Graph Theory Model Research Articles

Two-phase heuristic algorithms for full truckloads multi-depot capacitated vehicle routing problem in carrier collaboration

Collaborative transportation, as an emerging new mode, represents one of the major developing trends of transportation systems. Focusing on the full truckloads multi-depot capacitated vehicle routing problem in carrier collaboration, this paper proposes a mathematical programming model and its corresponding graph theory model, with the objective of minimizing empty vehicle movements. A two-phase greedy algorithm is given to solve practical large-scale problems. In the first phase, a set of directed cycles is created to fulfil the transportation orders. In the second phase, chains that are composed of cycles are generated. Furthermore, a set of local search strategies is put forward to improve the initial results. To evaluate the performance of the proposed algorithms, two lower bounds are developed. Finally, computational experiments on various randomly generated problems are conducted. The results show that the proposed methods are effective and the algorithms can provide reasonable solutions within an acceptable computational time.

Detection of dynamic brain networks modulated by acupuncture using a graph theory model

Neuroimaging studies involving acute acupuncture manipulation have already demonstrated significant modulatory effects on wide limbic/paralimbic nuclei, subcortical gray structures and the neocortical system of the brain. Due to the sustained effect of acupuncture, however, knowledge on the organization of such large-scale cortical networks behind the active needle stimulation phase is lacking. In this study, we originally adopted a network model analysis from graph theory to evaluate the functional connectivity among multiple brain regions during the post-stimulus phase. Evidence from our findings clearly supported the existence of a large organized functional connectivity network related to acupuncture function in the resting brain. More importantly, acupuncture can change such a network into a functional state underlying both pain perception and modulation, which is exhibited by significant changes in the functional connectivity of some brain regions. This analysis may help us to better understand the long-lasting effects of acupuncture on brain function, as well as the potential benefits of clinical treatments.

Graph labellings with variable weights, a survey

Graph labellings form an important graph theory model for the channel assignment problem. An optimum labelling usually depends on one or more parameters that ensure minimum separations between frequencies assigned to nearby transmitters. The study of spans and of the structure of optimum labellings as functions of such parameters has attracted substantial attention from researchers, leading to the introduction of real number graph labellings and λ -graphs. We survey recent results obtained in this area.

Clustering, community partition and disjoint spanning trees

Clustering method is one of the most important tools in statistics. In a graph theory model, clustering is the process of finding all dense subgraphs. A mathematically well-defined measure for graph density is introduced in this article as follows. Let G = ( V , E ) be a graph (or multi-graph) and H be a subgraph of G . The dynamic density of H is the greatest integer k such that min ∀ P {| E ( H / P )|/| V ( H / P )| − 1} > k where the minimum is taken over all possible partitions P of the vertex set of H , and H / P is the graph obtained from H by contracting each part of P into a single vertex. A subgraph H of G is a level- k community if H is a maximal subgraph of G with dynamic density at least k . An algorithm is designed in this paper to detect all level- h communities of an input multi-graph G . The worst-case complexity of this algorithm is upper bounded by O (| V ( G )| 2 h 2 ). This new method is one of few available clustering methods that are mathematically well-defined, supported by rigorous mathematical proof and able to achieve the optimization goal with polynomial complexity. As a byproduct, this algorithm also can be applied for finding edge-disjoint spanning trees of a multi-graph. The worst-case complexity is lower than all known algorithms for multi-graphs.

System mathematic model of epicyclic gear trains

On the basis of analyzing structure character and state character of epicyclical gear trains (EGTs),topological graph is i mproved. It is realized that gear trains is a physics system,and net problem is presented that is a systems graph theory model applied to solve physics character of EGTs.It is found that topological graph accords with Kirchhoff's law.On the basis of analyzing physics and geometry character of topological graph,branch-characteristics matrix and cut-characteristics matrix are presented,the two are applied to create a systems mathematic model,namely loop-equations and cut-equations.A new fundamental theory is presented for structure characteristics and state characteristics of EGTs based on systems research.And it is applied to velocity,torque and bifurcated power characteristics of EGTs with an example.

Power Domination in Product Graphs

The power system monitoring problem asks for as few as possible measurement devices to be put in an electric power system. The problem has a graph theory model involving power dominating sets in graphs. The power domination number $\gamma_P(G)$ of $G$ is the minimum cardinality of a power dominating set. Dorfling and Henning [Discrete Appl. Math., 154 (2006), pp. 1023-1027] determined the power domination number of the Cartesian product of paths. In this paper the power domination number is determined for all direct products of paths except for the odd component of the direct product of two odd paths. For instance, if $n$ is even and $C$ a connected component of $P_m\times P_n$, where $m$ is odd or $m\geq n$, then $\gamma_P(C)=\left\lceil n/4 \right\rceil$. For the strong product we prove that $\gamma_P(P_n \boxtimes P_m) = \max\{\lceil n/3\rceil, \lceil (n+m-2)/4\rceil\}$, unless $3m-n-6 \equiv 4\pmod 8$. The power domination number is also determined for an arbitrary lexicographic product.

