Graph planar

A graph is planar if it can be drawn on a plane without edges crossing, with intersections only at the edges (independently of how it is drawn). For example, cubane can be drawn as a planar graph. 

It is feasible to carry out Hartree-Fock calculations on our available computer resources (an SGI Crimson Elan Workstation) using an STO-3G basis set with full geometry optimization of CeoMu but only partial geometry optimisations of the ChoMu isomers. Fig. 1 shows planar graphs of Ceo and C70 with the carbon atoms suitably labelled for future reference. 

Equilibrium data are thus necessary to estimate compositions of both extract and raffinate when the time of extraction is sufficiently long. Phase equilibria have been studied for many ternary systems and the data can be found in the open literature. However, the position of the envelope can be strongly affected by other components of the feed. Furthermore, the envelope line and the tie lines are a function of temperature. Therefore, they should be determined experimentally. The other shapes of the equilibrium line can be found in literature. Equilibria in multi-component mixtures cannot be presented in planar graphs. To deal with such systems lumping of consolutes has been done to describe the system as pseudo-ternary. This can, however, lead to considerable errors in the estimation of the composition of the phases. A more rigorous thermodynamic approach is needed to regress the experimental data on equilibria in these systems. 

Figure 9.31 Some varients of allocation of PBUs fora primitive cubic packing (Pc) (a) PBU/C (b) PBU as a 3D graph and (c) as 2D planar graph (d) PBU/P black circles (b) and (c) are the sites of the lattice, that correspond to the centers of particles (or atoms), the solid lines are the bonds between them, the dotted lines are the bonds with the sites from the neighbor cells.

Eqn. (3) is the known Euler equation connecting the number of vertices, edges and cycles in a planar graph. 

Figure 33. Any non-planar graph contains one of these graphs.

Theorem 4.3.6 is an analog of the Steinitz theorem for 3-connected planar graphs. 

BrMK06] G. Brinkmann and B. D. McKay, Fast generation of planar graphs, MATCH 58-2 (2007) 333-367. 

CDG97] V. Chepoi, M. Deza, and V.P. Grishukhin, Clin d oeil on -embeddable planar graphs, Discrete Applied Mathematics 80 (1997) 3-19. 

Dut02] M. Dutour, HanGraph, a GAP package for planar graphs, www.liga.ens.fr/ dutour/ PlanGraph/, 2002. 

Fa48] I. Fary, On straight-line representation of planar graphs, Acta Urdu Szeged. Sect. ScL Math. 11 (1948) 229-233. 

GrSh87b] B. Griinbaum and G.C. Shephard, Edge-transitive planar graphs, Journal of Graph Theory 11-2 (1987) 141-155. 

Mal70] J. Malkevitch, Properties of planar graphs with uniform vertex and face structure, Memoirs cfthe American Mathematical Society, American Mathematical Society 1970. 

PSC90] K. F. Prisacaru, P. S. Soltan, and V. D. Chepoi, On embeddings of planar graphs into hypercubes, Proceedings of Moldavian Academy of Sciences, Mathematics 1 (1990) 43-50 (in Russian). 

A planar graph is a graph which can be drawn on a piece of paper in such a way that no edges cross. Most chemical molecules (with the possible exception of complex organic molecules) are planar (note that this does not mean planar in the sense of stereochemistry). For planar graphs the isomorphism problem also has an easy solution. 

Hopcroft, J. E. and R. E. TarjanT "Isomorphism of Planar Graphs," in Complexity of Computer Computations, ed. Raymond E. Miller and James W. Thatcher, Plenum Press, New York,... 

Conceptually, the representation of alternative process flowsheet(s) is based on elementary graph theory ideas. By representing each unit of the superstructure as a node, each input and output as a node, the interconnections among the process units as two-way arcs, the interconnections between the inputs and the process units as one-way arcs, the interconnections between the process units and the outputs as one-way arcs, and the interconnections between the inputs and the outputs as one-way arcs, then we have a bipartite planar graph that represents all options of the superstructure. 

This formula can easily be deduced from a theory due to P. W. Kasteleyn [4] (1961) which allows the number of 1-factors of any planar graph G with an even number of vertices to be expressed as the value of the Pfaffian PfS = j/det S of some skew-symmetric matrix S connected with G. Elementary proofs of Eq. (2) (not using Kasteleyn s formula) for plane graphs in which every face F is a (4k + 2)-gon (where k depends on F) were also given by D. Cvetkovic, I. Gutman and N. Trinajstic [5] (1972) and H. Sachs [6] (1986). 

The most general Kekule-structure-count method of the present type was devised by Kasteleyn [146], though there is slightly earlier work for different special cases [33,147]. This too involves certain matrices, most simply the graph adjacency matrices /4(G) with rows columns that are labelled by the sites of G and elements that are all 0 except those Aab=+ with a b adjacent sites in G. Then Kastelyn shows how for "planar" graphs to set up a "signed" version (G) of this matrix with half of its +1 elements replaced by -1 such that... 

Kasteleyn [146] describes how this "odd orientation" is readily achievable for any planar graph. For instance, if one inserts arrows on edges of G so that an arrow from a to b indicates SaA=+l while Sba= 1, then an example of one such odd orientation is... 

Intrinsically non-planar graphs are non-planar by virtue of their connectivity only. It has been shown [51] that they all contain one of the two basic non-planar graphs as a subgraph [60] the first and the second Kuratowski graphs K5 and... 

Extrinsically non-planar graphs are non-planar by virtue of their embedding in the 3D-space. They may be intrinsically planar at the same time (Fig. 6) this is the case of links (45) which are homeomorphic to disjoint sets of circuit graphs [71,72] (46) and knots (47a and b) which are homeomorphic to a circuit graph (48). 

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