Given a 2-generated finite group
G
, the non-generating graph of
G
has as vertices the elements of
G
and two vertices are adjacent if and only if they are distinct and do not generate
G
. We consider ...the graph
Σ
(
G
)
obtained from the non-generating graph of
G
by deleting the universal vertices. We prove that if the derived subgroup of
G
is not nilpotent, then this graph is connected, with diameter at most 5. Moreover, we give a complete classification of the finite groups
G
such that
Σ
(
G
)
is disconnected.
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EMUNI, FIS, FZAB, GEOZS, GIS, IJS, IMTLJ, KILJ, KISLJ, MFDPS, NLZOH, NUK, OILJ, PNG, SAZU, SBCE, SBJE, SBMB, SBNM, UKNU, UL, UM, UPUK, VKSCE, ZAGLJ
We prove that the graph obtained from the non-nilpotent graph of a finite group by deleting the isolated vertices is connected with diameter at most 3. This bound is the best possible.
The non‐F graph of a finite group Lucchini, Andrea; Nemmi, Daniele
Mathematische Nachrichten,
October 2021, 2021-10-00, 20211001, Volume:
294, Issue:
10
Journal Article
Peer reviewed
Given a formation F, we consider the graph whose vertices are the elements of G and where two vertices g,h∈G are adjacent if and only if g,h∉F. We are interested in the two following questions. Is ...the set of the isolated vertices of this graph a subgroup of G? Is the subgraph obtained by deleting the isolated vertices a connected graph?
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BFBNIB, FZAB, GIS, IJS, KILJ, NLZOH, NUK, OILJ, SAZU, SBCE, SBMB, UL, UM, UPUK
4.
On the soluble graph of a finite group Burness, Timothy C.; Lucchini, Andrea; Nemmi, Daniele
Journal of combinatorial theory. Series A,
February 2023, 2023-02-00, Volume:
194
Journal Article
Peer reviewed
Open access
Let G be a finite insoluble group with soluble radical R(G). In this paper we investigate the soluble graph of G, which is a natural generalisation of the widely studied commuting graph. Here the ...vertices are the elements in G∖R(G), with x adjacent to y if they generate a soluble subgroup of G. Our main result states that this graph is always connected and its diameter, denoted δS(G), is at most 5. More precisely, we show that δS(G)⩽3 if G is not almost simple and we obtain stronger bounds for various families of almost simple groups. For example, we will show that δS(Sn)=3 for all n⩾6. We also establish the existence of simple groups with δS(G)⩾4. For instance, we prove that δS(A2p+1)⩾4 for every Sophie Germain prime p⩾5, which demonstrates that our general upper bound of 5 is close to best possible. We conclude by briefly discussing some variations of the soluble graph construction and we present several open problems.
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Available for:
GEOZS, IJS, IMTLJ, KILJ, KISLJ, NLZOH, NUK, OILJ, PNG, SAZU, SBCE, SBJE, UILJ, UL, UM, UPUK, ZAGLJ, ZRSKP
The Engel graph of a finite group Detomi, Eloisa; Lucchini, Andrea; Nemmi, Daniele
Forum mathematicum,
01/2023, Volume:
35, Issue:
1
Journal Article
Peer reviewed
Open access
For a finite group
, we investigate the directed graph
, whose vertices are the non-hypercentral elements of
and where there is an edge
if and only if
for some
.
We prove that
is always weakly ...connected and is strongly connected if
is neither Frobenius nor almost simple.
The generating graph encodes how generating pairs are spread among the
elements of a group. For more than ten years it has been conjectured that this
graph is connected for every finite group. In ...this paper, we give evidence
supporting this conjecture: we prove that it holds for all but a finite number
of almost simple groups and give a reduction to groups without non-trivial
soluble normal subgroups. Let $d(G)$ be the minimal cardinality of a generating
set for $G$. When $d(G)\geq3$, the generating graph is empty and the conjecture
is trivially true. We consider it in the more general setting of the rank
graph, which encodes how pairs of elements belonging to generating sets of
minimal cardinality spread among the elements of a group. It carries
information even when $d(G)\geq3$ and corresponds to the generating graph when
$d(G)=2$. We prove that it is connected whenever $d(G)\geq3$, giving tools and
ideas that may be used to address the original conjecture.
Given a class $\mathfrak F$ of finite groups, we consider the graph
$\widetilde\Gamma_{\mathfrak F}(G)$ whose vertices are the elements of $G$ and
where two vertices $g,h\in G$ are adjacent if and ...only if $\langle
g,h\rangle\notin\mathfrak F$. Moreover we denote by $\mathcal{I}_{\mathfrak
F}(G)$ the set of the isolated vertices of $\widetilde\Gamma_{\mathfrak F}(G).$
We address the following question: to what extent the fact that
$\mathcal{I}_{\mathfrak F}(G)$ is a subgroup of $H$ for any $H\leq G,$ implies
that the graph $\Gamma_{\mathfrak F}G)$ obtained from
$\widetilde\Gamma_{\mathfrak F}(G)$ by deleting the isolated vertices is a
connected graph?
Given a 2-generated finite group \(G\), the non-generating graph of \(G\) has as vertices the elements of \(G\) and two vertices are adjacent if and only if they are distinct and do not generate ...\(G\). We consider the graph \(\Sigma(G)\) obtained from the non-generating graph of \(G\) by deleting the universal vertices. We prove that if the derived subgroup of \(G\) is not nilpotent, then this graph is connected, with diameter at most 5. Moreover we give a complete classification of the finite groups \(G\) such that \(\Sigma(G)\) is disconnected.
