Plane graph

Program THERM solves the dynamic model equations. The initial values of concentration and temperature in the reactor can be changed after each run using the ISIM interactive commands. The plot statement causes a composite phase-plane graph of concentration versus temperature to be drawn. Note that for comparison both programs should be used with the same parameter values. 

Choose a multiple steady-state case and try to upset the reactor by changing AO, F, TO or TJ interactively during a run. Only very small changes are required to cause the reactor to move to the other steady state. Plot as time and phase-plane graphs. 

For a connected plane graph G, denote its plane dual graph by G and define it on the set of faces of G with two faces being adjacent if they share an edge. Clearly, v(G ) = p(G) and p(G ) = v(G). 

A closed map, cell-complex of a polyhedron. It is a 5Rq, 4/ 2 plane graph (see Chapter 9)... 

A reader who is interested only in plane graphs, our main subject, can move now directly to Section 1.4. But, for foil understanding of the toroidal case, we need maps in all their generality. For reference on Map Theory see, for example, [BoLi95] and [MoThOl]. 

In Mi and M2, the closed path is homotopic to a null path. In Mi, this cycle is the boundary of a face, while in M2, the closed path is the boundary of all faces put together. More generally, a plane graph and a finite plane graph minus a face are simply connected. But the closed path in M3 is not homotopic to a null path. Actually, this closed path is a generator of the fundamental group Jti(Mf) — Z. 

We remind that an automorphism of a simple graph is a permutation of the vertices preserving adjacencies between vertices. For plane graphs, we require also that faces are sent to faces but for 3-connected graphs this condition is redundant. Recall that Aut(G) denotes the group of automorphisms of G. 

Denote by Bundlem, m > 2, the plane graph with two vertices and m edges between them (so, m 2-gonal faces). The plane graph Bundlem, which is dual to m-gon, has the symmetry group Dmh = 7 (2,2, m) and it is a regular map, which is not a cell-complex. 

For 2 < m < oo, denote by snub Prismm a 3-valent plane graph with two m-gonal faces separated by two m-rings of 5-gons. Its symmetry group is if m 5 and... 

Call a two-faced map and, specifically, ( a, b), k)-map any valent map with only a- and fc-gonal faces, for given integers 2 < a < b. We will also use terms ( a, b], k) sphere (moreover, ( a, b], k)-polyhedron if it is 3-connected) or ( a, b), k)-torus for maps on sphere S2 or torus K2, respectively. Call ( a, b], k)-plane any infinite /c-valent plane graph with a- and fe-gonal faces and without exterior faces. More generally, for R c N — 1, call (R,k)-map a k-valent map whose faces have gonalities i 6 R. 

We can interpret the quantity 2k — b(k — 2) as the curvature of the faces of gonality b Euler formula is the condition that the total curvature is a constant, equal to 4k, for fc-valent plane graphs. This curvature has an interpretation and applications in Computational Group Theory, see [Par06] and [LySc77, Chapter 9]. 

Theorem 2.0.2 ([DeDuOS]) For a 3-valent plane graph G with faces of gonality between 3 and 6, it holds ... 

The Goldberg-Coxeter construction takes a 3- or 4-valent plane graph Go, two integers k and l, and returns another 3- or 4-valent plane graph denoted by GC, /(Go)-This construction occurs in many contexts, whose (non-exhaustive) list (for the main case of G0 being Dodecahedron) is given below ... 

Triacon. He also called the Goldberg-Coxeter construction Breakdown of the initial plane graph Go. 

The Goldberg-Coxeter construction for 3- or 4-valent plane graphs can be seen, in algebraic tains, as the scalar multiplication by Eisenstein or Gaussian integers in the parameter space. More precisely, GC/y corresponds to multiplication by complex number k + l(o or k + li in the 3- or 4-valent case, respectively. 

The program CPF ([BDDH97]) generates 3-valent plane graphs with specified p-vector. 

The program ENU ([BHH03] and [Hei98]) does the same for 4-valent plane graphs. 

We recall that any finite plane graph has a unique exterior face an infinite plane graph can have any number of exterior faces, including zero and infinity. Denote by pr the number of interior faces for example, Dodecahedron on the plane has P5 = 11-... 

See in Figure 4.1 some examples of connected simple plane graphs that are not (r, < )-polycycles. 

Recall that an isomorphism between two plane graphs, G and G2, is a function

incidence relations. Two (r, )-polycycles, Pi and P2, are isomorphic if there is an... 

---