Cayley graph

Calculadons predict that a C120 molecular realization of this solid should exist. See [11, 12]. For more elaborate treatments, including the use of the Cayley graph, see [13, 14]. 

Cayley calls it kenogram (Biggs, N. L. Algebraic Graph Theory, Tracts in Mathematics Series, No. 67, Cambridge University Press, 1974, p. 61). 

One example of a tree-based separator system is shown below in Fig. 2.8 where the Bethe lattice or Cayley tree is shown (Wilson, 1996). This graph can be expanded to any number of levels and can function with dilferent types of columns and electrophoretic elements. This is not the only graph that can function as a complex multidimensional separator system. But it is an example of something with multiple... 

FIGURE 2.8 Separator systems cascaded to form a Bethe lattice or Cayley tree where the point of introduction is the graph vertex 0 and solute can be sampled from any of the outward nodes at position 1, 2, 3, 4, and so on. The sample loops and valves are not shown. 

Sir Arthur Cayley [3] introduced the application of graph theory to chemistry. Meanwhile chemical graph theory has developed into an important and rapidly growing branch of mathematical chemistry [4]. 

Beachhead. However, basically this discrepance does not disparage the explorations of the English mathematician. Cayley was the first to represent a molecule as a topological graph and to start developing the methods of isomer enumeration on the basis of the graph theory. Moreover, Cayley forwarded the notion of enumerative polynomial for rooted trees, i.e. the polynomial whose coefficients Ai specify the number of rooted trees with i vertices ... 

Although the term graph was first introduced into literature by mathematician Sylvester [Sylvester, 1877, 1878], who derived it from the contemporary chemical term graphical notation, used to denote the chemical structure of a molecule, the research field that is nowadays called chemical graph theory started some years before when the British mathematician Arthur Cayley published his works about trees [Cayley, 1857,1859] and then the paper On the mathematical theory of isomer [Cayley, 1874]. 

Gutman, 1., Vidovic, D. and Popovic, L. (1998) Graph representation of organic molecules. Cayley s plerograms vs. his kenograms. /. Chem. Soc. Faraday Trans., 94, 857-860. 

I. Gutman, D. Vidovic, L. Popovic, Graph representation of organic molecules Cayley s plerograms vs... 

Figure 11. Reproductions of tree graphs called "icenograrm" as used by Cayley (1874) in his enumeration of the alkyl radicab.

Graph theory was first applied to a chemical problem by Cayley when he proposed the concept of the tree [56] in 1857 and subsequently plied the concept to enumeration of hydrocarbons [57] in 1874. Since then, graph theory has been spiled to a range of chemical subjects such as reaction graphs, synthesis design, chemical documentation, and kinetics. However, relatively little work has been done until recently on application... 

Graph centret 6tcen(re centroid and bicentroid. It is well-known [7] that any tree possesses a centre this is either one vertex (when it is referred to as a centre) or a pair of adjacent vertices (when the pair is called a bicentre). To find the (bi)ccntre one prunes sequentially the vertices of degree one (endpoints) of the tree alternatively, one considers the eccentricities of vertices defined as the maximum topological distance to an endpoint (bi)centres have minimum eccentricities. Cayley used in this case the term (bi)centre of number. 

In this subsection, we briefly summarize results relevant to labeled graphs. We then survey the work on counting series by Cayley, Polya, Ffarary,... 

Consider the polycondensation of functional monomers of the type R AB/ i. The reaction is assumed to take place between the A group and B group only [1], Nonlinear polymers with a tree structure are formed by reaction. They may have intramolecular cycles, but to find the exact solution we consider only branched tree-type polymers which have no cycles (Figure 3.6). These are sometimes called Cayley trees, named after the mathematician who studied tree-type graphs. The approximation under this assumption of no intramolecular cycles is called the tree approximation. 

The term algebraic chemistry has in due course been replaced by the more general term mathematical chemistry, but a better term than graph has never been found. The seminal role of Cayley and Sylvester in the early development of mathematical chemistry in general and chemical graph theory in particular has been expertly reviewed by Dennis H. Rouvray (1989). It is important to point out why mathematical chemistry is relevant to chemistry. We could not do better than Jerome Karle, Nobel Prize Laureate 1985, who wrote Mathematical chanistry provides the framework and broad foundation on which chemical science proceeds (Karle, 1986). 

Clifford, Sylvester, and the term graph Enumeration, from Cayley to Polya... 

Molecular architectures can be structurally classified as being more comb-like or Cayley tree-like. Structure has impact on the radius of gyration, which is larger for linear molecules than for branched molecules of the same weight (number of monomer units), since the latter are more compact. The ratio between branched and linear radius is usually described by a contraction factor . Furthermore, Cayley tree-like structures are more compact than comb-like structures [33, 56]. We will show here how to obtain the contraction factor from the architectural information. The squared radius of gyration is expressed in monomer sizes. According to a statistical-mechanical model [55] it follows from the architecture as represented in graph theoretical terms, the KirchhofF matrix, K, which is derived from the incidence matrix, C [33] ... 

If structural formulae displayed the connections between atoms, the sequence of their mutual combination, it is not surprising that Cayley saw them as graphs, embodying topological information only. As such they do fail to provide the spatial positions of atoms, but only because they abstract away from particular spatial arrangements. 

---