Qualitative uncertainty principles for groups with finite dimensional irreducible representations

Let G be a locally compact group of type I and its dual space. Roughly speaking, qualitative uncertainty principles state that the concentration of a nonzero integrable function on G and of its operator-valued Fourier transform on is limited. Such principles have been established for locally compact abelian groups and for compact groups. In this paper we prove generalizations to the considerably larger class of groups with finite dimensional irreducible representations.     

Quasi-convex density and determining subgroups of compact abelian groups

For an abelian topological group G, let denote the dual group of all continuous characters endowed with the compact open topology. Given a closed subset X of an infinite compact abelian group G such that w(X)<w(G), and an open neighborhood U of 0 in T, we show that . (Here, w(G) denotes the weight of G.) A subgroup D of G determines G if the map defined by r(χ)=χ?D for , is an isomorphism between and . We prove that

     

Extension of Fourier algebra homomorphisms to duals of algebras of uniformly continuous functionals

For a locally compact group G, let XG be one of the following introverted subspaces of VN(G): , the C^∗-algebra of uniformly continuous functionals on A(G); , the space of weakly almost periodic functionals on A(G); or , the C^∗-algebra generated by the left regular representation on the measure algebra of G. We discuss the extension of homomorphisms of (reduced) Fourier-Stieltjes algebras on G and H to cb-norm preserving, weak^∗-weak^∗-continuous homomorphisms of into , where (XG,XH) is one of the pairs , , or . When G is amenable, these extensions are characterized in terms of piecewise affine maps.     

Metric aspects of noncommutative homogeneous spaces

For a closed cocompact subgroup Γ of a locally compact group G, given a compact abelian subgroup K of G and a homomorphism satisfying certain conditions, Landstad and Raeburn constructed equivariant noncommutative deformations of the homogeneous space G/Γ, generalizing Rieffel's construction of quantum Heisenberg manifolds. We show that when G is a Lie group and G/Γ is connected, given any norm on the Lie algebra of G, the seminorm on induced by the derivation map of the canonical G-action defines a compact quantum metric. Furthermore, it is shown that this compact quantum metric space depends on ρ continuously, with respect to quantum Gromov-Hausdorff distances.     

Pseudocompact group topologies with no infinite compact subsets

We show that every Abelian group satisfying a mild cardinal inequality admits a pseudocompact group topology from which all countable subgroups inherit the maximal totally bounded topology (we say that such a topology satisfies property ).Every pseudocompact Abelian group G with cardinality |G|≤2^2c satisfies this inequality and therefore admits a pseudocompact group topology with property . Under the Singular Cardinal Hypothesis (SCH) this criterion can be combined with an analysis of the algebraic structure of pseudocompact groups to prove that every pseudocompact Abelian group admits a pseudocompact group topology with property .We also observe that pseudocompact Abelian groups with property contain no infinite compact subsets and are examples of Pontryagin reflexive precompact groups that are not compact.     

Directed 3-cycle decompositions of complete directed graphs with quadratic leaves

Let G be a vertex-disjoint union of directed cycles in the complete directed graph D_t, let |E(G)| be the number of directed edges of G and suppose or if t=5, and if t=6. It is proved in this paper that for each positive integer t, there exist -decompositions for D_t−G if and only if .     

On signed cycle domination in graphs

Let G=(V,E) be a graph, a function f:E→{−1,1} is said to be an signed cycle dominating function (SCDF) of G if ∑_e∈E(C)f(e)≥1 holds for any induced cycle C of G. The signed cycle domination number of G is defined as is an SCDF of G}. In this paper, we obtain bounds on , characterize all connected graphs G with , and determine the exact value of for some special classes of graphs G. In addition, we pose some open problems and conjectures.     

Periodicity of Adams operations on the Green ring of a finite group

The Adams operations and on the Green ring of a group G over a field K arise from the study of the exterior powers and symmetric powers of KG-modules. When G is finite and K has prime characteristic p we show that and are periodic in n if and only if the Sylow p-subgroups of G are cyclic. In the case where G is a cyclic p-group we find the minimum periods and use recent work of Symonds to express in terms of .     

On conjugacy classes in metaplectic groups

Let E be a finite dimensional symplectic space over a local field of characteristic zero. We show that for every element in the metaplectic double cover of the symplectic group Sp(E), and are conjugate by an element of GSp(E) with similitude −1.     

On wreathed lexicographic products of graphs

This paper proves a necessary and sufficient condition for the endomorphism monoid of a lexicographic product G[H] of graphs G,H to be the wreath product of the monoids and . The paper also gives respective necessary and sufficient conditions for specialized cases such as for unretractive or triangle-free graphs G.     

