Composite Graph Research Articles

Harper-type lower bounds and the bandwidths of the compositions of graphs

We give a general Harper-type lower bound for the bandwidth of a graph which is a common generalization of several known results. As applications we get a lower bound for the bandwidth of the composition of two graphs. By using this we determine the bandwidths of some composition graphs such as ( P r × P s )[ H], ( P r × C s )[ H] (2 r ≠ s), ( C r × C s )[ H] (6 ⩽ 2 r ⩽ s), etc., for any graph H. Interestingly, the bandwidths of these graphs have nothing to do with the structure of H in general.

On hereditary properties of composition graphs

The composition graph of a family of n+1 disjoint graphs fHi : 0 i ng is the graph H obtained by substituting the n vertices of H0 respectively by the graphs H1; H2; :::; Hn. If H has some hereditary property P , then necessarily all its factors enjoy the same property. For some sort of graphs it is sucien t that all factors fHi : 0 i ng have a certain common P to endow H with this P . For instance, it is known that the composition graph of a family of perfect graphs is also a perfect graph (B. Bollobas, 1978), and the composition graph of a family of comparability graphs is a comparability graph as well (M.C. Golumbic, 1980). In this paper we show that the composition graph of a family of co-graphs (i.e., P4-free graphs), is also a co-graph, whereas for 1-perfect graphs (i.e., P4-free and C4-free graphs) and for threshold graphs (i.e., P4-free, C4-free and 2K2-free graphs), the corresponding factors fHi : 0 i ng have to be equipped with some special structure.

On the genus of joins and compositions of graphs

We will use a surgical technique to imbed the composition graph G[ H] when G has minimum degree 2 and H has even order. This imbedding is shown to be minimal when G is triangle-free. Along the way, we construct genus imbeddings of the join G + H when both factors have even order, G is empty or 1-regular, and G has order at least twice that of H. Variants and applications to specific graphs, such as complete tripartite graphs, are given.

Differences in problem solving by nonscience majors in introductory chemistry on paired algorithmic-conceptual problems

The purpose of this investigation was to identify and describe the differences in the methods used by faculty teaching introductory chemistry and students enrolled in an introductory chemistry course at the university level to solve paired algorithmic and conceptual problems. Of the 180 students involved, the problem-solving schemas of 20 selected students and 2 professors were evaluated using a graphical method to dissect their think-aloud interviews into episodes indicative of solutions to paired problems on density, stoichiometry, bonding, and gas laws. The interviewed students were classified into four different problem-solving categories (i.e., high algorithmic/high conceptual, high algorithmic/low conceptual, low algorithmic/high conceptual, and low algorithmic/low conceptual), and composite graphs of their problem-solving schemas were compared to those representative of members of the faculty experts' category. Results of these comparisons indicated that as the students' ability to solve both algorithmic and conceptual problems improved, less time and fewer transitions between episodes of the problem-solving schemas were required to complete the problems. Regardless of the students' problem-solving ability, algorithmic-mode problems always required more time and a greater number of transitions for completion than did the paired conceptual-mode problems. However, regardless of topic, all students more frequently correctly solved the algorithmic-mode problems than the corresponding paired conceptual-mode problems. © 1997 John Wiley & Sons, Inc. J Res Sci Teach 34: 905–923, 1997.

Rapid Automated Determination of Shallow Velocity–Depth Structure using First Breaks and the Generalised Reciprocal Method

The generalised reciprocal method (GRM) is a convenient and substantial enhancement to the capability of the refraction method in dealing with irregular refractor interfaces with lateral changes in refractor velocity. Processing approaches have been developed which implement a fast automated GRM-based approach and are capable of handling multifold refraction data such as that from long seismic reflection survey lines. A method is proposed for determining the resolution achievable with the velocity analysis technique as a function of the detector separation. This provides a means of optimising the resolution with spatially under-sampled data for most shallow exploration applications.There are three major stages in this approach: the recognition through stacking coherency of individual layers within the multifold data, and the production of a single composite travel-time graph; the determination of the seismic velocities within each refractor using an automatic curve recognition technique, using the velocity analysis function to obtain the optimum geophone migration separation and to locate the lateral velocity variations within a refractor; and depth inversion in which a modified depth conversion factor is proposed to control the trade off between the vertical resolution of velocity layers and changes in layer thickness. This factor reduces spurious details in inverted depths layers and is stable when a layer appears or disappears in the section.The approach has been tested successfully on both synthetic data and real data from the Kalgoorlie deep seismic reflection traverse. Results show that application of the techniques achieves better resolution than conventional manual interpretation, and provides a fast and direct inversion that appears to discount spurious detail.

