The number of spanning trees of multi-star and multi-complete related graphs based on cycles or paths

图G的生成树的数目是图论中一个重要的不变量，出现在大量应用中。其最显著的应用之一是在网络可靠性方面：我们通常用图来模拟网络中的连通模式，这个网络中所有的站点之间可以互相通讯，意味着图G中必须包含一个生成树。因此，图G具有的生成树的数目能够揭示相应的网络组合配置的复杂性，是评价该图（网络）可靠性的重要指标之一，最大化生成树的数目是加强网络可靠性的一个途径。因此，确定图类的生成树数目的计数公式是很有意义的。 图的生成树计数问题已经被国内外学者所广泛研究。我们的研究兴趣是：确定图G在完全图中的补图，即G的相关图的生成树数目的计数公式。到目前为止，图类的许多不同类型已经被研究，例如当G是1条边、m条...The number of spanning trees of a graph G is an important quantity in graph theory, and appears in a number of applications. One of its most notable applications is in the field of network reliability: we often use graphs to model connection patterns in a network, intercommunication between all nodes of the network implies that the graph must contain a spanning tree. Thus, the number of spanning t...学位：理学硕士院系专业：数学科学学院数学与应用数学系_应用数学学号：X200617001

Minimal instances for toric code ground states

A decade ago Kitaev's toric code model established the new paradigm of topological quantum computation. Due to remarkable theoretical and experimental progress, the quantum simulation of such complex many-body systems is now within the realms of possibility. Here we consider the question, to which extent the ground states of small toric code systems differ from LU-equivalent graph states. We argue that simplistic (though experimentally attractive) setups obliterate the differences between the toric code and equivalent graph states; hence we search for the smallest setups on the square- and triangular lattice, such that the quasi-locality of the toric code hamiltonian becomes a distinctive feature. To this end, a purely geometric procedure to transform a given toric code setup into an LC-equivalent graph state is derived. In combination with an algorithmic computation of LC-equivalent graph states, we find the smallest non-trivial setup on the square lattice to contain 5 plaquettes and 16 qubits; on the triangular lattice the number of plaquettes and qubits is reduced to 4 and 9, respectively.Comment: 14 pages, 11 figure

A Relation-Based Page Rank Algorithm for Semantic Web Search Engines

With the tremendous growth of information available to end users through the Web, search engines come to play ever a more critical role. Nevertheless, because of their general-purpose approach, it is always less uncommon that obtained result sets provide a burden of useless pages. The next-generation Web architecture, represented by the Semantic Web, provides the layered architecture possibly allowing overcoming this limitation. Several search engines have been proposed, which allow increasing information retrieval accuracy by exploiting a key content of Semantic Web resources, that is, relations. However, in order to rank results, most of the existing solutions need to work on the whole annotated knowledge base. In this paper, we propose a relation-based page rank algorithm to be used in conjunction with Semantic Web search engines that simply relies on information that could be extracted from user queries and on annotated resources. Relevance is measured as the probability that a retrieved resource actually contains those relations whose existence was assumed by the user at the time of query definitio

Spanning trees short or small

We study the problem of finding small trees. Classical network design problems are considered with the additional constraint that only a specified number $k$ of nodes are required to be connected in the solution. A prototypical example is the $k$MST problem in which we require a tree of minimum weight spanning at least $k$ nodes in an edge-weighted graph. We show that the $k$MST problem is NP-hard even for points in the Euclidean plane. We provide approximation algorithms with performance ratio $2\sqrt{k}$ for the general edge-weighted case and $O(k^{1/4})$ for the case of points in the plane. Polynomial-time exact solutions are also presented for the class of decomposable graphs which includes trees, series-parallel graphs, and bounded bandwidth graphs, and for points on the boundary of a convex region in the Euclidean plane. We also investigate the problem of finding short trees, and more generally, that of finding networks with minimum diameter. A simple technique is used to provide a polynomial-time solution for finding $k$-trees of minimum diameter. We identify easy and hard problems arising in finding short networks using a framework due to T. C. Hu.Comment: 27 page