On the Brauer group of a semisimple algebraic group

Let k be a field with algebraic closure , G a semisimple algebraic k-group, and a maximal torus with character group X(T). Denote Λ the abstract weight lattice of the roots system of G, and by and the n-torsion subgroup of the Brauer group of k and G, respectively. We prove that if chark does not divide n and n is prime to the order of Λ/X(T) then the natural homomorphism is an isomorphism.     

Some relations between rank, chromatic number and energy of graphs

The energy of a graph G, denoted by E(G), is defined as the sum of the absolute values of all eigenvalues of G. Let G be a graph of order n and be the rank of the adjacency matrix of G. In this paper we characterize all graphs with . Among other results we show that apart from a few families of graphs, , where n is the number of vertices of G, and χ(G) are the complement and the chromatic number of G, respectively. Moreover some new lower bounds for E(G) in terms of are given.     

Game colouring of the square of graphs

This paper studies the game chromatic number and game colouring number of the square of graphs. In particular, we prove that if G is a forest of maximum degree Δ≥9, then , and there are forests G with . It is also proved that for an outerplanar graph G of maximum degree Δ, , and for a planar graph G of maximum degree Δ, .     

Facet-inducing web and antiweb inequalities for the graph coloring polytope

For a graph G and its complement , we define the graph coloring polytope P(G) to be the convex hull of the incidence vectors of star partitions of . We examine inequalities whose support graphs are webs and antiwebs appearing as induced subgraphs in G. We show that for an antiweb in G the corresponding inequality is facet-inducing for P(G) if and only if is critical with respect to vertex colorings. An analogous result is also proved for the web inequalities.     

Note on the energy of regular graphs

For a simple graph G, the energy E(G) is defined as the sum of the absolute values of all the eigenvalues of its adjacency matrix A(G). Let n,m, respectively, be the number of vertices and edges of G. One well-known inequality is that , where λ₁ is the spectral radius. If G is k-regular, we have . Denote . Balakrishnan [R. Balakrishnan, The energy of a graph, Linear Algebra Appl. 387 (2004) 287-295] proved that for each ?>0, there exist infinitely many n for each of which there exists a k-regular graph G of order n with k<n-1 and , and proposed an open problem that, given a positive integer n?3, and ?>0, does there exist a k-regular graph G of order n such that . In this paper, we show that for each ?>0, there exist infinitely many such n that . Moreover, we construct another class of simpler graphs which also supports the first assertion that .     

Generalization of transitive fraternal augmentations for directed graphs and its applications

Let be a directed graph. A transitive fraternal augmentation of is a directed graph with the same vertex set, including all the arcs of and such that for any vertices x,y,z,

1.

if and then or (fraternity);

2.

if and then (transitivity).

In this paper, we explore some generalization of the transitive fraternal augmentations for directed graphs and its applications. In particular, we show that the 2-coloring number col₂(G)≤O(∇₁(G)∇₀(G)²), where ∇_k(G) (k≥0) denotes the greatest reduced average density with depth k of a graph G; we give a constructive proof that ∇_k(G) bounds the distance (k+1)-coloring number col_k+1(G) with a function f(∇_k(G)). On the other hand, ∇_k(G)≤(col_2k+1(G))^2k+1. We also show that an inductive generalization of transitive fraternal augmentations can be used to study nonrepetitive colorings of graphs.     

Covering line graphs with equivalence relations

An equivalence graph is a disjoint union of cliques, and the equivalence number of a graph G is the minimum number of equivalence subgraphs needed to cover the edges of G. We consider the equivalence number of a line graph, giving improved upper and lower bounds: . This disproves a recent conjecture that is at most three for triangle-free G; indeed it can be arbitrarily large.To bound we bound the closely related invariant σ(G), which is the minimum number of orientations of G such that for any two edges e,f incident to some vertex v, both e and f are oriented out of v in some orientation. When G is triangle-free, . We prove that even when G is triangle-free, it is NP-complete to decide whether or not σ(G)≤3.     

On certain cohomological invariants of groups

Let R be a left and right ℵ₀-Noetherian ring. We show that if all projective left and all projective right R-modules have finite injective dimension, then all injective left and all injective right R-modules have finite projective dimension. Using this result, we prove that the invariants and , which were introduced by Gedrich and Gruenberg (1987) [15], are equal for any group G. As an application of the latter equality, we show that a group G is finite if and only if , where is the generalized cohomological dimension of groups introduced by Ikenaga (1984) [21].     

Maximum bipartite subgraphs of cubic triangle-free planar graphs

Thomassen recently proved, using the Tutte cycle technique, that if G is a 3-connected cubic triangle-free planar graph then G contains a bipartite subgraph with at least edges, improving the previously known lower bound . We extend Thomassen’s technique and further improve this lower bound to .     

On the circumference of a graph and its complement

Let G be a graph of order n and circumference c(G). Let be the complement of G. We prove that and show sharpness of this bound.