46427487
The Clar Covering Polynomial of Hexagonal Systems I
Abstract In this paper the Clar covering polynomial of a hexagonal system is introduced. In fact it is a kind of F polynomial [4] of a graph, and can be calculated by recurrence relations. We show that the number of aromatic sextets (in a Clar formula), the number of Clar formulas, the number of Kekule structures and the first Herndon number for any Kekulean hexagonal system can be easily obtained by its Clar covering polynomial. In addition, we give some theorems to calculate the Clar covering polynomial of a hexagonal system. As examples we finally derive the explicit expressions of the Clar covering polynomials for some small hexagonal systems and several types of catacondensed hexagonal systems. A relation between the resonance energy and the Clar covering polynomial of a hexagonal system is considered in the next paper.
41201857
A Note on the Number of Perfect Matchings of Bipartite Graphs
Abstract Let G be a bipartite graph with 2n vertices, A its adjacency matrix and K the number of perfect matchings. For plane bipartite graphs each interior face of which is surrounded by a circuit of length 4s + 2, s ϵ {1, 2, …}, an elegant formula, i.e. det A = (−1)nK2, had been rigorously proved by Cvetkovic et al. (1982). For general bipartite graphs, this note contains a necessary and sufficient condition for the above relation to hold. A fast algorithm to check if a plane bipartite graph has such a relation is given.
29766053
Construction for bicritical graphs and k-extendable bipartite graphs
A graph G is said to be bicritical if G-u-v has a perfect matching for every choice of a pair of points u and v. Bicritical graphs play a central role in decomposition theory of elementary graphs with respect to perfect matchings. As Plummer pointed out many times, the structure of bicritical graphs is far from completely understood. This paper presents a concise structure characterization on bicritical graphs in terms of factor-critical graphs and transversals of hypergraphs. A connected graph G with at least 2k+2 points is said to be k-extendable if it contains a matching of k lines and every such matching is contained in a perfect matching. A structure characterization for k-extendable bipartite graphs is given in a recursive way. Furthermore, this paper presents an O(mn) algorithm for determining the extendability of a bipartite graph G, the maximum integer k such that G is k-extendable, where n is the number of points and m is the number of lines in G.
17799789
Cell rotation graphs of strongly connected orientations of plane graphs with an application
The cell rotation graph D(G) on the strongly connected orientations of a 2-edge-connected plane graph G is defined. It is shown that D(G) is a directed forest and every component is an in-tree with one root; if T is a component of D(G), the reversions of all orientations in T induce a component of D(G), denoted by T-, thus (T,T-) is called a pair of in-trees of D(G); G is Eulerian if and only if D(G) has an odd number of components (all Eulerian orientations of G induce the same component of D(G)); the width and height of T are equal to that of T-, respectively. Further it is shown that the pair of directed tree structures on the perfect matchings of a plane elementary bipartite graph G coincide with a pair of in-trees of D(G). Accordingly, such a pair of in-trees on the perfect matchings of any plane bipartite graph have the same width and height.
6097933
The Clar covering polynomial of hexagonal systems III
Abstract The Clar covering polynomial is a recently proposed concept of hexagonal systems by which some important topological indices such as perfect matching count, Clar number, first Herndon number, etc., can be easily obtained. In this paper we establish a relationship between the Clar covering polynomial and sextet polynomial. A lower bound of the Clar number and some properties of coefficients of the Clar covering polynomial are thus deduced. It is mentioned that the summation of coefficients of Clar covering polynomial can be used to calculate the number of perfect matchings of a certain kind of polyominoes relating to crystal physics.
207679368
A Distributive Lattice on the Set of Perfect Matchings of a Plane Bipartite Graph
Let G be a plane bipartite graph and M(G) the set of perfect matchings of G. The Z-transformation graph of G is defined as a graph on M(G): M,M′∈M(G) are joined by an edge if and only if they differ only in one cycle that is the boundary of an inner face of G. A property that a certain orientation of the Z-transformation graph of G is acyclic implies a partially ordered relation on M(G). An equivalent definition of the poset M(G) is discussed in detail. If G is elementary, the following main results are obtained in this article: the poset M(G) is a finite distributive lattice, and its Hasse diagram is isomorphic to the Z-transformation digraph of G. Further, a distributive lattice structure is established on the set of perfect matchings of any plane bipartite graph.
