The signless Laplacian spectral radius of bicyclic graphs with prescribed degree sequences

In this paper, we characterize all extremal connected bicyclic graphs with the largest signless Laplacian spectral radius in the set of all connected bicyclic graphs with prescribed degree sequences. Moreover, the signless Laplacian majorization theorem is proved to be true for connected bicyclic graphs. As corollaries, all extremal connected bicyclic graphs having the largest signless Laplacian spectral radius are obtained in the set of all connected bicyclic graphs of order n (resp. all connected bicyclic graphs with a specified number of pendant vertices, and all connected bicyclic graphs with given maximum degree).     

Connected Domatic Number in Planar Graphs

A dominating set in a graph G is a connected dominating set of G if it induces a connected subgraph of G. The connected domatic number of G is the maximum number of pairwise disjoint, connected dominating sets in V(G). We establish a sharp lower bound on the number of edges in a connected graph with a given order and given connected domatic number. We also show that a planar graph has connected domatic number at most 4 and give a characterization of planar graphs having connected domatic number 3.     

Pseudo definably connected definable sets

In o‐minimal structures, every cell is definably connected and every definable set is a finite union of its definably connected components. In this note, we introduce pseudo definably connected definable sets in weakly o‐minimal structures having strong cell decomposition, and prove that every strong cell in those structures is pseudo definably connected. It follows that every definable set can be written as a finite union of its pseudo definably connected components. We also show that the projections of pseudo definably connected definable sets are pseudo definably connected. Finally, we compare pseudo definable connectedness with (recently introduced) weak definable connectedness of definable sets in weakly o‐minimal structures.     

Local Connectedness Made Uniform

The uniformly locally connected reflection for a locally connected uniform frame is constructed. Applications of this construction to the theory of locally connected completely regular frames are given. One such application is that if a completely regular frame is locally connected and pseudocompact then every compactification of it is locally connected.     

Optimality conditions and duality for semi-infinite programming involving B-arcwise connected functions

In this paper, a class of functions called B-arcwise connected (BCN) and strictly B-arcwise connected (STBCN) functions are introduced by relaxing definitions of arcwise connected function (CN) and B-vex function. The differential properties of B-arcwise connected function (BCN) are studied. Their two extreme properties are proved. The necessary and sufficient optimality conditions are obtained for the nondifferentiable nonlinear semi-infinite programming involving B-arcwise connected (BCN) and strictly B-arcwise connected (STBCN) functions. Mond-Weir type duality results have also been established.     

A Class of Spaces Containing all Connected and all Locally Connected Spaces

A simultaneous generalization of connectedness and local connectedness, called sum connectedness, is introduced. The category of sum connected spaces forms the smallest co-reflective subcategory of TOP containing all connected spaces. A product theorem, analogous to that for locally connected spaces, is proved for sum connected spaces and a necessary and sufficient condition for the STONE -?ECH compactification of a TYCHONOFF space to be sum connected is obtained.     

相通连续Domain的若干特征定理

首先引入了相通连续Domain的概念,利用主理想及连通闭集刻画了相通连续Domain; 其次考察了相通完备偏序集的定向完备化,得到了一些好的结果.     

On perfectly two-edge connected graphs

This paper studies the graphs for which the 2-edge connected spanning subgraph polytope is completely described by the trivial inequalities and the so-called cut inequalities. These graphs are called perfectly 2-edge connected. The class of perfectly 2-edge connected graphs contains for instance the class of series-parallel graphs. We introduce a new class of perfectly 2-edge connected graphs. We discuss some structural properties of graphs which are (minimally with respect to some reduction operations) nonperfectly 2-edge connected. Using this we give sufficient conditions for a graph to be perfectly 2-edge connected.     

Berechnung des Kernwiderstandes kreisringförmiger Platten

Summary From measurements of two transfer impedances the resistivity of arbitrarily shaped discs may be determined for simply connected, but not for multiply connected regions. As an example of a doubly connected region the transfer impedance of a circular ring is computed. All doubly connected regions may be mapped conformally onto a circular ring.     

Graph-like continua, augmenting arcs, and Menger’s theorem

We show that an adaptation of the augmenting path method for graphs proves Menger’s Theorem for wide classes of topological spaces. For example, it holds for locally compact, locally connected, metric spaces, as already known. The method lends itself particularly well to another class of spaces, namely the locally arcwise connected, hereditarily locally connected, metric spaces. Finally, it applies to every space where every point can be separated from every closed set not containing it by a finite set, in particular to every subspace of the Freudenthal compactification of a locally finite, connected graph. While closed subsets of such a space behave nicely in that they are compact and locally connected (and therefore locally arcwise connected), the general subspaces do not: They may be connected without being arcwise connected. Nevertheless, they satisfy Menger’s Theorem. This work was carried out while Antoine Vella was a Marie Curie Fellow at the Technical University of Denmark, as part of the research project TOPGRAPHS (Contract MEIF-CT-2005-009922), under the supervision of Carsten Thomassen.     

