Kinds Of Graphs Research Articles

Domatic partitions of computable graphs

Given a graph G, we say that a subset D of the vertex set V is a dominating set if it is near all the vertices, in that every vertex outside of D is adjacent to a vertex in D. A domatic k-partition of G is a partition of V into k dominating sets. In this paper, we will consider issues of computability related to domatic partitions of computable graphs. Our investigation will center on answering two types of questions for the case when k = 3. First, if domatic 3-partitions exist in a computable graph, how complicated can they be? Second, a decision problem: given a graph, how difficult is it to decide whether it has a domatic 3-partition? We will completely classify this decision problem for highly computable graphs, locally finite computable graphs, and computable graphs in general. Specifically, we show the decision problems for these kinds of graphs to be $${\Pi^{0}_{1}}$$ ? 1 0 -, $${\Pi^{0}_{2}}$$ ? 2 0 -, and $${\Sigma^{1}_{1}}$$ Σ 1 1 -complete, respectively.

Choosability with Separation of Complete Multipartite Graphs and Hypergraphs

AbstractFor a hypergraph G and a positive integer s, let be the minimum value of l such that G is L‐colorable from every list L with for each and for all . This parameter was studied by Kratochvíl, Tuza, and Voigt for various kinds of graphs. Using randomized constructions we find the asymptotics of for balanced complete multipartite graphs and for complete k‐partite k‐uniform hypergraphs.

Compact representation of Web graphs with extended functionality

The representation of large subsets of the World Wide Web in the form of a directed graph has been extensively used to analyze structure, behavior, and evolution of those so-called Web graphs. However, interesting Web graphs are very large and their classical representations do not fit into the main memory of typical computers, whereas the required graph algorithms perform inefficiently on secondary memory. Compressed graph representations drastically reduce their space requirements while allowing their efficient navigation in compressed form. While the most basic navigation operation is to retrieve the successors of a node, several important Web graph algorithms require support for extended queries, such as finding the predecessors of a node, checking the presence of a link, or retrieving links between ranges of nodes. Those are seldom supported by compressed graph representations.This paper presents the k2-tree, a novel Web graph representation based on a compact tree structure that takes advantage of large empty areas of the adjacency matrix of the graph. The representation not only retrieves successors and predecessors in symmetric fashion, but also it is particularly efficient to check for specific links between nodes, or between ranges of nodes, or to list the links between ranges. Compared to the best representations in the literature supporting successor and predecessor queries, our technique offers the least space usage (1–3 bits per link) while supporting fast navigation to predecessors and successors (2–8μs per neighbor retrieved) and sharply outperforming the others on the extended queries. The representation is also of general interest and can be used to compress other kinds of graphs and data structures.

A Design of Graph Generator of Laser Shake Mirror Based on C8051F020 Microprocessor

A design method of graph generator of laser shake mirror based on C8051F020 microprocessor is put forward and introduces hardware structure and circuit scheme. Through the modularized design method of software, kinds of graph can be outputed. Through test, this system can achieve its functions.

Visual displays: analytical study and applications to graphs and real data

Principal axis methods such as principal component analysis (PCA) and correspondence analysis (CA) are useful for identifying structures in data through interesting planar graphic displays. However, some kinds of data sets can be dealt alternatively with PCA or CA. This paper focuses on methods, such as PCA and CA, and on visual displays. Our aim is to illustrate the implications for a potential user of selecting either method, and its advantages and disadvantages, from an applied point of view. This is a matter covered broadly in textbooks and elsewhere considering theoretical arguments. Our purpose is to contribute to the comparison between these methods, over the same data set, in order to illustrate them for the practitioner. In the first part of this paper we present a novel analytical study of a binary matrix associated with a non-oriented axis-symmetric graph and show that CA outperforms standardized PCA for the reconstitution and visualization of such kind of graphs. In the second part we present a case using real data dealing with the distribution of employees in different economic sectors for the countries of the European Union, analyzed by means of standardized PCA and two-way CA, in order to see the differences between the two methods in practice.

Finding maximal homogeneous clique sets

Many datasets can be encoded as graphs with sets of labels associated with the vertices. We consider this kind of graphs and we propose to look for patterns called maximal homogeneous clique sets, where such a pattern is a subgraph that is structured in several large cliques and where all vertices share enough labels. We present an algorithm based on graph enumeration to compute all patterns satisfying user-defined constraints on the number of separated cliques, on the size of these cliques, and on the number of labels shared by all the vertices. Our approach is tested on real datasets based on a social network of scientific collaborations and on a biological network of protein---protein interactions. The experiments show that the patterns are useful to exhibit subgraphs organized in several core modules of interactions. Performances are reported on real data and also on synthetic ones, showing that the approach can be applied on different kinds of large datasets.

