94 research outputs found
Identifying derivations through the spectra of their values
We consider the relationship between derivations and of a Banach
algebra that satisfy \s(g(x)) \subseteq \s(d(x)) for every ,
where \s(\, . \,) stands for the spectrum. It turns out that in some basic
situations, say if , the only possibilities are that , , and,
if is an inner derivation implemented by an algebraic element of degree 2,
also . The conclusions in more complex classes of algebras are not so
simple, but are of a similar spirit. A rather definitive result is obtained for
von Neumann algebras. In general -algebras we have to make some
adjustments, in particular we restrict our attention to inner derivations
implemented by selfadjoint elements. We also consider a related condition
for all selfadjoint elements from a
-algebra , where and is normal.Comment: 12 page
A local-global principle for linear dependence of noncommutative polynomials
A set of polynomials in noncommuting variables is called locally linearly
dependent if their evaluations at tuples of matrices are always linearly
dependent. By a theorem of Camino, Helton, Skelton and Ye, a finite locally
linearly dependent set of polynomials is linearly dependent. In this short note
an alternative proof based on the theory of polynomial identities is given. The
method of the proof yields generalizations to directional local linear
dependence and evaluations in general algebras over fields of arbitrary
characteristic. A main feature of the proof is that it makes it possible to
deduce bounds on the size of the matrices where the (directional) local linear
dependence needs to be tested in order to establish linear dependence.Comment: 8 page
Kernelization and Parameterized Algorithms for 3-Path Vertex Cover
A 3-path vertex cover in a graph is a vertex subset such that every path
of three vertices contains at least one vertex from . The parameterized
3-path vertex cover problem asks whether a graph has a 3-path vertex cover of
size at most . In this paper, we give a kernel of vertices and an
-time and polynomial-space algorithm for this problem, both new
results improve previous known bounds.Comment: in TAMC 2016, LNCS 9796, 201
The generalized 3-edge-connectivity of lexicographic product graphs
The generalized -edge-connectivity of a graph is a
generalization of the concept of edge-connectivity. The lexicographic product
of two graphs and , denoted by , is an important graph
product. In this paper, we mainly study the generalized 3-edge-connectivity of
, and get upper and lower bounds of .
Moreover, all bounds are sharp.Comment: 14 page
Bucolic Complexes
We introduce and investigate bucolic complexes, a common generalization of
systolic complexes and of CAT(0) cubical complexes. They are defined as simply
connected prism complexes satisfying some local combinatorial conditions. We
study various approaches to bucolic complexes: from graph-theoretic and
topological perspective, as well as from the point of view of geometric group
theory. In particular, we characterize bucolic complexes by some properties of
their 2-skeleta and 1-skeleta (that we call bucolic graphs), by which several
known results are generalized. We also show that locally-finite bucolic
complexes are contractible, and satisfy some nonpositive-curvature-like
properties.Comment: 45 pages, 4 figure
The variety of domination games
Domination game (Brešar et al. in SIAM J Discrete Math 24:979–991, 2010) and total domination game (Henning et al. in Graphs Comb 31:1453–1462 (2015) are by now well established games played on graphs by two players, named Dominator and Staller. In this paper, Z-domination game, L-domination game, and LL-domination game are introduced as natural companions of the standard domination games. Versions of the Continuation Principle are proved for the new games. It is proved that in each of these games the outcome of the game, which is a corresponding graph invariant, differs by at most one depending whether Dominator or Staller starts the game. The hierarchy of the five domination games is established. The invariants are also bounded with respect to the (total) domination number and to the order of a graph. Values of the three new invariants are determined for paths up to a small constant independent from the length of a path. Several open problems and a conjecture are listed. The latter asserts that the L-domination game number is not greater than 6 / 7 of the order of a graph. © 2019, Springer Nature Switzerland AG
Nonrepetitive Colouring via Entropy Compression
A vertex colouring of a graph is \emph{nonrepetitive} if there is no path
whose first half receives the same sequence of colours as the second half. A
graph is nonrepetitively -choosable if given lists of at least colours
at each vertex, there is a nonrepetitive colouring such that each vertex is
coloured from its own list. It is known that every graph with maximum degree
is -choosable, for some constant . We prove this result
with (ignoring lower order terms). We then prove that every subdivision
of a graph with sufficiently many division vertices per edge is nonrepetitively
5-choosable. The proofs of both these results are based on the Moser-Tardos
entropy-compression method, and a recent extension by Grytczuk, Kozik and Micek
for the nonrepetitive choosability of paths. Finally, we prove that every graph
with pathwidth is nonrepetitively -colourable.Comment: v4: Minor changes made following helpful comments by the referee
The groupoid approach to Leavitt path algebras
When the theory of Leavitt path algebras was already quite advanced, it was discovered that some of the more difficult questions were susceptible to a new approach using topological groupoids. The main result that makes this possible is that the Leavitt path algebra of a graph is graded isomorphic to the Steinberg algebra of the graph’s boundary path groupoid. This expository paper has three parts: Part 1 is on the Steinberg algebra of a groupoid, Part 2 is on the path space and boundary path groupoid of a graph, and Part 3 is on the Leavitt path algebra of a graph. It is a self-contained reference on these topics, intended to be useful to beginners and experts alike. While revisiting the fundamentals, we prove some results in greater generality than can be found elsewhere, including the uniqueness theorems for Leavitt path algebras
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