Graphs Skeleton

Figure 8-2. Edge graph of the carbon skeleton of 2-methylbutane (1).

Procedure. Using graph paper with Fig. 4-5 as a guide, construct the approximate carbon atom skeletons of cis- and rra./r.v-2-butene. 

Finally, it should be emphasized, that the skeleton of v = 3 vertices (shown by larger black dots and numbered in figure 3) form graphs which are identical to the topological, colored graphs which... 

According to Graph Theory, a molecule may be represented by its skeletal molecular graph which, from a mathematical point of view, is the union of a set of points, symbolising the atoms other than hydrogens, and a set of lines, symbolising bonds. Its properties can be then expressed in terms of graph-theoretical invariants, Njj, which have been defined as "the number of distinct ways in which skeleton i... 

So, the number of connections can be determined either from the "bond graph" [7], or even better, from the number of "propanes" that can be derived from the skeleton of the molecule under consideration. 

We begin with the way chemists perceive similarity between two molecules. This process involves, consciously or unconsciously, comparing several types of structural features present in the molecules. For example, considering the five aliphatic alcohols (represented by their H-suppressed molecular graphs) in Figure 1, we note both similarities and differences they are all four-carbon alcohols a, b, c and d are acyclic, whereas e has a ring a and b are primary alcohols, c and e are secondary alcohols and d is a tertiary alcohol b and c have the same skeleton, but for the labeling of points (atoms), while the other skeletons are distinct etc. 

We will consider only 3-dimensional polytopes they are called polyhedra. Their 0-dimensional faces are called vertices and the 2-dimensional faces are called just faces. Two vertices are called adjacent if there exist an edge, i.e. a 1-dimensional face containing both of them. The skeleton of a polyhedron is the graph formed by all its vertices with two vertices being adjacent if they share an edge. This graph is 3-connected and admits a plane embedding. 

MedfT etrahedron) = Octahedron and MediCube) = Cuboctahedron. The skeleton of Med(G) is the line graph of the skeleton of G if G is a 3-valent map... 

The skeleton of a polycycle is the edge-vertex graph defined by it, i.e. we forget the faces. By Theorem 4.3.6, except for five Platonic ones, the skeleton has a unique polycyclic realization, i.e. a polycycle for which it is the skeleton. 

Theorem 43.6 Given a graph G that is the skeleton of an (r, q)-polycycle different from (one of 5) elliptic [r, q], then die polycyclic realization is completely determined byG. 

Given an (r, -polycycle P, its major skeleton Maj(P) is the plane graph formed by its elementary components with two components being adjacent if they share an open edge. A tree is a connected graph with no cycles. 

If b = 12, then, by the same reasoning, the gonality of faces of the major skeleton is at most 6. Since it is a plane graph, there exists a face of gonality at most 5. Such a face is incident to a vertex v, corresponding to a (5,3)-polycycle C3 or Cj. So, the original ( 5,12, 3)-sphere contains the pattern below ... 

The n-electronic energy is obtained by adding up the solutions of the characteristic, or secular, polynomial of the graph representing the carbon atom skeleton of the molecule. Namely, for I and II we have... 

A graph with an odd number of points is non-Kekulean by definition. No benzenoid hydrocarbon molecule or radical corresponding to non-Kekulean graph has ever been synthesized. The phenalene skeleton, XHIa, is the smallest non-Kekulean benzenoid. However, it is regrettably true that even phenalenyl radical is... 

An obvious first question for a given carbon skeleton is how many distinct Heilbronner modes and distortions are possible Given the form of the model, where the freedoms involve the values of /3 on the edges of the graph of the carbon framework, and the constraints are imposed at the (multivalent) vertices, it is reasonable to expect n(8), the number of independent Heilbronner modes, to depend on a difference function of edges and vertices. The next section deals with this connection. 

Visual devices are also useful for indicating order. Primary among them are lines. Lines form the skeletons of trees and graphs, both of which are commonly used to display ordered concepts, to indicate asymmetry on a variety of relations, including kind of, part of, reports to, and derived from. Examples include hierarchical displays, as in linguistic trees, evolutionary trees, and organizational charts. Other visual and spatial devices used to display order rest on the metaphor of salience. More salient features have more of the relevant property. Such features include size, color,... 

A number of options are provided for the user to adjust the constitutional formulas being output to his special desires. Atom numbers or atomic symbols, or both, can be put onto the nodes of the constitutional graph. Further, the entire constitutional formula with all hydrogen atoms, or only the skeleton of the molecule, can be drawn. In a few cases of molecules with complex polycyclic ring structures only partially complete formulas are output. Then, messages pointing to the deficiences are given which usually allow the completion of the constitutional formulas by hand. 

The mathematical chemist will recognize a as one of the Kekule valence-bond structures of the hydrocarbon, while is the corres nd-ing molecular graph Model is called an inner dual or dualist [2], is a caterpillar tree [3], and is called a Clar graph [4]. The latter two models are apparently quite different from the original skeleton, however, as it will turn out later, the topological properties of this benzenoid system are best modeled by either d or e-... 

Graphs and electricity. After Euler the graph theory was rediscovered several times. In 1847 G. Kirchhoff developed the theory of trees to solve the problem about currents flowing in each conductor and in each loop of an electric circuit. Kirchhoff replaced the electric circuit by the graph and developed the procedure to solve the system of equations defining the current intensity, analysing only the so-called skeleton trees of the initial graph. 

In 1955 the US scientist N. Rashevsky introduced a concept of topological information of a graph Itop)- His theory is based on subdivision of the set of N vertices of the graph into classes of topologically equivalent vertices. However, the quantity I top turned out to be unsuitable for characterization of branching of alkane s carbon skeleton. 

The topology of a polyhedron can be described by a graph, called the 1-skeleton of the polyhedron. The vertices and edges of the 1-skeleton correspond to the vertices and edges, respectively, of the underlying polyhedron. Of fundamental importance are relationships between possible numbers and... 

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