Spanning trees with variable degree bounds

In this paper, we introduce and study a generalization of the degree constrained minimum spanning tree problem where we may install one of several available transmission systems (each with a different cost value) in each edge. The degree of the endnodes of each edge depends on the system installed on the edge. We also discuss a particular case that arises in the design of wireless mesh networks (in this variant the degree of the endnodes of each edge depend on the transmission system installed on it as well as on the length of the edge). We propose three classes of models using different sets of variables and compare from a theoretical perspective as well as from a computational point of view, the models and the corresponding linear programming relaxations. The computational results show that some of the proposed models are able to solve to optimality instances with 100 nodes and different scenarios.     

传递子集的性质及其在符号动力系统中的应用

讨论了传递子集的一些性质,并且应用这些性质研究符号动力系统的弱混合子集和传递子集之间的关系,给出了符号动力系统的传递子集是弱混合子集的一个充分条件.     

Relations,models and a memetic approach for three degree-dependent spanning tree problems

In this paper we take into account three different spanning tree problems with degree-dependent objective functions. The main application of these problems is in the field of optical network design. In particular, we propose the classical Minimum Leaves Spanning Tree problem as a relevant problem in this field and show its relations with the Minimum Branch Vertices and the Minimum Degree Sum Problems. We present a unified memetic algorithm for the three problems and show its effectiveness on a wide range of test instances.     

A COUNTER-EXAMPLE CONCERNING THE ANALYTIC RADON-NIKODYM PROPERTY

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The number of spanning trees of an infinite family of outerplanar,small-world and self-similar graphs

In this paper we give an exact analytical expression for the number of spanning trees of an infinite family of outerplanar, small-world and self-similar graphs. This number is an important graph invariant related to different topological and dynamic properties of the graph, such as its reliability, synchronization capability and diffusion properties. The calculation of the number of spanning trees is a demanding and difficult task, in particular for large graphs, and thus there is much interest in obtaining closed expressions for relevant infinite graph families. We have also calculated the spanning tree entropy of the graphs which we have compared with those for graphs with the same average degree.     

The number of triangular islands on a triangular grid

The aim of the present paper is to carry on the research of Czédli in determining the maximum number of rectangular islands on a rectangular grid. We estimate the maximum of the number of triangular islands on a triangular grid.      

广义度量S-KKM映射的性质及其对变分不等式的应用

引入了超S-γ-广义拟凸（凹）函数,建立了广义度量S-KKM映射与超S-γ-广义拟凸（凹）函数的关系.作为应用,获得了超凸度量空间中的新的KyFan极大极小不等式和鞍点定理.     

Degree-bounded minimum spanning trees

Given n points in the Euclidean plane, the degree-δ minimum spanning tree (MST) problem asks for a spanning tree of minimum weight in which the degree of each vertex is at most δ. The problem is NP-hard for 2≤δ≤3, while the NP-hardness of the problem is open for δ=4. The problem is polynomial-time solvable when δ=5. By presenting an improved approximation analysis for Chan’s degree-4 MST algorithm [T. Chan, Euclidean bounded-degree spanning tree ratios, Discrete & Computational Geometry 32 (2004) 177-194], we show that, for any arbitrary collection of points in the Euclidean plane, there always exists a degree-4 spanning tree of weight at most times the weight of an MST.     

Bounded-degree spanning tree problems: models and new algorithms

Given a connected graph G, a vertex v of G is said to be a branch vertex if its degree is greater than 2. We consider two problems arising in the context of optical networks: