Identifying derivations through the spectra of their values

We consider the relationship between derivations $d$ and $g$ of a Banach algebra $B$ that satisfy \s(g(x)) \subseteq \s(d(x)) for every $x\in B$, where \s(\, . \,) stands for the spectrum. It turns out that in some basic situations, say if $B=B(X)$, the only possibilities are that $g=d$, $g=0$, and, if $d$ is an inner derivation implemented by an algebraic element of degree 2, also $g=-d$. 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 $C^*$-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 $\|[b,x]\|\leq M\|[a,x]\|$ for all selfadjoint elements $x$ from a $C^*$-algebra $B$, where $a,b\in B$ and $a$ 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 $C$ such that every path of three vertices contains at least one vertex from $C$. The parameterized 3-path vertex cover problem asks whether a graph has a 3-path vertex cover of size at most $k$. In this paper, we give a kernel of $5k$ vertices and an $O^*(1.7485^k)$-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 $k$-edge-connectivity $\lambda_k(G)$ of a graph $G$ is a generalization of the concept of edge-connectivity. The lexicographic product of two graphs $G$ and $H$, denoted by $G\circ H$, is an important graph product. In this paper, we mainly study the generalized 3-edge-connectivity of $G \circ H$, and get upper and lower bounds of $\lambda_3(G \circ H)$. 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 $k$-choosable if given lists of at least $k$ 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 $\Delta$ is $c\Delta^2$-choosable, for some constant $c$. We prove this result with $c=1$ (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 $k$ is nonrepetitively $O(k^{2})$-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