B-rep object description from multiple range views

We present a system which computes an integrated description of an object from multiple range images. The object description is in the form of B-rep (boundary representation), which has not been achieved by the computer vision community. To do so, we emphasize the inherent difficulties and ambiguities in the low to mid level vision, and present novel techniques of resolving them. In this system, each view of the object is represented as an attributed graph, where nodes correspond to the surfaces (vertices) and links represent the relationship between surfaces. The main issue in surface extraction is contour closure, which is formulated as a dynamic network. The underlying principle for this network is weak smoothness and geometric cohesion, and is modeled as the interaction between long and short term variables. Long term variables represent the initial boundary grouping computed from the low level surface features, and short term variables represent the competing hypotheses that cooperate with the long term variables. The matching problem involves matching visible surfaces and vertices, and provides the necessary basis for volumetric reconstruction from multiple views. The matching strategy is a two step process, where in each step uses the Hopfield network. At each step, we specify a set of local, adjacency and global constraints, and define an appropriate energy function to be minimized. At the first level of this hierarchy, surface patches are matched and the rigidity transformation is computed. At the second level, the mapping is refined by matching the corresponding vertices, and the transformation is verified. The multiple-view reconstruction consists of two steps. First, we build a composite graph that contains the bounding surfaces and their corresponding attributes, and then intersect these surfaces so that the edges and vertices corresponding to the B-rep description are identified. We present results on objects with planar, as well as quadratically-curved, surfaces.

The Szeged and the Wiener index of graphs

The Szeged index Sz is a recently introduced graph invariant, having applications in chemistry. In this paper, a formula for the Szeged index of Cartesian product graphs is obtained and some other composite graphs are considered. We also prove that for all connected graphs, Sz is greater than or equal to the sum of distances between all vertices. A conjecture concerning the maximum value of Sz is put forward.

A Comparison of Perceived Musical Tension and Aesthetic Response

This study used the Continuous Response Digital Interface (CRDI) to measure musicians' and non-musicians' perceptions of ongoing "aesthetic response" in the first movement of Haydn's Symphony No. 104, in an attempt to determine if musical "tension" might relate to the aesthetic experience. Subjects (musicians n = 43, non-musicians n = 40) indicated their aesthetic responsiveness using the CRDI. Analysis focuses on composite graphs of mean group responses and comparisons to results of a previous study (Madsen and Fredrickson, 1993) which replicated the work of Frede V. Neilsen entitled "Oplevelse af Musikalsk Spending (The Experience of Musical Tension)." Results of this study indicated some co-variation between perceived tension graphs of subject responses from the previous study and recorded aesthetic response, primarily in the musicians. Overall, tension responses were found to be more variable than the aesthetic response, with the latter being consistently positive and somewhat higher. Non-musicians' responses were consistently more positive than musicians' responses throughout, which conforms to the findings in previous studies using the CRDI with these populations.

On the sum of all distances in composite graphs

Let G 1 × G 2, G 1 + G 2, G 1[ G 2], G 1 o G 2 and G1{ G 2} be the product, join, composition, corona and cluster, respectively, of the graphs G 1 and G 2. We compute the sum of distances between all pairs of vertices in these composite graphs.

Neutron and X-ray effects on small intestine summarized by using a mathematical model or paradigm

The responses of intestinal tissues to ionizing radiation can be described by comparing irradiated cell populations qualitatively or quantitatively with corresponding controls. This paper describes quantitative data obtained from resin-embedded sections of neutron-irradiated mouse small intestine at different times after treatment. Information is collected by counting cells or structures present per complete circumference. The data are assessed by using standard statistical tests, which show that early mitotic arrest precedes changes in goblet, absorptive, endocrine and stromal cells and a decrease in crypt numbers. The data can also produce ratios of irradiated: control figures for cells or structural elements. These ratios, along with tissue area measurements, can be used to summarize the structural damage as a composite graph and table, including a total figure, known as the Morphological Index. This is used to quantify the temporal response of the wall as a whole and to compare the effects of different qualities of radiation, here X-ray and cyclotron-produced neutron radiations. It is possible that such analysis can be used predictively along with other reference data to identify the treatment, dose and time required to produce observed tissue damage.

Derived thermodynamic design data for rankine power cycle systems operating on R22

Theoretical Rankine power cycle efficiencies ν R and the pressure ( PR) have been presented for rankine power cycles operating on R22. These values are listed in tabular form for temperature drops of 5–75°C and for boiler temperatures 40–85°C in 5°C increments. Composite graph showing the relationship between ν R, T BO, ( PR) and temperature drop ( T BO - T CO) illustrates the feasible operating range for R22 power cycle systems. The derived thermodynamic data can be used for the rapid preliminary design of the Rankine power cycle systems operating on R22.

Derived thermodynamic design data for rankine power cycle systems operating on R142b

Theoretical Rankine power cycle efficiencies ν R and the ( PR) have been presented for Rankine power cycles operating on R142b. These values are listed in tabular form for temperature drops of 5–75°C and for boiler temperatures 25–125°C in 5°C increments. A composite graph showing the relationship between ν R, T BO, ( PR) and temperature drop ( T BO − T CO) illustrates the feasible operating range for R142b power cycle systems. The derived thermodynamic data can be used for the rapid preliminary design of the Rankine power cycle systems operating on R142b.