A new fuzzy genetic algorithm for the dynamic bi-objective cell formation problem considering passive and active strategies

This paper deals with manufacturing cell formation considering the dynamic behavior of the production system. First, we discuss the importance of taking into account the dynamic aspect of the problem that has been poorly studied in the related literature. We argue that by considering a multi-periodic planning horizon, we can tackle the problem according to two strategies: passive and active. The first strategy consists of maintaining the same composition of machines during the overall planning horizon, while the second allows performing a different composition for each period. We present a graph theory model for the problem. For the passive cell formation problem (PCFP), we prove that it amounts to a cell formation problem in a static system. In order to solve the active cell formation problem (ACFP), we propose a Shortest Path heuristic (SP) and a Genetic Algorithm (GA) based method. When the decision maker wants to choose the most adequate strategy for its environment, we state that the decision problem between active and passive strategies can be solved by solving ACFP. However, the complexity of ACFP justifies the need to control the presented solving approaches. In this situation, we propose a new fuzzy logic enhancement to the GA. The results, using this enhancement, are better than those obtained using the SP method or the GA alone.

A new multimembership clustering method

Clustering method is one of the most important tools in statistics. In a graph theory model, clustering is the process of finding all dense subgraphs. In this paper, a new clustering method is introduced. One of the most significant differences between the new method and other existing methods is that this new method constructs a much smaller hierarchical tree, which clearly highlights meaningful clusters. Another important feature of the new method is the feature of <em> overlapping clustering or multi-membership</em>. The property of multi-membership is a concept that has recently received increased attention in the literature (Palla, Derényi, Farkas and Vicsek, <em> (Nature 2005)</em>; Pereira-Leal, Enright and Ouzounis, <em>(Bioinformatics, 2004)</em>; Futschik and Carlisle, <em> (J. Bioinformatics and Computational Biology 2005)</em>)

Observing local and global properties of metabolic pathways: ‘load points’ and ‘choke points’ in the metabolic networks

The local and global aspects of metabolic network analyses allow us to identify enzymes or reactions that are crucial for the survival of the organism(s), therefore directing us towards the discovery of potential drug targets. We demonstrate a new method ('load points') to rank the enzymes/metabolites in the metabolic network and propose a model to determine and rank the biochemical lethality in metabolic networks (enzymes/metabolites) through 'choke points'. Based on an extended form of the graph theory model of metabolic networks, metabolite structural information was used to calculate the k-shortest paths between metabolites (the presence of more than one competing path between substrate and product). On the basis of these paths and connectivity information, load points were calculated and used to empirically rank the importance of metabolites/enzymes in the metabolic network. The load point analysis emphasizes the role that the biochemical structure of a metabolite, rather than its connectivity (hubs), plays in the conversion pathway. In order to identify potential drug targets (based on the biochemical lethality of metabolic networks), the concept of choke points and load points was used to find enzymes (edges) which uniquely consume or produce a particular metabolite (nodes). A non-pathogenic bacterial strain Bacillus subtilis 168 (lactic acid producing bacteria) and a related pathogenic bacterial strain Bacillus anthracis Sterne (avirulent but toxigenic strain, producing the toxin Anthrax) were selected as model organisms. The choke point strategy was implemented on the pathogen bacterial network of B.anthracis Sterne. Potential drug targets are proposed based on the analysis of the top 10 choke points in the bacterial network. A comparative study between the reported top 10 bacterial choke points and the human metabolic network was performed. Further biological inferences were made on results obtained by performing a homology search against the human genome. The load and choke point modules are introduced in the Pathway Hunter Tool (PHT), the basic version of which is available on http://www.pht.uni-koeln.de.

