For a group G, let Γ(G) denote the graph defined on the elements of G in such a way that two distinct vertices are connected by an edge if and only if they generate G. Let Γ∗(G) be the subgraph of Γ(G) that is induced by all the vertices of Γ(G) that are not isolated. We prove that if G is a 2-generated noncyclic abelian group, then Γ∗(G) is connected. Moreover, diam(Γ∗(G)) = 2 if the torsion subgroup of G is nontrivial and diam(Γ∗(G)) = ∞ otherwise. If F is the free group of rank 2, then Γ∗(F) is connected and we deduce from diam(Γ∗(Z × Z)) = ∞ that diam(Γ∗(F)) = ∞

The generating graph of infinite abelian groups / Acciarri, C.; Lucchini, A.. - In: BULLETIN OF THE AUSTRALIAN MATHEMATICAL SOCIETY. - ISSN 0004-9727. - 100:1(2019), pp. 68-75. [10.1017/S0004972718001466]

### The generating graph of infinite abelian groups

#### Abstract

For a group G, let Γ(G) denote the graph defined on the elements of G in such a way that two distinct vertices are connected by an edge if and only if they generate G. Let Γ∗(G) be the subgraph of Γ(G) that is induced by all the vertices of Γ(G) that are not isolated. We prove that if G is a 2-generated noncyclic abelian group, then Γ∗(G) is connected. Moreover, diam(Γ∗(G)) = 2 if the torsion subgroup of G is nontrivial and diam(Γ∗(G)) = ∞ otherwise. If F is the free group of rank 2, then Γ∗(F) is connected and we deduce from diam(Γ∗(Z × Z)) = ∞ that diam(Γ∗(F)) = ∞
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The generating graph of infinite abelian groups / Acciarri, C.; Lucchini, A.. - In: BULLETIN OF THE AUSTRALIAN MATHEMATICAL SOCIETY. - ISSN 0004-9727. - 100:1(2019), pp. 68-75. [10.1017/S0004972718001466]
Acciarri, C.; Lucchini, A.
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Utilizza questo identificativo per citare o creare un link a questo documento: `https://hdl.handle.net/11380/1255519`