Automorphism group and Laplacian spectrum of a graph over Brandt semigroups (Q6601647)

A Brandt semigroup is a completely \(0\)-simple semigroup in which the idempotents commute.The finite Brandt semigroup over a trivial group with \(n^2+1\) elements is denoted \(B_n\). The paper presents results on the enhanced power graph \(\mathcal{P}_e(B_n)\), whose vertices are the elements of \(B_n\) and in which two distinct vertices are adjacent when they are powers of the same element. For \(B_n\), this is a rather obvious graph with idempotents as isolated points, the remaining vertices forming a star centered at \(0\). It does not require much thought to see that there is not much of interest one can say about such a graph. In particular, it is obvious what its automorphism group is; yet, this is presented as one of the main results in the paper. The author goes on to compute the Laplacian matrix, its spectrum and its distance spectrum, which amounts to exercises in linear algebra. Unfortunately, the paper is also not very well written, with plenty of typographical errors, notation that is used before being introduced and even a meaningless statement (Lemma~7.2).\N\NFor the entire collection see [Zbl 1515.20018].