The convexity spectra of graphs

Let D be a connected oriented graph. A set S⊆V(D) is convex in D if, for every pair of vertices x,y∈S, the vertex set of every x-y geodesic (x-y shortest dipath) and y-x geodesic in D is contained in S. The convexity numbercon(D) of a nontrivial oriented graph D is the maximum cardinality of a proper convex set of D. Let G be a graph. We define that S_C(G)={con(D):D is an orientation of G} and S_SC(G)={con(D):D is a strongly connected orientation of G}. In the paper, we show that, for any n?4, 1?a?n-2, and a≠2, there exists a 2-connected graph G with n vertices such that S_C(G)=S_SC(G)={a,n-1} and there is no connected graph G of order n?3 with S_SC(G)={n-1}. Then, we determine that S_C(K₃)={1,2}, S_C(K₄)={1,3}, S_SC(K₃)=S_SC(K₄)={1}, S_C(K₅)={1,3,4}, S_C(K₆)={1,3,4,5}, S_SC(K₅)=S_SC(K₆)={1,3}, S_C(K_n)={1,3,5,6,…,n-1}, S_SC(K_n)={1,3,5,6,…,n-2} for n?7. Finally, we prove that, for any integers n, m, and k with , 1?k?n-1, and k≠2,4, there exists a strongly connected oriented graph D with n vertices, m edges, and convexity number k.     

On the direct image of intersections in exact homological categories

Given a regular epimorphism f:X?Y in an exact homological category C, and a pair (U,V) of kernel subobjects of X, we show that the quotient (f(U)∩f(V))/f(U∩V) is always abelian. When C is nonpointed, i.e. only exact protomodular, the translation of the previous result is that, given any pair (R,S) of equivalence relations on X, the difference mappingδ:Y/f(R∩S)?Y/(f(R)∩f(S)) has an abelian kernel relation. This last result actually holds true in any exact Mal'cev category. Setting Y=X/T, this result says that the difference mapping determined by the inclusion T∪(R∩S)?(T∪R)∩(T∪S) has an abelian kernel relation, which casts a new light on the congruence distributive property.     

Fixed points on model solvmanifold pairs

In this paper we define model solvmanifold pairs and their diagonal type selfmaps in the tradition of Heath and Keppelmann. We derive an explicit formula for computing the relative Nielsen number N(F;X,A) on these spaces and selfmaps F:(X,A)→(X,A). We find that model solvmanifold pairs often exhibit interesting Schirmer theory, meaning N(F;X,A)>max{N(F),N(F|_A)}.     

The construction of all locally compact extensions

The paper presents one of the ways to construct all the locally compact extensions of a given Tychonoff space T. First, there proved the “local” variant of the Stone-C?ech theorem on “completely regular” Riesz spaces X(T) of continuous bounded functions on T with no unit function, in general, but with a collection of local units. In Theorem 1 it is proved that all the functions from X(T) can be “completely regularly” extended on the largest locally compact extension β_xT. Theorem 3 states, that β_xT are presenting, in fact, all the locally compact extensions of T.     

The eccentric connectivity index of nanotubes and nanotori

Let G be a molecular graph. The eccentric connectivity index ξ^c(G) is defined as ξ^c(G)=∑_u∈V(G)deg_G(u)ε_G(u), where deg_G(u) denotes the degree of vertex u and ε_G(u) is the largest distance between u and any other vertex v of G. In this paper exact formulas for the eccentric connectivity index of TUC₄C₈(S) nanotube and TC₄C₈(S) nanotorus are given.     

Principal topologies and transformation semigroups

For a given set X, the set F(X) of all maps from X to X forms a semigroup under composition. A subsemigroup S of F(X) is said to be saturated if for each x∈X there exists a set Ox⊆X with x∈Ox such that . It is shown that there exists a one-to-one correspondence between principal topologies on X and saturated subsemigroups of F(X). Some properties of principal topologies on X and the corresponding properties of their associated saturated subsemigroups of F(X) are discussed.     

The disconnection number of a graph

The disconnection number d(X) is the least number of points in a connected topological graph X such that removal of d(X) points will disconnect X (Nadler, 1993 [6]). Let Dn denote the set of all homeomorphism classes of topological graphs with disconnection number n. The main result characterizes the members of Dn+1 in terms of four possible operations on members of Dn. In addition, if X and Y are topological graphs and X is a subspace of Y with no endpoints, then d(X)?d(Y) and Y obtains from X with exactly d(Y)−d(X) operations. Some upper and lower bounds on the size of Dn are discussed.The algorithm of the main result has been implemented to construct the classes Dn for n?8, to estimate the size of D₉, and to obtain information on certain subclasses such as non-planar graphs (n?9) and regular graphs (n?10).     

