Walk counts, labyrinthicity, and complexity of acyclic and cyclic graphs and molecules

Walks in molecular graphs and their counts for a long time have found applications in theoretical chemistry. These are based on the fact that the (i, j)-entry of the kth power of the adjacency matrix is equal to the number of walks starting at vertex i, ending at vertex j, and having length k. In recent papers (refs 13, 18, 19) the numbers of all walks of length k, called molecular walk counts, mwc(k), and their sum from k = 1 to k = n - 1, called total walk count, twc, were proposed as quantities suitable for QSPR studies and capable of measuring the complexity of organic molecules. We now establish a few general properties of mwc's and twc among which are the linear dependence between the mwc's and linear correlations between the mwc's and twc, the spectral decomposition of mwc's, and various connections between the walk counts and the eigenvalues and eigenvectors of the molecular graph. We also characterize the graphs possessing minimal and maximal walk counts.

Multi-way graph partition by stochastic probe

The number of spanning trees (or complexity) of a graph can be used to measure the reliabilities of network and circuit design. Although many NP hard problems are related to the complexity of graphs, there are explicit formulas for the complexity of some specific graph classes. This paper mainly studies the explicit expression of the complexity of graphs obtained by graph operations. Through classical graph operations, we obtain some operation graphs generated by cycle and complete graph, and get the closed formulas for the complexity in these operation graphs. Compared with Daoud’s results, we obtain some interesting algebraic properties about the complexity of operation graphs.

Graph-structured databases have numerous recent applications including the Semantic Web, biological databases and XML, among many others. In this paper, we study the maintenance problem of a popular structural index, namely bisimulation, of a possibly cyclic data graph. In comparison, previous work mainly focuses on acyclic graphs. In the context of database applications, it is natural to compute minimal bisimulation with merging algorithms. First, we propose a maintenance algorithm for a minimal bisimulation of a cyclic graph, in the style of merging. Second, to prune the computation on non-bisimilar SCCs, we propose a feature-based optimization. The features are designed to be constructed and used more efficiently than bisimulation minimization. Third, we conduct an experimental study that verifies the effectiveness and efficiency of our algorithm. Our features-based optimization pruned 50% (on average) unnecessary bisimulation computation. Our experiment verifies that we extend the current state-of-the-art with a capability on handling cyclic graphs in maintenance of minimal bisimulation.KeywordsSpan TreeMinimum Span TreeData GraphStrongly Connect ComponentCircuit BaseThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

ChemInform is a weekly Abstracting Service, delivering concise information at a glance that was extracted from about 100 leading journals. To access a ChemInform Abstract of an article which was published elsewhere, please select a “Full Text” option. The original article is trackable via the “References” option.

Self-avoiding walks interacting with a surface are a useful model of polymer adsorption and one can choose to weight the walk according to the number of vertices or the number of edges in the surface. We consider a combined model of self-avoiding walks on the d-dimensional hypercubic lattice with weights associated with the numbers of vertices and edges in the surface and investigate its phase diagram. We also derive inequalities relating the critical points in the edge and vertex weighting schemes.

Exact and heuristic algorithms for the weighted total domination problem

Brain Electroencephalography (EEG) classification is widely applied to analyze cerebral diseases in recent years. Unfortunately, invalid/noisy EEGs degrade the diagnosis performance and most previously developed methods ignore the necessity of EEG selection for classification. To this end, this paper proposes a novel maximum weight clique-based EEG selection approach, named mwcEEGs, to map EEG selection to searching maximum similarity-weighted cliques from an improved Fréchet distance-weighted undirected EEG graph simultaneously considering edge weights and vertex weights. Our mwcEEGs improves the classification performance by selecting intra-clique pairwise similar and inter-clique discriminative EEGs with similarity threshold δ. Experimental results demonstrate the algorithm effectiveness compared with the state-of-the-art time series selection algorithms on real-world EEG datasets.

In this paper we propose to generalize Jaccard and related measures, often used as similarity coefficients between two sets. We define Jaccard, Dice-Sorensen and Tversky edge weights on a graph and generalize them to account for vertex weights. We develop an efficient parallel algorithm for computing Jaccard edge and PageRank vertex weights. We highlight that the weights computation can obtain more than 10X speedup on the GPU versus CPU on large realistic data sets. Also, we show that finding a minimum balanced cut for modified weights can be related to minimizing the sum of ratios of the intersection and union of nodes on the boundary of clusters. Finally, we show that the novel weights can improve the quality of the graph clustering by about 15% and 80% for multi-level and spectral graph partitioning and clustering schemes, respectively.

