1,142 research outputs found
Domination Number of the Non-commuting Graph of Finite Groups
Let G be a non-abelian group. The non-commuting graph of group G, shown by ΓG, is a graph with the vertex set G \ Z(G), where Z(G) is the center of group G. Also two distinct vertices of a and b are adjacent whenever ab ≠ba. A set S ⊆ V(Γ) of vertices in a graph Γ is a dominating set if every vertex v ∈ V(Γ) is an element of S or adjacent to an element of S. The domination number of a graph Γ denoted by γ(Γ), is the minimum size of a dominating set of Γ. </p><p>Here, we study some properties of the non-commuting graph of some finite groups. In this paper, we show that \gamma(\Gamma_G)<\frac{|G|-|Z(G)|}{2}. Also we charactrize all of groups G of order n with t = ∣Z(G)∣, in which $\gamma(\Gamma_{G})+\gamma(\overline{\Gamma}_{G})\in \{n-t+1,n-t,n-t-1,n-t-2\}.
Determinantal probability measures
Determinantal point processes have arisen in diverse settings in recent years
and have been investigated intensively. We study basic combinatorial and
probabilistic aspects in the discrete case. Our main results concern
relationships with matroids, stochastic domination, negative association,
completeness for infinite matroids, tail triviality, and a method for extension
of results from orthogonal projections to positive contractions. We also
present several new avenues for further investigation, involving Hilbert
spaces, combinatorics, homology, and group representations, among other areas.Comment: 50 pp; added reference to revision. Revised introduction and made
other small change
Finiteness of outer automorphism groups of random right-angled Artin groups
We consider the outer automorphism group Out(A_Gamma) of the right-angled
Artin group A_Gamma of a random graph Gamma on n vertices in the Erdos--Renyi
model. We show that the functions (log(n)+log(log(n)))/n and
1-(log(n)+log(log(n)))/n bound the range of edge probability functions for
which Out(A_Gamma) is finite: if the probability of an edge in Gamma is
strictly between these functions as n grows, then asymptotically Out(A_Gamma)
is almost surely finite, and if the edge probability is strictly outside of
both of these functions, then asymptotically Out(A_Gamma) is almost surely
infinite. This sharpens results of Ruth Charney and Michael Farber from their
preprint "Random groups arising as graph products", arXiv:1006.3378v1.Comment: 29 pages. Mostly rewritten, results tightened, statements corrected,
gaps fille
Unbounded operators, Lie algebras, and local representations
We prove a number of results on integrability and extendability of Lie
algebras of unbounded skew-symmetric operators with common dense domain in
Hilbert space. By integrability for a Lie algebra , we mean that
there is an associated unitary representation of the
corresponding simply connected Lie group such that is the
differential of . Our results extend earlier integrability results
in the literature; and are new even in the case of a single operator. Our
applications include a new invariant for certain Riemann surfaces.Comment: 20 pages, 2 Figure
Symplectic structures on right-angled Artin groups: between the mapping class group and the symplectic group
We define a family of groups that include the mapping class group of a genus
g surface with one boundary component and the integral symplectic group
Sp(2g,Z). We then prove that these groups are finitely generated. These groups,
which we call mapping class groups over graphs, are indexed over labeled
simplicial graphs with 2g vertices. The mapping class group over the graph
Gamma is defined to be a subgroup of the automorphism group of the right-angled
Artin group A_Gamma of Gamma. We also prove that the kernel of the map Aut
A_Gamma to Aut H_1(A_Gamma) is finitely generated, generalizing a theorem of
Magnus.Comment: 45 page
- …