Derived thermodynamic design data for rankine power cycle systems operating on R502

Theoretical Rankine power cycle efficiencies ν R and the pressure ( PR) have been presented for Rankine power cycles operating on R502. These values are listed in tabular form for temperature drops of 5–75°C and for boiler temperatures 35–80°C in 5°C increments. A composite graph showing the relationship between ν R, T BO, ( PR) and temperature drop ( T BO - T CO illustrates the feasible operating range for R502 power cycle systems. The derived thermodynamic data can be used for the rapid preliminary design of Rankine power cycle systems operating on R502.

Topological formulas for subnetwork extraction theorems and multiterminal feedback networks

The concepts of modified graphs of a composite graph with respect to two vertexpairs and a hyperedge-decomposition are introduced, respectively. By applying them and the directed hypergraph theory, the topological formulas for the parameter-extraction theorem and subnetwork-extraction theorems are derived, and then the topological formulas for multiterminal feedback networks are presented. In these formulas the parameters of the feedback subnetwork are separated from that of the fundamental subnetwork, so that it is convenient to find out the effect of the feedback parameters. Furthermore, since one network is decomposed into two smaller subnetworks, the computing time complexity and space complexity can be reduced.

Generalizing the duality theorem of graph embeddings

Let ω: Ḡ→G be a wrapped covering of graphs and let G be enbedded in a surface S. It is known that if the embedding of G in S is orientable, then this embedding may be lifted to an orientable embedding of Ḡ in S̄. In this paper, we generalize this result by showing that the embedding of G in S may be lifted to an embedding of Ḡ in S̄, where S̄ has the same orientability characteristic as that of S. Furthermore, ω extends to a branched covering B:S̄→S of surfaces and the restriction of B to the dual of Ḡ is a wrapped covering onto the dual of G. These results were applied to obtain genus embeddings of composition graphs G[K̄n] from embeddings of the graph G.

72.16 Composite functions and graph sketching

72.16 Composite functions and graph sketching

Primary subgraph method for linear active network analysis

This paper introduces the concept of the primary subgraph of a composite graph associated with a linear active network, and presents a new and efficient topological method, called the primary subgraph method, for linear active network analysis. This method is an improvement and unification of the complete tree method and directed tree method. It thoroughly solves the sign evaluation problem and cancellation term problem.

Automatic gas chromatographic evaluation of the performance of membranes used for permeation processes

The separation of liquid mixtures by permeation through membranes frm a liquid stream to a gas phase is influenced by the permeability and the selectivity of the polymeric layer. These propeties depend on various parameters: time, temperature, swelling of the membrane in water and in organic solvents, compositiion of the permeating mixture, etc. The evaluation of a given membrane is accomplished by measuring the distribution of the compounds to be separated over the liquid and the gas phase. This procedure is time-consuming because many analyses of both phases must be performed in order to investigate the effect of the various parameters. Full automation of the analysis and of the evaluation of the results has been achieved by connecting an external events module to a digital integrator in order to connect an automatic gas-sampling valve and a small-volume liquid-sample valve to the carrier gas source and to the permeating streams, respectively. Each valve is automatically filled with samples coming from the liquid or gas circuit and activated to deliver the sample to the gas chromatographic unit, through a low-dead-volume injector, suitable for on-column introduction of both liquid and gaseous samples. The gas chromatographic columns were packed with various amounts of porous polymer-bead stationary phases (Porapak or Chromosorb “Century Series”) in order to separate water efficiently fromthe organic components of the permeating mixture (alcohols, glycols). A thermal conductivity detector was used. Computer programs were developed for automatic evaluation of the quantitative results, recording of the data, calculation of the retention-index values, plotting of the time—composition graphs, and calculation of the membrane selectivity as a function of time, temperature and composition of the permeating mixture. Low-cost personal computers are suitable for this applicatioin. With minor modifications and by using other columns and detectors, the system can be used for automatic analysis of any combination of liquid or gas streams.

Biparietal diameter and menstrual age in the black population attending Edendale Hospital.

The relationship between fetal biparietal diameter measured by echography and menstrual age for blacks was studied. Predicted values and their confidence limits are presented in tables and graphs. The biparietal diameter was significantly smaller from 28 weeks than American and European studies, demonstrating that a composite graph was not applicable to the Zulus. The echographic results were checked at term by external cephalometry. Occipitofrontal circumference and birth weights for newborns were found similar to other studies. The smaller BPD in late pregnancy could be explained by the head's shape. The new charts are valuable in assessing fetal growth in African blacks.

Recognizing Composite Graphs is Equivalent to Testing Graph Isomorphism

We consider composition, a graph multiplication operator defined by Harary and Sabidussi, from a complexity theoretic point of view. If G and H are undirected graphs without self-loops, then the composite graph $G[H]$ has vertex set $V(G) \times V(H)$ and edge set $\{ (g_1 ,h_1 ) \text{---} (g_2 ,h_2 ):g_1 \text{---} g_2 \in E(G){\text{ or }}g_1 = g_2 {\text{ and }}h_1 \text{---} h_2 \in E(H)\} $. We show that the complexity of testing whether an arbitrary graph can be written nontrivially as the composition of two smaller graphs is the same, to within polynomial factors, as the complexity of testing whether two graphs are isomorphic.