Communication with evidence in the lab

A graph with a vertex set is said to be a prime graph if there exists a bijective mapping , where denotes the number of vertices in , such that for any two adjacent vertices and in have . Tensor Product graph is a way to combine (compose) two graphs into one larger and more complex graph. The result is a new graph that reflects the connection properties of the two original graphs, but in a very specific and more complex way than other graph operations. Therefore, this research aims to determine whether there is prime labeling in the class of graphs resulting from the Tensor Product of the path graph and the cycle graph . The research employed analytical and exploratory methods with a trial-and-error strategy to determine the labeling that possesses a prime property. The results of this study prove that two classes of the Tensor Product graph for , and graph , for are prime graph. This finding expands the results on classes of graphs that admit prime labeling and provides a basis for further research on graph labeling in other graph operations

A weighted, oriented topological structure, denoted by <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">G</tex> and called a flow graph, is associated with a set of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m</tex> equations in <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</tex> variables, denoted by <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">KX = 0</tex> , such that <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">K</tex> is a connection matrix and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">X</tex> a vertex weight matrix of the associated graph. This same set of equations can be written as <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">A_{-v:}^{-} C(A+)'X = 0</tex> where <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">A_{-}^{-v:}</tex> and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">A^{+}</tex> are negative and positive incidence matrices and where <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">C</tex> and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">X</tex> are respectively branch and vertex weight matrices of the graph. By familiar algebraic procedures, an expression for the weight <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">x_p</tex> , of a nonreference vertex of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">G</tex> is obtained as a linear combination of the weights of the reference vertices (vertices with zero negative order) and can be written as <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">x_p = \Sigma_{j=1}^{s} \zeta p{\dot}r_{j}x_{r_j}</tex> . To these algebraic results there correspond topological expressions in terms of subgraphs of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">G</tex> for the coefficients, <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\zeta P{\dot}r_j</tex> . A similar correspondence is obtained between the topological operation of deleting a vertex from the flow graph and the algebraic operation of eliminating a variable from the set of equations. These results are derived from the algebraic equations written in terms of the incidence and weight matrices of the graph. They are similar to those given for the familiar Signal-Flow-Graph, although they are more convient to use, since the topological properties of the flow graph depend only upon the algebraic properties of the set of equations. A flow graph can be drawn directly from an electric network diagram, and the flow-graph properties, used to obtain a solution of the network equations. Examples of this for two types of feedback networks are shown.

A modular strategy for scheduling iterative computations is proposed. Iterative computations are represented using cyclic task graphs that are transformed into acyclic task graphs. These acyclic task graphs are subsequently scheduled using one of the many well known and high quality static scheduling strategies from the literature. Graph unfolding is not employed and the generated schedules therefore require less memory than schedules generated through graph unfolding. Further, the number of iterations does not need to be known at compile time. The paper experimentally quantifies how the task transformation affects the make span of the schedules and the effectiveness of the approach is compared to other methods including a graph unfolding strategy. A new more intuitive graph unfolding formulation is also presented.

Every molecular structure can be regarded as a graph, and each graph can be transformed into characteristic scalar – topological index. There are many, over 1000 topological indices used by chemists, but they are not equally successful in solving specific problems. For the prediction of stability of coordination compounds valence molecular connectivity index of the 3rd order (3χv) proved best. Stability of these compounds was expressed as log K (K stays for stability constant), which is proportional to the Gibbs free energy of reaction between metal (or metal complex) and ligand. Various molecular graphs were checked, from a simple graph of ligand, to the graphs of various metal-ligand complexes (aqua complex, complex with additional bonds etc.) leading to conclusion that chemically sounder graphs generally yield better models. The best results were obtained for the systems of one metal, mesured at the same experimental conditions. However, our graph-theoretical approach enables development of models for the simulateneous prediction of stability constants of few metals.

Let G = (V, E, w) be a directed graph, where w : V ← R is an arbitrary weight function defined on its vertices. The bottleneck weight, or the capacity, of a path is the smallest weight of a vertex on the path. For two vertices u, v the bottleneck weight, or the capacity, from u to v, denoted c(u, v), is the maximum bottleneck weight of a path from u to v. In the All-Pairs Bottleneck Paths (APBP) problem we have to find the bottleneck weights for all ordered pairs of vertices. Our main result is an O(n2.575) time algorithm for the APBP problem. The exponent is derived from the exponent of fast matrix multiplication. Our algorithm is the first sub-cubic algorithm for this problem. Unlike the sub-cubic algorithm for the all-pairs shortest paths (APSP) problem, that only applies to bounded (or relatively small) integer edge or vertex weights, the algorithm presented for APBP problem works for arbitrary large vertex weights.The APBP problem has numerous applications, and several interesting problems that have recently attracted attention can be reduced to it, with no asymptotic loss in the running times of the known algorithms for these problems. Some examples are a result of Vassilevska and Williams [STOC 2006] on finding a triangle of maximum weight, a result of Bender et al. [SODA 2001] on computing least common ancestors in DAGs and a result of Kowaluk and Lingas [ICALP 2005] on finding maximum witnesses for boolean matrix multiplication. Thus, the APBP problem provides a uniform framework for these applications. For some of these problems, we can in fact show that their complexity is equivalent to that of the APBP problem.A slight modification of our algorithm enables us to compute shortest paths of maximum bottleneck weight. Let d(u, v) denote the (unweighted) distance from u to v, and let sc(u, v) denote the maximum bottleneck weight of a path from u to v having length d(u, v). The All-Pairs Bottleneck Shortest Paths (APBSP) problem is to compute sc(u, v) for all ordered pairs of vertices. We present an algorithm for the APBSP problem whose running time is O(n2.86).