An algorithm based on graphs for solving a fair division problem

In this paper we propose an algorithm for solving a fair division problem: We want to find a distribution of a set of divisible items among a set of agents such that the agent who is getting the lowest amount obtains his maximum possible value. Even more, according to the found distribution, each agent obtains the same amount, and this is as high as possible. We model this situation as a linear programming problem and we use its dual problem for solving it. For doing this, we associate a bipartite graph with each set of dual variables. The algorithm presented uses the intrinsic relations in this kind of graphs for searching the optimal solution for the primal and dual problems. Finally, we show the convergence of the proposed algorithm.

A Class of L-Cospectral Graphs are Isomorphic

The spectral graph theory is a theory in which graphs are studied by means of their adjacency eigenvalues, Laplacian eigenvalues and signless Laplacian eigenvalues of associated graph matrix. What kind of graphs are determined by their spectra is a difficult problem in graph theory. Let 1 2 3 { , ( , , )} n q H C T n n n be a graph of order n obtained by identifying a vertex of cycle q C with the vertex of degree three of T-shape three 1 2 3 ( , , ) T n n n . In this paper, we will prove that for q even, all the L-cospectral graphs 1 2 3 { , ( , , )} n q H C T n n n are isomorphic.

Routing-efficient CDS construction in Disk-Containment Graphs

In wireless networks, Connected Dominating Sets (CDSs) are widely used as virtual backbones for communications. On one hand, reducing the backbone size will reduce the maintenance overhead. So how to minimize the CDS size is a critical issue. On the other hand, when evaluating the performance of a wireless network, the hop distance between two communication nodes, which reflect the energy consumption and response delay, is of great importance. Hence how to minimize the routing cost is also a key problem for constructing the network virtual backbone. In this paper, we study the problem of constructing applicable CDS in wireless networks in terms of size and routing cost. We formulate a wireless network as a Disk-Containment Graph (DCG), which is a generalization of the Unit-Disk Graph (UDG), and we develop an efficient algorithm to construct CDS in such kind of graphs. The algorithm contains two parts and is flexible to balance the performance between the two metrics. We also analyze the algorithm theoretically. It is shown that our algorithm has provable performance in minimizing the CDS size and reducing the communication distance for routing.

CO-MAXIMAL IDEAL GRAPHS OF COMMUTATIVE RINGS

In this paper, a new kind of graph on a commutative ring R with identity, namely the co-maximal ideal graph is defined and studied. We use [Formula: see text] to denote this graph, with its vertices the proper ideals of R which are not contained in the Jacobson radical of R, and two vertices I1 and I2 are adjacent if and only if I1 + I2 = R. We show some properties of this graph. For example, this graph is a simple, connected graph with diameter less than or equal to three, and both the clique number and the chromatic number of the graph are equal to the number of maximal ideals of the ring R.

Super-connectivity of Kronecker products of split graphs, powers of cycles, powers of paths and complete graphs

The Kronecker product of two connected graphs G 1 , G 2 , denoted by G 1 × G 2 , is the graph with vertex set V ( G 1 × G 2 ) = V ( G 1 ) × V ( G 2 ) and edge set E ( G 1 × G 2 ) = { ( u 1 , v 1 ) ( u 2 , v 2 ) : u 1 u 2 ∈ E ( G 1 ) , v 1 v 2 ∈ E ( G 2 ) } . The k th power G k of G is the graph with vertex set V ( G ) such that two distinct vertices are adjacent in G k if and only if their distance apart in G is at most k . A connected graph G is called super- κ if every minimal vertex cut of G is the set of neighbors of some vertex in G . In this note, we consider the super-connectivity of the Kronecker products of several kinds of graphs and complete graphs. We show that D = G × K m is super- κ for m ≥ 3 and G satisfying one of the following conditions: (1) G is a non-complete split graph with | C | ≥ 5 ; (2) G is a power graph of a path P n k such that n ≥ 2 k ; (3) G is a power graph of a cycle C n r such that n ≥ m and n ≥ 2 r + 1 .

The competition number of a graph and the dimension of its hole space

The competition graph of a digraph D is a (simple undirected) graph which has the same vertex set as D and has an edge between x and y if and only if there exists a vertex v in D such that ( x , v ) and ( y , v ) are arcs of D . For any graph G , G together with sufficiently many isolated vertices is the competition graph of some acyclic digraph. The competition number k ( G ) of G is the smallest number of such isolated vertices. In general, it is hard to compute the competition number k ( G ) for a graph G and it has been one of the important research problems in the study of competition graphs to characterize a graph by its competition number. Recently, the relationship between the competition number and the number of holes of a graph has been studied. A hole of a graph is a cycle of length at least 4 as an induced subgraph. In this paper, we conjecture that the dimension of the hole space of a graph is not smaller than the competition number of the graph. We verify this conjecture for various kinds of graphs and show that our conjectured inequality is indeed an equality for connected triangle-free graphs.

