Clar graph

Fig. 5. A Gutman tree, T, an unbranched benzenoid system, B, a Clar graph, A, a king polyomino-graphy, P, and a rook board, Pr...

It is almost trivial to see that an equivalence relation exists between the elements of such sets of graphs as the one shown in Fig. 5, i.e., between what is called here equinumerical graphs. Let this relation be R. Then R is evidently reflexive because every graph is related to itself. Also R is symmetric because for every Gutman tree T we can generate a Clar graph A. In fact T and A are related as... 

The article reports investigations of the topological properties of benzenoid molecules which the author has performed in the last 20 years. Emphasis is given on recent developments and other scientists contributions to these researches. Topics covered in recent books and reviews are avoided. The article outlines spectral properties, some aspects of the study of Kekule and Clar structures, the Wiener index as well as a number of graphs derived from benzenoid systems (inner dual, excised internal structure, Clar graph, Gutman tree, coral and its dual). 

Define the Clar graph C(B) of the benzenoid system B as a graph whose vertices correspond to the hexagons of B. Two vertices of C(B) are adjacent if, and only if, it is not possible to simultaneously arrange aromatic sextets (=circles in the generalized Clar formulas) in the respective hexagons of B. 

It turns out that in some cases the Clar graph is a line graph. This fact was conceived even before the discovery of the Clar-graph concept (of course in a somewhat different form). We may thus reformulate the result originally obtained in 1977 as follows ... 

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-... 

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... 

Line graph of T [17] Clar graph Branched and unbranched systems... 

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 ... 

Fig 11 A set of twelve gtaphs comprising four types a caterpillar, a polyhex graph, a Clar graph and a polyomino graph-... 

Fig- 17 A plot of tnolecular connectivity index, jg(B) s of a homologous series rf polyhex graphs against the corresponding quantities, %(A) s, of the corresponding Clar graphs-... 

In addition, these author claimed that every Clar structure which forms the basis for VB calculations corresponds to a maximum independent set of a Clar graph, which however is not the case, as outlined in ref 793. As a consequence, enumerations of Clar structures by El-Basil et are in error. 

---