Hexagons, graph

Fig. 8. Left graph example of a boundary node map for a disc in a 2D hexagonal lattice. Right graph illustration of the bounce-back rule on an enlarged section of the boundary. The distribution at site r that moves at time t into direction i, instead of arriving at the (virtual) site s, is bounced at the boundary node, and thus arrives back at site r at time t + 8t, but is now headed in the opposite direction i.

At least one toroidal polyhex that is cell-complex exists for all numbers of vertices v > 14. The unique cell-complex toroidal fullerene at v = 14 is a realization of the Heawood graph. It is GC2,i(hexagon) in terms of Goldberg-Coxeter construction and is the dual of K7, which itself realizes the 7-color map on the torus. This map and its dual are shown in Figure 3.1. 

In the theory known nowadays as the Clar theory of the aromatic sextet [12] a benzenoid system is represented by a Clar structure which is obtained by drawing circles in some of the hexagons of the corresponding benzenoid graph. These circles represent the aromatic sextets in the hydrocarbon. We consider here only Clar structures containing the maximum number of circles which are some times referred to as proper (or correct) Clar formulas. The rules for constructing such Clar structures are as follows [13]. 

Namely A is the line graph [30] of T. Other less transparent relations can be envisaged for other pairs of the set. For example let 6 be an operator which maps the edges of T into hexagons of B so that two incident edges in T will always correspond to two nonresonant hexagons in B. Formally... 

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. 

Fig 7 Four equivalent graphs, arbitrarily labeled There is a one-tooone correspondence between edges in T, vertices of A, hexagons of B and the squares making board P ... 

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