In this paper, we prove that the n-simple braid divisible by the generators xi for all 2 ≤ i ≤ n-2 has trivial simple centralizer. Consequently, the commuting graph defined on the set of simple braids is disconnected. We also prove that the graph has one major component.
Affine monoids are the considered as natural discrete analogues of the finitely generated cones. The interconnection between these two objects has been an active area of research since last decade. Star network is one of the most common in computer network topologies. In this work, we study star topology Sn and associate a Coxeter structure of affine type on it. We find a recurrence relation and the Hilbert series of the associated right-angled monoid MSn∞. We observe that the growth rate of the monoid MSn∞ is unbounded.
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