and spectra, including explicit computations for null semigroups.

Findings

The degree formula relates graph degree to counting invariants.

Regularity of the graph corresponds to constancy of a specific function Q.

Explicit spectrum and energy are computed for null semigroups.

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.