Multiple edges

Intuitively, a graph can be realized geometrically in a three-dimensional Euclidean space vertices arc represented by points and edges are represented either by lines (in the case of undirected graphs) or arrows (in the case of directed graphs). In this book, we will be concerned with both kinds of graphs multiple edges i.e. when vertices arc connected by more than one line or arrow), however, are not allowed. 

Di Cicco, A., Taglienti, M., Minicucci, M., and Filipponi, A., Short-range structure of solid and liquid AgBr determined by multiple-edge X-ray absorption spectroscopy, Phys. Rev. B, 62, 12001-12013, 2000. 

For multiple edges (unsaturated or aromatic systems) fractional distances are introduced in the distance matrix so that fractional distance sums result on summation if the bond order between vertices i, j is b, then 1/b is the entry in the row/... 

The quotient of a map can be a map with loops and multiple edges. Consider, for example, the 4-valent plane tiling 4,4 (see Section 1.5) formed by 4-gons and the group Z2 acting by translations on it. There is one orbit of vertices, two orbits of edges, and one orbit of faces under Z2 so the quotient 4,4)/ 2 is a torus represented by a single vertex and two loops. 

For the concept of a graph see any textbook. All graphs considered in this paper are finite and have no loops and no multiple edges. 

Multiple edges are taken as several single edges, and the sign of the edge is retained in loops.)... 

Graph having no arcs, no multiple edges or loops. 

Graph having no arcs or loops, but including multiple edges. 

Multiple edges are considered as independent vertices in the line graph. 

An illustrative example is given below. The number of connected simple graphs (that is, graphs without loops and multiple edges)11 with N vertices is given by ... 

Proceeding from the linkage type of two KG cycles, another kind of class, is defined when the two cycles are connected by two or more disjoint vertices. This class corresponds to the introduc tion of a cycle multiple edge) connecting two supergraph vertices. 

The RMC method is more general than this simple algorithm in that any set(s) of data that can be directly related to the structure can be modelled. It can be applied to isotopic substitution in neutron diffraction, or equivalently to anomalous scattering in X-ray diffraction, to EXAFS (multiple edges) and possibly to NMR data. All data sets can be modelled simultaneously simply by adding the respective y1 values. 

Two vertices are adjacent when they are connected by an edge. The edge and these two vertices are said to be incideni to one another Two edges are said to be adjacent provided they have a vertex in common. Single edges of a multiple edge are always adjacent. 

A path, Wr i Is a walk, Ur , in which each of its vertices occurs only once. If in multiple edges are absent, the path Wr G is uniquely defined by the corresponding sequence of vertices. Thus, wjk G nnd are denoted as fcJlows ... 

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