Embedding Distributions and Chebyshev Polynomials

The history of genus distributions began with J. Gross et al. in 1980s. Since then, a lot of study has given to this parameter, and the explicit formulas are obtained for various kinds of graphs. In this paper, we find a new usage of Chebyshev polynomials in the study of genus distribution, using the overlap matrix, we obtain homogeneous recurrence relation for rank distribution polynomial, which can be solved in terms of Chebyshev polynomials of the second kind. The method here can find explicit formula for embedding distribution of some other graphs. As an application, the well known genus distributions of closed-end ladders and cobblestone paths (Furst et al. in J Combin Ser B 46:22–36, 1989) are derived. The explicit formula for non-orientable embedding distributions of closed-end ladders and cobblestone paths are also obtained.

Solar coronal magnetic field diagnostics through polarimetric forward modelling of the Hanle effect

Context. Progress in the solution to some of the most outstanding open problems of solar physics, such as coronal heating, solar wind acceleration, the generation and triggering of explosive events like flares and CMEs, hinges on the provision of a more stringent estimate of the solar magnetic field coordinates. Aims. We seek a way to infer the magnetic field of the solar atmosphere. A very promising way of doing this is by using the Hanle effect in resonance scattering in the Lα line of the solar atmosphere. Methods. By forward modelling the known scattering effects in the presence of magnetic fields, i.e. rotation of the plane of polarisation and depolarisation of the linear polarisation parameters, and by comparing them to observations, one could potentially uncover the magnetic morphology and restrict its intensity range. We simulate the effects of simple dipole configurations along the coordinate axes and analyse the outcome through two kinds of graphs (i.e. the difference in angle of the plane of linear polarisation with respect to the field-free case, and the relative depolarisation). Results. The graphs are either symmetric, anti-symmetric or asymmetric with respect to the (y,z) plane. This is explained by invoking two symmetry operations and taking into account that the magnetic field is a pseudovector. We also show the polarimetric effects of active regions and use them pairwise with the magnetic field due to dipoles to analyse the polarimetric signatures of magnetic field line loops. Inspired by the famous TRACE image, we finally show what one could expect from polarimetry performed on the region of the solar atmosphere displayed in the image. Conclusions. By combining the two complementary remote sensing techniques, i.e. the Zeeman and the Hanle effect, in all thinkable ways with tracers such as the images revealed by TRACE, SOHO, STEREO, etc., we hope one day to be able to infer the solar magnetic field coordinates. Much theoretical and instrumental work still lies ahead, however.

Metabolic Networks: Beyond the Graph

Drugs are devised to enter into the metabolism of an organism in order to produce a desired effect. From the chemical point of view, cellular metabolism is constituted by a complex network of reactions transforming metabolites one in each other. Knowledge on the structure of this network could help to develop novel methods for drug design, and to comprehend the root of known unexpected side effects. Many large-scale studies on the structure of metabolic networks have been developed following models based on different kinds of graphs as the fundamental image of the reaction network. Graphs models, however, comport wrong assumptions regarding the structure of reaction networks that may lead into wrong conclusions if they are not taken into account. In this article we critically review some graph-theoretical approaches to the analysis of centrality, vulnerability and modularity of metabolic networks, analyzing their limitations in estimating these key network properties, consider some proposals explicit or implicitly based on directed hypergraphs regarding their ability to overcome these issues, and review some recent implementation improvements that make the application of these models in increasingly large networks a viable option.

Directed graphs and combinatorial properties of completely regular semigroups

In this paper, we investigate the divisibility graphs and power graphs of completely regular semigroups. We give the structures of these two kinds of graphs and describe a combinatorial property of completely regular semigroups defined in terms of divisibility graphs and power graphs, respectively.

Studies on Dimultigraph and Prograph Based Applications of Graph Theory in Computer Science

Graph Theory has gained an impulsion in the past few years in rapacious dimensions. Day by day with rapid advancements of the technology, its demand for proper utilization of resources is increasing at a drastic rate. Simultaneously, it has emerged as one of the most powerful tools for the illustration and solution of the problems that are basically chronological in nature. And many seemingly diverse problems in computer science have been worked out with the help of this thought. A special approach (DPBA model) for different kinds of graphs and their applications in computer science is proposed in this paper. It introduces dimultigraph by considering digraph. The approaches of dimultigraph are enumerated in simple manner. Introduction of prograph in performance evaluation in the model reflects its freshness as well as widely acceptance both theoretically and in implementation.

