Caterpillar tree

Certain structures of groups of chemical interest which may be difficult to see, can be easily envisaged by using caterpillar trees as models. We use as an example the composition [9] of two permutation groups St and Sj (involving the permutation of i and j objects respectively) which is denoted by S,[SJ (and is read S around Sj). This group is known as the wreath product or the Gruppenkranz . As a simple illustration we show how a caterpillar tree can be used to model the elements of S2[S3]. Then we have two sets, viz.,... 

The enumeration of Ld-sequences is equivalent to the enumeration of Gutman trees, which in the mathematical literature are called caterpillar trees (or caterpillars). These objects correspond to special trees in the graph-theoretical sense. 

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-... 

One of the present authors [7] used caterpillar trees to model Wreath product groups while Balasubramanian [8] used a particle in-a box model to represent the same type of permutation ... 

In this review we will focus on polyhex graphs, caterpillar trees, Clar graphs and several related polyomino gra s [19] In addition sets of graphs obeying certain types of recursive relations, called "Fibonacci Graphs" will be discussed particularly from the point of view of their computational importance ... 

First we will focus attention on selected topics relating to the equivalence between benzenoid hydrocarbons, and special types of graphs and other mathematical objects that we can associate with benzenoids- In particular we will explore relations involving caterpillar trees [3] associated with catacondensed benzenoids and their line graphs [17] called, as already mentioned, Clar graphs [4]. Also relations involving "boards" (known technically as polyominos) of special properties such as those associated with "king" and "rook" pieces of chess... 

Caterpillar tree, T Young diagram Unbranched benzenoid... 

Let us consider more closely the set of objects of Fig 7 associated with the same adjacency matrix A We will refer to these as the set T, A, B, P standing, respectively, for a caterpillar tree, a Clar graph (As L(T)) a benzenoid graph and a king polyomino The grai invariants that we will consider in each case are as follows ... 

Polyhex graphs in which no vertex is common to more than two hexagons are equivalent to caterpillar trees and vice-versa-... 

Then the resonance-relations of the hexagons e B exactly match the edge-incidence relations of the-edges T Since a caterpillar tree T (mj, m2, —, m ) is constructed by the addition of m. monovalent vertices to v. of a path P the proposition is proved, i e T (m,m., > -------------------------------------.. ... 

Fie- 19 A plot of logarithms of Kekule counts, In K(B) s of a homologous series of benzenoids and the connectivity indices of the corresponding caterpillar trees (L -Lg). The equivalent king boards are drawn to the left of the° correlation line The numbers in parentheses are the individual Kekul counts ... 

Fig. 10. Ordering of the set of Young diagrams containing 6 boxes. The corresponding tree graphs are shown. Underlined graphs are non-caterpillar trees...

A remarkable type of tree graph is called a Caterpillar El-Basil (1987) (or a caterpillar tree) P, (m1 m2,. .., m,) which is obtained by the addition of mi monovalent vertices to the first vertex iq of path P,, m2 monovalent vertices to n2 of Pj and so on. The three tree graphs shown in Fig. 10 are all caterpillar trees and may be designated respectively as ... 

Fig. 12. An unbranched benzenoid hydrocarbon, the corresponding caterpillar tree T, alkane hydrocarbon skeleton and Young diagram. T is a non-caterpillar tree...

The corresponding caterpillar tree is composed by the addition of mi monovalent vertices to Ui, m2 monovalent vertices to i>2,. .., j monovalent vertices to j- vertex of path P, (on j vertices). 

Clar E (1972) The Aromatic Sextet. John Wiely sons, London Coleman AJ (1968) The Symmetric Group Made Easy, in Advances in Quantum Chemistry 4. Ed. Lowdin PO, Academic Press Inc New York p 86 Cyvin SJ, Gutman I (1988) Kekule Structures in Benzenoid Hydrocarbons, Lecture Notes in Chemistry 46. Springer-Verlag Berlin El-Basil S (1987) Applications of Caterpillar Trees in Chemistry and Physics. J MathChem 1 153-174 See also ... 

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