The security number of lexicographic products

A subset S of vertices of a graph G is a secure set if |N [X] ∩ S| ≥ |N [X] ? S| holds for any subset X of S, where N [X] denotes the closed neighborhood of X. The minimum cardinality s(G) of a secure set in G is called the security number of G. We investigate the security number of lexicographic product graphs by defining a new concept of tightly-securable graphs. In particular we derive several exact results for different families of graphs which yield some general results.     

Automatic continuity of biseparating homomorphisms defined between groups of continuous functions

Let C(X,T) be the group of continuous functions of a compact Hausdorff space X to the unit circle of the complex plane T with the pointwise multiplication as the composition law. We investigate how the structure of C(X,T) determines the topology of X. In particular, which group isomorphisms H between the groups C(X,T) and C(Y,T) imply the existence of a continuous map h of Y into X such that H is canonically represented by h. Among other results, it is proved that C(X,T) determines X module a biseparating group isomorphism and, when X is first countable, the automatic continuity and representation as Banach-Stone maps for biseparating group isomorphisms is also obtained.     

Universal factorization property of certain polycyclic groups

Let H be a torsion-free strongly polycyclic (torsion-free virtually polycyclic, resp.) group. Let G be any group with maximal condition. We show that there exists a torsion-free strongly polycyclic (torsion-free virtually polycyclic, resp.) group and an epimorphism such that for any homomorphism ?:G→H, it factors through , i.e., there exists a homomorphism such that . We show that this factorization property cannot be extended to any finitely generated group G. As an application of factorization, we give necessary and sufficient conditions for N(f,g)=R(f,g) to hold for maps f,g:X→Y between closed orientable n-manifolds where π₁(X) has the maximal condition, Y is an infra-solvmanifold, N(f,g) and R(f,g) denote the Nielsen and Reidemeister coincidence numbers, respectively.     

Topologizing homeomorphism groups of rim-compact spaces

Let X be a Tychonoff space, H(X) the group of all self-homeomorphisms of X with the usual composition and the evaluation function. Topologies on H(X) providing continuity of the evaluation function are called admissible. Topologies on H(X) compatible with the group operations are called group topologies. Whenever X is locally compact T₂, there is the minimum among all admissible group topologies on H(X). That can be described simply as a set-open topology, further agreeing with the compact-open topology if X is also locally connected. We show the same result in two essentially different cases of rim-compactness. The former one, where X is rim-compact T₂ and locally connected. The latter one, where X agrees with the rational number space Q equipped with the euclidean topology. In the first case the minimal admissible group topology on H(X) is the closed-open topology determined by all closed sets with compact boundaries contained in some component of X. Moreover, whenever X is also separable metric, it is Polish. In the rational case the minimal admissible group topology on H(Q) is just the closed-open topology. In both cases the minimal admissible group topology on H(X) is closely linked to the Freudenthal compactification of X. The Freudenthal compactification in rim-compactness plays a key role as the one-point compactification does in local compactness. In the rational case we investigate whether the fine or Whitney topology on H(Q) induces an admissible group topology on H(Q) stronger than the closed-open topology.     

Characterization of double domination subdivision number of trees

In a graph G, a vertex dominates itself and its neighbors. A subset S⊆V(G) is a double dominating set of G if S dominates every vertex of G at least twice. The double domination numberdd(G) is the minimum cardinality of a double dominating set of G. The double domination subdivision numbersd_dd(G) is the minimum number of edges that must be subdivided (each edge in G can be subdivided at most once) in order to increase the double domination number. In this paper first we establish upper bounds on the double domination subdivision number for arbitrary graphs in terms of vertex degree. Then we present several different conditions on G which are sufficient to imply that sd_dd(G)?3. We also prove that 1?sd_dd(T)?2 for every tree T, and characterize the trees T for which sd_dd(T)=2.     

