On Omega and Sadhana Polynomials of Leapfrog Fullerenes
نویسنده
چکیده
A fullerene graph is a cubic 3-connected plane graph with (exactly 12) pentagonal faces and hexagonal faces. Let Fn be a fullerene graph with n vertices. By the Euler formula one can see that Fn has 12 pentagonal and n/2 – 10 hexagonal faces. Let G = (V, E) be a connected graph with the vertices set V = V(G) and the edges set E = E(G), without loops and multiple edges. The distance d(x,y) between x and y is defined as the length of a minimum path between x and y. Two edges e = ab and f = xy of G are called codistant, “e co f”, if and only if d(a,x) = d(b,y) = k and d(a,y) = d(b,x) = k+1 or vice versa, for a non-negative integer k. It is easy to see that the relation “co” is reflexive and symmetric but it is not necessary to be transitive. Set ( ) { ( ) co } C e f E G f e = ∈ . If the relation “co” is transitive on C(e) then C(e) is called an orthogonal cut “oc” of the graph G. The graph G is called cograph if and only if the edge set E(G) a union of disjoint orthogonal cuts. If any two consecutive edges of an edge-cut sequence are topologically parallel within the same face of the covering, such a sequence is called a quasi-orthogonal cut qoc strip. The Omega polynomial has been defined on the ground of qoc strips [1-5]:
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