Faster Algorithms for Frobenius Numbers

The Frobenius problem, also known as the postage-stamp problem or the money-changing problem, is an integer programming problem that seeks nonnegative integer solutions to $ x_1 a_1 +\cdots + x_n a_n =M$, where $ a_i $ and $M$ are positive integers. In particular, the Frobenius number $f(A)$, where $A=\{ a_i \}$, is the largest $M$ so that this equation fails to have a solution. A simple way to compute this number is to transform the problem to a shortest-path problem in a directed weighted graph; then Dijkstra's algorithm can be used. We show how one can use the additional symmetry properties of the graph in question to design algorithms that are very fast. For example, on a standard desktop computer, our methods can handle cases where $n=10$ and $ a_1 = 10^7$. We have two main new methods, one based on breadth-first search and another that uses the number theory and combinatorial structure inherent in the problem to speed up the Dijkstra approach. For both methods we conjecture that the average-case complexity is $O( a_1 \sqrt{n} )$. The previous best method is due to Böcker and Lipták and runs in time $O( a_1 n)$. These algorithms can also be used to solve auxiliary problems such as (1) find a solution to the main equation for a given value of $M$; or (2) eliminate all redundant entries from a basis. We then show how the graph theory model leads to a new upper bound on $f(A)$ that is significantly lower than existing upper bounds. We also present a conjecture, supported by many computations, that the expected value of $f(A)$ is a small constant multiple of $ \left({1\over2} \, n! \, \Pi A\right)^{1/(n-1)} -\Sigma A$.

A natural graph-theory model for partition and kinetic coefficients

A model of six partition coefficients and two metabolic kinetic parameters of a class of halogenated compounds has been performed with the aid of molecular connectivity concepts, which are based on complete graphs and general graphs. The complete graphs are used to encode the core electrons into the main parameter of molecular connectivity theory, the valence delta number, δv. The present model, which is solely based on graph concepts, confirms the central importance of a complete graph conjecture, which is based on an odd number of vertices for the core electrons. Two slightly different algorithms for δv, both centered on this conjecture, compete in deriving an optimal model for the six sets of partition coefficients. An algorithm, the Kp-(p-odd) algorithm, is valid for a model based on a linear combination of four connectivity and pseudoconnectivity basis indices. This linear combination is able to model, in a satisfactory way, all the six sets of partition coefficients. The other algorithm, the Kp-(pp-odd) algorithm, is, instead, able to derive a molecular connectivity term, which is able to adequately model four sets of partition coefficients, and in a less satisfactory way another set of partition coefficients. The two metabolic kinetic constants are, instead, optimally modeled by different basis indices based on the Kp-(pp-odd) algorithm. Underlying are also (i) the importance, in nearly all cases, of the 1χv basis index as best single descriptor, and (ii) the overall improvement the model undergoes when the only cis-compound of the class of chemicals is deleted form the class. Preliminary results show the possibility of improving the model with a δv algorithm, which encodes also the bonded hydrogens.

A graph theory analysis of renal glomerular microvascular networks

A graph theory model and its invariants are used to compare previously published renal glomerular networks of six adult rats, one adult uremic rat, and one newborn rat. Invariants calculated include order, size, cycle rank, eccentricity, root distance, planarity, and vertex degree distribution. These invariants enabled the differentiation of six normal adult glomerular microvascular networks from that of the uremic glomerulus and from that of the normal newborn glomerulus. These invariants might then be used to differentiate between normal and pathological vascular networks. Also proposed are graph theory invariants that might be used to develop a quantitative model for angiogenesis.

Network analysis modeling towards GIS based on object-relation database

This paper compares the differences between the mathematical model in graph theory and GIS network analysis model. Thus it claims that the GIS network analysis model needs to solve. Then this paper introduces the spatial data management methods in object-relation database for GIS and discusses its effects on the network analysis model. Finally it puts forward the GIS network analysis model based on the object-relation database. The structure of the model is introduced in detail and research is done to the internal and external memory data structure of the model. The results show that it performs well in practice.

El problema del conjunto independiente en la selección de horarios de cursos

El proceso de inscripción para alumnos de la Universidad Autónoma Metropolitana tiene como fundamento la libertad de cada alumno de seleccionar las asignaturas que cursará, así como los grupos en los que quedará inscrito. El éxito de este sistema, medido en términos del porcentaje de alumnos que obtienen inscripción en los cursos que seleccionaron, depende en gran medida tanto de las características de la oferta de grupos, relativas principalmente a la cantidad y a la variedad de horarios, como de la posibilidad por parte de los alumnos de hacer una selección adecuada de horarios para los cursos por los que optaron. Una selección adecuada de horarios de cursos es aquella en la que las asignaturas seleccionadas tienen horarios dos a dos compatibles. El problema de selección de horarios consiste en la obtención de una selección de horarios adecuada de cardinalidad máxima. En este trabajo se presentará un modelo de Teoría de Gráficas para este problema así como un algoritmo de solución para el mismo.&#x0D;

