Ordering Of Vertices Research Articles

A new approach for devising local graph invariants: Derived topological indices with low degeneracy and good correlation ability

A new approach is presented for obtaining graph invariants which have very high discriminating ability for different vertices within a graph. These invariants are obtained as the solution set (local invariant set, LOIS) of a system of linear equationsQ · X = R, whereQ is a topological matrix derived from the adjacency matrix of the graph, andR is a column vector which encodes either a topological property (vertex degree, number of vertices in the graph, distance sum) or a chemical property (atomic number). Twenty examples of LOOIs are given and their degeneracy and ordering ability of vertices is discussed. Interestingly, in some cases the ordering of vertices obtained by means of these invariants parallels closely the ordering from an entirely different procedure based on Hierarchically Ordered Extended Connectivities which was recently reported. New topological indices are easily constructed from LOISs. Excellent correlations are obtained for the boiling points and vaporization enthalpies of alkanesversus the topological index representing the sum of local vertex invariants. Les spectacular correlations with NMR chemical shifts, liquid phase density, partial molal volumes, motor octane numbers of alkanes or cavity surface areas of alcohols emphasize, however, the potential of this approach, which remains to be developed in the near future.

A geometric hidden-line processing algorithm

This paper presents a new geometric hidden-line algorithm applicable to complex three-dimensional vector graphics scenes. The algorithm has advantages of generality, coding simplicity, and is particularly appropriate to parallel processing. The algorithm works accurately for any axonometric or oblique projection and can be adapted to work with the central projection. It applies to any nuimber of three-dimensional polyhedral objects which may have holes partially or fully through them, or other shape complexities. The algorithm does not require any particular ordering of polygon vertices and operates in both object and image space. An application of this algorithm to a computer aided learning package for engineering students is described.

Reachable grasps on a polygon: The convex rope algorithm

An algorithm that finds the externally visible vertices of a polygon is described. This algorithm generates a new geometric construction, termed the convex ropes of each visible vertex. The convex ropes give the range of angles from which each vertex is visible, and they give all the pairs of vertices which are reachable by a straight robot finger. All of the convex ropes can be found in expected time order n, where n is the number of vertices of the polygon. We discuss the application of this geometric construction to automated grasp planning. The algorithm may also be useful in image interpretation and graphics where efficient computation of visible points is important. The direct application of the algorithm is restricted to two dimension since sequential ordering of vertices is required. Extension to three dimension would rely on well chosen intersecting or projective planes.

Numerical experiences with graph coloring algorithms

We consider the problem to color the vertices of a graph with a minimal number of colors. Sequential vertex (Sv) algorithms are mostly used for approximate solutions. For the selection of the colors all the approximation Sv-algorithms proposed in literature use the Min rule. In this paper a look-ahead rule is applied to determine a most ‘suitable’ color. Furthermore, the influence of the vertex order at the beginning of the Sv algorithm ( Lf or Sl) in investigated. The tests are made on a sample of 3000 random graphs with up to 300 vertices.

On color critical graphs

It is shown that the minimal number of edges which have to be omitted from a ( k + 1)-critical graph on n vertices in order to make it bipartite is at least k 2 for n large enough. This bound is best possible. Various related questions are considered.

Comparability graphs and molecular properties. A novel approach to the ordering of isomers

A graph-theoretical approach to the problem of hierarchical ordering of isomeric structures, which is relevant to their properties, has been developed. The key to this approach is the so-called comparability graph which is based on a number of structural rules derived in advance. These rules express the regular change in the different topological indices occurring upon certain systematic changes in the structure of isomeric molecules. They are presented as partial comparability graphs where the vertices correspond to the isomers while the directed edges correspond to the isomer interconversions. The total comparability graph for a complete set of isomers is built by superimposing the partial graphs. The paths on such an oriented graph specify the ordering of vertices. Each path contains comparable isomers which may be regarded as an optimized correlation sample for various structure–property correlations. The isomers from different graph paths are viewed as non-comparable, owing to different alterations in their topology. Such isomers should not be included in the same correlation sample. To illustrate the approach, the importance of the comparability graphs of C7- and C8-alkane isomers has been examined with regard to 20 molecular properties.

Analysis of the efficiency of graph coloring algorithms

This paper discusses the computational efficiency and the number of colors used by the following algorithms for coloring vertices of graphs: sequential coloring and sequential coloring with interchange algorithms for a largest-first and a smallest-last orderings of vertices, the coloring-pairs algorithm, and the approximately maximum independent set algorithm. Each algorithm is supplied with a Pascal-like program, time complexity in terms of the size of a graph, and worst-case behaviour. In conclusion, some computational results are included with support the estimations and suggest the sequential coloring with interchange algorithm for a largest-first vertex ordering as a method which uses the least number of colors for uniformly distributed random graphs.

Left-right double trees

This paper refers to trees and rooted trees the vertices of which are (simply) ordered by means investigated in [2] Chapter IV. On the basis of the relationships between the edges and the ordering of the vertices, the various notions are introduced and discussed, particulary a certain pair of rooted trees.

Homology of Deleted Products of Contractible 2-Dimensional Polyhedra. I

If X is a space and , the kth deleted product space Xk* of X is the topological product X ⨯ X ⨯ . . . ⨯ X of copies of X minus the set of all points of the form (x, x, . . . , x), where x ∊ X. In (4), the author shows that the homology groups of Xk*, where X is a tree, produce as much information about trees as counting the orders of vertices.

On a certain ordering of the vertices of a tree

On a certain ordering of the vertices of a tree

On an ordering of the vertices of a graph

On an ordering of the vertices of a graph