2-Filteredness and The Point of Every Galois Topos

A connected locally connected topos is a Galois topos if the Galois objects generate the topos. We show that the full subcategory of Galois objects in any connected locally connected topos is an inversely 2-filtered 2-category, and as an application of the construction of 2-filtered bi-limits of topoi, we show that every Galois topos has a point.     

Every connected,locally connected graph is upper embeddable

In this Note it is proved that every connected, locally connected graph is upper embeddable. Moreover, a lower bound for the maximum genus of the square of a connected graph is given.     

序列连通空间

序列连通性具有许多相似于连通的性质．本文讨论了拓扑空间的序列连通性，并给出了序列连通空间的刻面及其性质．     

Induction of Geometric Actions

We prove that, in some situations, an induced action from a normal subgroup preserves a geometric structure. Combined with known geometric rigidity results, this result implies certain rigidity statements concerning the full diffeomorphism group of a manifold. It also provides many examples of actions on Lorentz manifolds. Combining these with a small number of well-known actions, we get the full list of connected, simply connected Lie groups admitting a locally faithful, orbit nonproper action by isometries of a connected Lorentz manifold. We give an example of a connected nilpotent Lie group with no complicated action on a Lorentz manifold. We show that, if a connected Lie group has a normal closed subgroup isomorphic to a (two-dimensional) cylinder, then it admits a locally faithful, orbit nonproper action by isometries of a connected Lorentz manifold.     

Der Verband der zusammenhängenden Untergruppen einer kommutativen algebraischen Gruppe

We classify the connected commutative algebraic groups having as lattice of connected closed subgroups a projective space and determine the coordinate fields of these projective spaces. Furthermore we determine the not necessarily commutative connected algebraic groups with a distributive or boolean lattice of connected closed subgroups.

Received: 3 December 2004     

A Characterization of the Interval Function of a (Finite or Infinite) Connected Graph

By the interval function of a finite connected graph we mean the interval function in the sense of H. M. Mulder. This function is very important for studying properties of a finite connected graph which depend on the distance between vertices. The interval function of a finite connected graph was characterized by the present author. The interval function of an infinite connected graph can be defined similarly to that of a finite one. In the present paper we give a characterization of the interval function of each connected graph.     

On spanning subgraphs of a connected bridgeless graph and their application to DT-graphs

The graphs considered are connected and bridgeless. For such graphs the existence of two types of connected spanning subgraphs is proved. Applying these results to a connected bridgeless DT-graph (i.e., every line is incident to a point of degree two), G, one obtains the existence of specific Hamiltonian cycles and Hamiltonian paths in G². In addition it is proved that the square of a connected bridgeless DT-graph is Hamiltonian connected.     

SUBGROUP DISTORTIONS IN NILPOTENT GROUPS

The explicit formula for the distortion function of a connected Lie subgroup in a connected simply connected nilpotent Lie group is obtained. In particular, we prove that a function f: N → R can be realized (up to equivalence) as the distortion function of a connected Lie subgroup in a connected simply connected nilpotent Lie group if and only if f ～ n^r for some nonnegative r ∈ Q. Considering lattices in Lie groups, we establish the analogous results for finitely generated nilpotent groups.     

Second-order optimality conditions for cone-subarcwise connected set-valued optimization problems

The concept of a cone subarcwise connected set-valued map is introduced. Several examples are given to illustrate that the cone subarcwise connected set-valued map is a proper generalization of the cone arcwise connected set-valued map, as well as the arcwise connected set is a proper generalization of the convex set, respectively. Then, by virtue of the generalized second-order contingent epiderivative, second-order necessary optimality conditions are established for a point pair to be a local global proper efficient element of set-valued optimization problems. When objective function is cone subarcwise connected, a second-order sufficient optimality condition is also obtained for a point pair to be a global proper efficient element of set-valued optimization problems.     

A characterization of weakly four‐connected graphs

A graph G = (V, E) is called weakly four‐connected if G is 4‐edge‐connected and G ? x is 2‐edge‐connected for all x ∈ V. We give sufficient conditions for the existence of ‘splittable’ vertices of degree four in weakly four‐connected graphs. By using these results we prove that every minimally weakly four‐connected graph on at least four vertices contains at least three ‘splittable’ vertices of degree four, which gives rise to an inductive construction of weakly four‐connected graphs. Our results can also be applied in the problem of finding 2‐connected orientations of graphs. © 2006 Wiley Periodicals, Inc. J Graph Theory 52: 217–229, 2006