Autopoiesis (self-production) in SME networks

Small and medium enterprise networks (SME Networks) are becoming an integral part of the Network Economy. From the ‘industrial districts’ of the Terza Italia to the entrepreneurial clusters of the Silicon Valley, SME's are a significant driving force of economic growth, job creation, disinflation and productivity enhancement in most industrial countries. After decades of research, these local industrial systems are still poorly understood in terms of their sustained processes of innovation, network interaction and competitive adjustments. While there could be some external economies due to agglomeration, division of labor, specialization and lowered transaction costs, differential innovation, interaction and adjustment capabilities are not fully explained by these mechanisms. A theoretical construct of local industrial system is missing. However, no mechanical or graph theory model of network architecture can substitute for what actually makes people in the network interact in order to become technologically innovative and capable of ongoing adjustment to their competitors. Counting the nodes and edges of graphs would be a poor substitute for understanding SME networks as dynamic (‘living’) organic systems they are. In this paper we propose a theoretical construct of network production, renewal and adaptation based on autopoiesis (self-production) of living systems.

Quasi-static resource allocation with interference avoidance for fixed wireless systems

We propose a novel intercell interference management technique, called quasi-static resource allocation with interference avoidance (QRA-IA), for fixed broadband wireless systems with narrow sectors or equipped with fixed or adaptive array antennas. The basic idea of QRA-IA is for every base to periodically turn off each of its beams (sectors) for a certain amount of time. This periodic turn off introduces a predictable nonuniformity in a terminal's performance and therefore permits each terminal to identify a preferred time period for transmission. QRA-IA requires that each terminal sense the interference over time, select the preferred transmission periods, and report these to the base. The resource allocation algorithm at the base can then use this information for scheduling transmissions to the terminals. The base can also use this information for obtaining an appropriate beam-off sequence. A graph theory model is used to show that an acceptable beam-off sequence exists for each base. A simple, distributed, and measurement-based beam-switching algorithm is designed for the system to find these sequences autonomously. We demonstrate via simulations that a system with QRA-IA provides a marked improvement in the packet error rate performance of terminals over the same system without QRA-IA. This improvement translates directly to improved coverage and throughput and also reduces the burden on higher layer protocols to ensure fairness and quality of service.

Microscopic analysis of macroscopic transport properties of single natural fractures using graph theory algorithms

Given an aperture distribution for a single fracture, a graph theory model has been employed to simulate the transport properties of the fracture. In the processes of imbibition and drainage, the connectivity of each phase at a particular capillary pressure can be sought out by the priority‐first search algorithm of graph theory. Then, the flow path in each phase can be identified, and flow equations can be established and solved in an optimal manner. It was observed that in a simulated rough fracture, a few flow “channels” can dominate the permeability of the whole system.

A graph theory model of the semantic structure of attitudes.

The semantic structure underlying the attitudes of pretreatment and posttreatment drug addicts was modeled using a network analysis of free word associations. Measures of graph theoretic properties were used to assess structural differences in the associative networks of the two populations. These measures modeled the information processes of associative networks proposed in the spreading activation theory of semantic processing. As expected based on graph theory, the structure of the associative networks of posttreatment subjects was more dense, less constrained, and more hierarchically organized by the self concept. In a test of the network model, the subjects' evaluations of concepts in the associative network were found to be a function of their evaluations of semantically similar concepts. Although preliminary and limited, the results suggest that graph theory may provide a broad mathematical foundation for diverse models of cognitive systems.

Rationale for using multiple regression analysis with complex interferences.

Non-specificities and interferences may become complex when they involve the analyte as well as other interfering substances. These non-specificities and interferences are known as analyte-dependent and multi-interferent interferences. Multiple regression analysis has proven valuable in analysing this type of interference, but the theoretical foundation for using multiple regression analysis to study the basic mechanisms of interference has not been explicitly demonstrated. Graph theory can depict and model the basic mechanisms of interferences and the possible interactions. The relationship between the analyte, the interferents, and the response of the instrument to these entities can be approximated by a polymial of order three, which includes partial derivatives and cross-terms. The partial derivatives relate to the different interactions found with the graph theory model. Further, the partial derivatives can be associated with the coefficients in the multiple regression analysis when the respective values of the three variables (analyte, interferent one, and interferent two) are multiplied by one another. One can decide to retain or discard the coefficient of a variable, based on the statistical significance of the coefficient. The respective interactions in the graphic model can then be assembled and the framework of the interference mechanism established.

Two metrics in a graph theory modeling of organic chemistry

Two different metrics in the graph theory model of organic chemistry are defined. The chemical distance between two graphs (molecules) taken from the same family of isomeric graphs F pq (where p is a number of vertices and q is a number of edges and loops) is based on the maximal common subgraph. The reaction distance assigned to a transformation G ⇒ G′, where G, G′ ε F pq , is equal to the minimal number of the so-called elementary transformations that are necessary for the transformation of G onto G′. Both the suggested metrics reflect formally many known fundamental chemical rules, in particular, the principle of minimal structural change.