Sharp bounds for the generalized connectivity [formula omitted

Let G be a nontrivial connected graph of order n and let k be an integer with 2 ≤ k ≤ n . For a set S of k vertices of G , let κ ( S ) denote the maximum number ℓ of edge-disjoint trees T 1 , T 2 , … , T ℓ in G such that V ( T i ) ∩ V ( T j ) = S for every pair i , j of distinct integers with 1 ≤ i , j ≤ ℓ . A collection { T 1 , T 2 , … , T ℓ } of trees in G with this property is called an internally disjoint set of trees connecting S . Chartrand et al. generalized the concept of connectivity as follows: The k - connectivity, denoted by κ k ( G ) , of G is defined by κ k ( G ) = min { κ ( S ) } , where the minimum is taken over all k -subsets S of V ( G ) . Thus κ 2 ( G ) = κ ( G ) , where κ ( G ) is the connectivity of G . For general k , the investigation of κ k ( G ) is very difficult. We therefore focus on the investigation on κ 3 ( G ) in this paper. We study the relation between the connectivity and the 3 -connectivity of a graph. First we give sharp upper and lower bounds of κ 3 ( G ) for general graphs G , and construct two kinds of graphs which attain the upper and lower bound, respectively. We then show that if G is a connected planar graph, then κ ( G ) − 1 ≤ κ 3 ( G ) ≤ κ ( G ) , and give some classes of graphs which attain the bounds. In the end we give an algorithm to determine κ 3 ( G ) for general graphs G . This algorithm runs in a polynomial time for graphs with a fixed value of connectivity, which implies that the problem of determining κ 3 ( G ) for graphs with a small minimum degree or connectivity can be solved in polynomial time, in particular, the problem whether κ ( G ) = κ 3 ( G ) for a planar graph G can be solved in polynomial time.

Managing Data-Objects in Dynamically Reconfigurable Caches

The widening gap between processor and memory speeds makes cache an important issue in the computer system design. Compared with work set of programs, cache resource is often rare. Therefore, it is very important for a computer system to use cache efficiently. Toward a dynamically reconfigurable cache proposed recently, DOOC (Data-Object Oriented Cache), this paper proposes a quantitative framework for analyzing the cache requirement of data-objects, which includes cache capacity, block size, associativity and coherence protocol. And a kind of graph coloring algorithm dealing with the competition between data-objects in the DOOC is proposed as well. Finally, we apply our approaches to the compiler management of DOOC. We test our approaches on both a single-core platform and a four-core platform. Compared with the traditional caches, the DOOC in both platforms achieves an average reduction of 44.98% and 49.69% in miss rate respectively. And its performance is very close to the ideal optimal cache.

Stata Tip 78: Going Gray Gracefully: Highlighting Subsets and Downplaying Substrates

In graphics, as in life, going gray is often forced upon us, yet it is also occasionally a deliberate choice. Journals may enforce publication of your graphs in black and white whenever full-blown color is prohibited, or else prohibitively expensive. Even when allowed, color may prove problematic for various and quite different reasons, ranging from physiology and psychology to sociology and aesthetics. For example, many people are red-green color-blind, while the spectral or rainbow sequence from red to violet is not in fact perceived as a monotonic scale. Wilkinson (2005) and Ware (2008) give good introductions to the use of color in visualization. Fortner and Meyer (1997) give a more detailed discussion. Brewer (2005) gives many specific suggestions on color schemes, which are as appropriate for statistical graphics as they are for cartography. Choosing differing shades of gray is also worth consideration for positive reasons. Expressing qualitative contrasts just with gray can be both effective and attractive. A previous Stata tip (Cox 2005) showed how distinct values on an ordered scale could be shown separately on scatterplots by markers of different gray-scale colors. This tip expands the theme with further examples. Naturally, black and white themselves qualify as extreme gray shades and may work very well. Here I will emphasize the use of intermediate shades. Let me underline that the examples here use the sj scheme. See (G) schemes intro for more information. Consider the highlighting of subsets. You may want to show the distribution of a sub- set with the distribution of the complete set as context. Unwin, Theus, and Hofmann (2006); Chen, Hardle, and Unwin (2008); and Myatt and Johnson (2009) include sev- eral examples of this device for various kinds of graphs. On a histogram with a frequency scale, this can be done by laying down the dis- tribution of the complete set first and plotting the distribution of the subset on top. Display of the subset can never occlude the display of the complete set, because at most all the observations in any bin belong to the subset.