Let R be a non-commutative ring. The commuting graph of R denoted by (R), is a graph with vertex set R \ Z(R), and two distinct vertices a and b are adjacent if ab = ba. In this paper we investigate some properties of (R), whenever R is a finite semisimple ring. For any finite field F, we obtain minimum degree, maximum degree and clique number of (M n (F )). Also it is shown that for any two finite semisimple rings R and S, if (R) (S), then there are commutative semisimple rings R 1 and S 1 and semisimple ring T such that R T × R 1 , S T × S 1 and |R 1 | = |S 1 |.
Using techniques of non-abelian harmonic analysis, we construct an explicit,
non-zero cyclic derivation on the Fourier algebra of the real $ax+b$ group. In
particular this provides the first proof that this algebra is not weakly
amenable. Using the structure theory of Lie groups, we deduce that the Fourier
algebras of connected, semisimple Lie groups also support non-zero, cyclic
derivations and are likewise not weakly amenable. Our results complement
earlier work of Johnson (JLMS, 1994), Plymen (unpublished note) and
Forrest--Samei--Spronk (IUMJ 2009). As an additional illustration of our
techniques, we construct an explicit, non-zero cyclic derivation on the Fourier
algebra of the reduced Heisenberg group, providing the first example of a
connected nilpotent group whose Fourier algebra is not weakly amenable.Comment: v4: AMS-LaTeX, 26 pages. Final version, to appear in JFA. Includes an
authors' correction added at proof stag
Abstract. Rajchman measures of locally compact Abelian groups are studied for almost a century now, and they play an important role in the study of trigonometric series. Eymard's influential work allowed generalizing these measures to the case of non-Abelian locally compact groups G. The Rajchman algebra of G, which we denote by B 0 (G), is the set of all elements of the Fourier-Stieltjes algebra that vanish at infinity.In the present article, we characterize the locally compact groups that have amenable Rajchman algebras. We show that B 0 (G) is amenable if and only if G is compact and almost Abelian. On the other extreme, we present many examples of locally compact groups, such as non-compact Abelian groups and infinite solvable groups, for which B 0 (G) fails to even have an approximate identity.
Abstract. Consider a random graph process where vertices are chosen from the interval [0,1], and edges are chosen independently at random, but so that, for a given vertex x, the probability that there is an edge to a vertex y decreases as the distance between x and y increases. We call this a random graph with a linear embedding.We define a new graph parameter Γ * , which aims to measure the similarity of the graph to an instance of a random graph with a linear embedding. For a graph G, Γ * (G) = 0 if and only if G is a unit interval graph, and thus a deterministic example of a graph with a linear embedding.We show that the behaviour of Γ * is consistent with the notion of convergence as defined in the theory of dense graph limits. In this theory, graph sequences converge to a symmetric, measurable function on [0, 1] 2 . We define an operator Γ which applies to graph limits, and which assumes the value zero precisely for graph limits that have a linear embedding. We show that, if a graph sequence {Gn} converges to a function w, then {Γ * (Gn)} converges as well. Moreover, there exists a function w * arbitrarily close to w under the box distance, so that limn→∞ Γ * (Gn) is arbitrarily close to Γ(w * ).
This paper proves that if G is a graph (parallel edges allowed) of maximum degree 3, then 0 c (G) 11=3 provided that G does not contain H 1 or H 2 as a subgraph, where H 1 and H 2 are obtained by subdividing one edge of K 3 2 (the graph with three parallel edges between two vertices) and K 4 , respectively. As
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