13233829
6 DOF long-range precision tracking system
A novel design of a 6 DOF tracking system intended for large-scale dynamic robot measurement is presented. By integrating data from three-laser tracking systems and an inertial measurement unit, the system provides high frequency, precision position and attitude data. The drifting convergence problem of the inertial measurement method is corrected by using the laser tracking systems. And the integration of laser tracking systems and inertial measurement unit also increase the overall precision of the combined measurement system. Even if the measurements of laser tracker are intermittent and are obtained at a low frequency, the experimental result indicates the method proposed in this paper is able to reach an accuracy of a few hundred micron. Finally, real experiments performed with a parallel robot are presented and analyzed.
2843129
Full Bayesian network classifiers
The structure of a Bayesian network (BN) encodes variable independence. Learning the structure of a BN, however, is typically of high computational complexity. In this paper, we explore and represent variable independence in learning conditional probability tables (CPTs), instead of in learning structure. A full Bayesian network is used as the structure and a decision tree is learned for each CPT. The resulting model is called full Bayesian network classifiers (FBCs). In learning an FBC, learning the decision trees for CPTs captures essentially both variable independence and context-specific independence. We present a novel, efficient decision tree learning, which is also effective in the context of FBC learning. In our experiments, the FBC learning algorithm demonstrates better performance in both classification and ranking compared with other state-of-the-art learning algorithms. In addition, its reduced effort on structure learning makes its time complexity quite low as well.
17695045
A Multi-Agent Approach for Peer-to-Peer Based Information Retrieval System
This paper develops and analyzes distributed search techniques for use in a peer-to-peer (P2P) network-based Information Retrieval (IR) system. In the absence of a centralized mediator with global knowledge that directs requests to appropriate agents, agents must cooperate to forward the queries among themselves so as to find appropriate agents, and return and merge the results in order to fulfill the information retrieval task in a distributed environment. In our approach, the agent society is connected through an agent-view structure maintained by each agent. Initially, the agent-view structures are formed by agents connecting to each other randomly. However, we show that such an approach can be significantly enhanced by dynamically reorganizing the underlying agent-view topology and deploying contextsensitive distributed search algorithms. Experimental results indicate that appropriate organizational structures and distributed search mechanisms can have a positive influence on system performance.
147783
Availability in BitTorrent Systems
In this paper, we investigate the problem of highly available, massive-scale file distribution in the Internet. To this end, we conduct a large-scale measurement study of BitTorrent, a popular class of systems that use swarms of actively downloading peers to assist each other in file distribution. The first generation of BitTorrent systems used a central tracker to enable coordination among peers, resulting in low availability due to the tracker's single point of failure. Our study analyzes the prevalence and impact of two recent trends to improve BitTorrent availability: (i) use of multiple trackers, and (ii) use of Distributed Hash Tables (DHTs), both of which also help to balance load better. The study considered more than 1,400 trackers and 24,000 DHT nodes (extracted from about 20,000 torrents) over a period of two months. We find that both trends improve availability, but for different and somewhat unexpected reasons. Our findings include: (i) multiple trackers improve availability, but the improvement largely comes from the choice of a single highly available tracker, (ii) such improvement is reduced by the presence of correlated failures, (iii) multiple trackers can significantly reduce the connectivity of the overlay formed by peers, (iv) the DHT improves information availability, but induces a higher response latency to peer queries.
3748150
Mesh blending
A new method for smoothly connecting different patches on triangle meshes with arbitrary connectivity, called mesh blending, is presented. A major feature of mesh blending is to move vertices of the blending region to a virtual blending surface by choosing an appropriate parameterization of those vertices. Once blending is completed, the parameterization optimization is performed to perfect the final meshes. Combining mesh blending with multiresolution techniques, an effective blending technique for meshes is obtained. Our method has several advantages: (1) the user can intuitively control the blending result using different blending radii, (2) the shape of cross-section curves can be adjusted to flexibly design complex models, and (3) the resulting mesh has the same connectivity as the original mesh. In this paper, some examples about smoothing, sharpening, and mesh editing show the efficiency of the method.