27,455 research outputs found
Signless Laplacian energies of non-commuting graphs of finite groups and related results
The non-commuting graph of a non-abelian group with center is a
simple undirected graph whose vertex set is and two vertices
are adjacent if . In this study, we compute Signless
Laplacian spectrum and Signless Laplacian energy of non-commuting graphs of
finite groups. We obtain several conditions such that the non-commuting graph
of is Q-integral and observe relations between energy, Signless Laplacian
energy and Laplacian energy. In addition, we look into the energetic hyper- and
hypo-properties of non-commuting graphs of finite groups. We also assess
whether the same graphs are Q-hyperenergetic and L-hyperenergetic.Comment: 39 page
Criterion of unrecognizability of a finite group by its Gruenberg-Kegel graph
The Gruenberg-Kegel graph associated with a finite group has
as vertices the prime divisors of , with an edge from to if and
only if contains an element of order . This graph has been the subject
of much recent interest; one of our goals here is to give a survey of some of
this material, relating to groups with the same Gruenberg-Kegel graph. However,
our main aim is to prove several new results. Among them are the following.
- There are infinitely many finite groups with the same Gruenberg-Kegel graph
as the Gruenberg-Kegel of a finite group if and only if there is a finite
group with non-trivial solvable radical such that .
- There is a function on the natural numbers with the property that if a
finite -vertex graph whose vertices are labelled by pairwise distinct primes
is the Gruenberg-Kegel graph of more than finite groups, then it is the
Gruenberg-Kegel graph of infinitely many finite groups. (The function we give
satisfies , but this is probably not best possible.)
- If a finite graph whose vertices are labelled by pairwise distinct
primes is the Gruenberg-Kegel graph of only finitely many finite groups, then
all such groups are almost simple; moreover, has at least three
pairwise non-adjacent vertices, and is non-adjacent to at least one odd
vertex.
- Groups whose power graphs, or commuting graphs, are isomorphic have the
same Gruenberg-Kegel graph.
- The groups and are uniquely determined by the
isomorphism types of their Gruenberg-Kegel graphs.
In addition, we consider groups whose Gruenberg-Kegel graph has no edges.
These are the groups in which every element has prime power order, and have
been studied under the name \emph{EPPO groups}; completing this line of
research, we give a complete list of such groups.Comment: 29 page
A probabilistic approach to a classical result of ore
The subgroup commutativity degree sd(G) of a finite group G was introduced almost ten years ago and deals with the number of commuting subgroups in the subgroups lattice L(G) of G. The extremal case sd(G) = 1 detects a class of groups classified by Iwasawa in 1941 (in fact sd(G) represents a probabilistic measure which allows us to understand how far is G from the groups of Iwasawa). Among them we have sd(G) = 1 when L(G) is distributive, that is, when G is cyclic. The characterization of a cyclic group by the distributivity of its lattice of subgroups is due to a classical result of Ore in 1938. Therefore sd(G) is strongly related to structural properties of L(G). Here we introduce a new notion of probability gsd(G) in which two arbitrary sublattices S(G) and T(G) of L(G) are involved simultaneously. In case S(G) = T(G) = L(G), we find exactly sd(G). Upper and lower bounds in terms of gsd(G) and sd(G) are among our main contributions, when the condition S(G) = T(G) = L(G) is removed. Then we investigate the problem of counting the pairs of commuting subgroups via an appropriate graph. Looking at the literature, we noted that a similar problem motivated the permutability graph of non–normal subgroups ΓN (G) in 1995, that is, the graph where all proper non– normal subgroups of G form the vertex set of ΓN (G) and two vertices H and K are joined if HK = KH. The graph ΓN (G) has been recently generalized via the notion of permutability graph of subgroups Γ(G), extending the vertex set to all proper subgroups of G and keeping the same criterion to join two vertices. We use gsd(G), in order to introduce the non–permutability graph of subgroups ΓL(G) ; its vertices are now given by the set L(G) − CL(G)(L(G)), where CL(G)(L(G)) is the smallest sublattice of L(G) containing all permutable subgroups of G, and we join two vertices H, K of ΓL(G) if HK 6= KH. We finally study some classical invariants for ΓL(G) and find numerical relations between the number of edges of ΓL(G) and gsd(G)
The diameter of the commuting graph of a finite group with trivial centre
The commuting graph of a finite group with trivial centre is examined. It is
shown that the connected components of the commuting graph have diameter at
most 10
On maximal distances in a commuting graph
We study maximal distances in the commuting graphs of matrix algebras defined
over algebraically closed fields. In particular, we show that the maximal
distance can be attained only between two nonderogatory matrices. We also
describe rank-one and semisimple matrices using the distances in the commuting
graph
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