Broué's conjecture for non-principal 3-blocks of finite groups

In representation theory of finite groups, there is a well-known and important conjecture due to M. Broué. He conjectures that, for any prime p, if a p-block A of a finite group G has an abelian defect group D, then A and its Brauer correspondent p-block B of N_G(D) are derived equivalent. We demonstrate in this paper that Broué's conjecture holds for two non-principal 3-blocks A with elementary abelian defect group D of order 9 of the O'Nan simple group and the Higman-Sims simple group. Moreover, we determine these two non-principal block algebras over a splitting field of characteristic 3 up to Morita equivalence.     

On the space of subgroups of a compact group II

Let G be a compact group. Let S(G), C(G), N(G) be the spaces of closed subgroups, cosets of closed subgroups, normal closed subgroups (respectively) of G, with the Vietoris topology.Then: (1) S(G) and C(G) are never connected; (2) N(G) is always totally disconnected; (3) C(G) is totally disconnected if and only if G is totally disconnected; and (4) S(G) is totally disconnected if and only if G/Z(G) is totally disconnected.Further: for totally disconnected G (equivalently, profinite G) (5) S(G), C(G) and N(G) are κ-metrisable; (6) S(G), C(G) and N(G) are Dugundji compact if G has small weight; and (7) consequences for field extensions are derived.     

Brill-Gordan loci, transvectants and an analogue of the Foulkes conjecture

The hypersurfaces of degree d in the projective space Pn correspond to points of PN, where . Now assume d=2e is even, and let X(n,d)⊆PN denote the subvariety of two e-fold hyperplanes. We exhibit an upper bound on the Castelnuovo regularity of the ideal of X(n,d), and show that this variety is r-normal for r?2. The latter result is representation-theoretic, and says that a certain GLn+1-equivariant morphism

Sr(S2e(Cn+1))→S₂(Sre(Cn+1))     

Generators and relations for subsemigroups via boundaries in Cayley graphs

Given a finitely generated semigroup S and subsemigroup T of S, we define the notion of the boundary of T in S which, intuitively, describes the position of T inside the left and right Cayley graphs of S. We prove that if S is finitely generated and T has a finite boundary in S then T is finitely generated. We also prove that if S is finitely presented and T has a finite boundary in S then T is finitely presented. Several corollaries and examples are given.     

Above the Glauberman correspondence

Let S be a finite solvable group, and suppose S acts on the finite group N, and they have coprime orders. Then, the celebrated Glauberman correspondence provides a natural bijection from the set IrrS(N) of irreducible characters of N which are invariant under the action of S to the set Irr(CN(S)) of all irreducible characters of the centralizer of S in N. Suppose, further, that the semidirect product SN is a normal subgroup of a finite group G. Let θ∈IrrS(N), and let ψ∈Irr(CN(S)) be its Glauberman correspondent. We prove that there is a bijection with good compatibility properties from the set Irr(G,θ) of the irreducible characters of G above θ to Irr(NG(S),ψ) such that, in the case when S is a p-group for some prime p, it preserves fields of values and Schur indices over Qp, the field of p-adic numbers. Using this result, we also prove a strengthening of the McKay Conjecture for all p-solvable groups.     

Maximum nullity of outerplanar graphs and the path cover number

Let G=(V,E) be a graph with V={1,2,…,n}. Define S(G) as the set of all n×n real-valued symmetric matrices A=[a_ij] with a_ij≠0,i≠j if and only if ij∈E. By M(G) we denote the largest possible nullity of any matrix A∈S(G). The path cover number of a graph G, denoted P(G), is the minimum number of vertex disjoint paths occurring as induced subgraphs of G which cover all the vertices of G.There has been some success with relating the path cover number of a graph to its maximum nullity. Johnson and Duarte [5], have shown that for a tree T,M(T)=P(T). Barioli et al. [2], show that for a unicyclic graph G,M(G)=P(G) or M(G)=P(G)-1. Notice that both families of graphs are outerplanar. We show that for any outerplanar graph G,M(G)?P(G). Further we show that for any partial 2-path G,M(G)=P(G).     

Localization of André-Quillen-Goodwillie towers, and the periodic homology of infinite loopspaces

Let K(n) be the nth Morava K-theory at a prime p, and let T(n) be the telescope of a v_n-self map of a finite complex of type n. In this paper we study the K(n)_*-homology of Ω^∞X, the 0th space of a spectrum X, and many related matters.We give a sampling of our results.Let PX be the free commutative S-algebra generated by X: it is weakly equivalent to the wedge of all the extended powers of X. We construct a natural map

s_n(X):L_T(n)P(X)→L_T(n)Σ^∞(Ω^∞X)₊