Abstract
Given a connected graph Δ, a group G can be constructed in such a way that Δ is often isomorphic to a subgraph of the commuting graph KG(Δ) of G. We show that, with one exception, KG(Δ) is connected, and in this latter case its diameter is at most that of Δ. If Δ is a path of length n>2, then diam(KG(Δ))=n.
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