MATCH Commun. Math. Comput. Chem. 60 (2008) 905-916 MATCH Communications in Mathematical and in Computer Chemistry ISSN 0340 - 6253 Computing PI and Omega Polynomials of an Infinite Family of Fullerenes A. R. Ashrafi,1 M. Jalali,1 M. Ghorbani1 and M. V. Diudea2 1Institute of Nanoscience and Nanotechnology, University of Kashan, Kashan 87317-51167, I R Iran 2Faculty of Chemistry and Chemical Engineering, Babes-Bolyai University, Arany Janos 11, 400028 Cluj, Romania (Received September 16, 2007 ) Counting of non-equidistant edges in a graph can be achieved in the formalism of PI-type polynomials. At least three ways of evaluating the equidistance of edges are discussed and the corresponding conditions established. Two versions of polynomials counting non-equidistant edges (PI-type polynomials) and two polynomials counting equidistant edges (Theta and Omega polynomials) are discussed and analytical formulas for an infinite series of tubulenes/fullerenes, derived from the dodecahedron C20, are developed. 1. Introduction Fullerenes are molecules in the form of cage-like polyhedra, consisting solely of carbon atoms. Fullerenes Cn can be drawn for n = 20 and for all even n ≥ 24. They have n carbon atoms, 3n/2 bonds, 12 pentagonal and n/2-10 hexagonal faces. The 1,2 most important member of the family of fullerenes is C60. Let G = (V,E) be a connected bipartite graph with the vertex set V = V(G) and the edge set E = E(G), without loops and multiple edges. In 1988, Hosoya3 introduced what he termed the Wiener (and latter called Hosoya) polynomial of a graph as HGx(,)=⋅∑ mGk (,) xk , where m(G,k) is the number of pairs of vertices in G 1<<kl that are distance k apart, and l is the maximum value of k or the diameter of G. Sagan, Yeh and Zhang4 produced a treatment apparently independent of Hosoya's. Perhaps the most interesting property of H(G,x) is the first derivative, evaluated at x = 1, which equals the Wiener index: HG′(,1)= WG (). Ashrafi, Manoochehrian and - 906 - Yousefi-Azari5,6 continued the line of the mentioned paper of Sagan et al. to introduce the notion of PI polynomial of a molecular graph G as: Nuv(,) PI(,) G x= ∑ x (1) (,)uv=∈ e EG ( ) =+ where Nuv(,) neu (| eG ) n ev (| eG )and neGeu (| )is the number of edges lying closer to u than v (i.e., the non-equidistant edges) while the number of edges equidistant to the edge euvEG=∈(,) ( ) is given by: N()eEGNuv=− ( ) (,) 2 PI() G==− PI′ (,1) G() E∑ N () e (2) e which is just the formula proposed by John et al.8 to calculate the PI index. Two edges euv= (,) and f = (,xy )of a graph G are called equidistant if the two ends of one edge show the same distance to the other edge. However, the distance between edges can be defined in several modes, as presented below. (a) The distance from a vertex z to an edge euv= (,) is taken as the minimum distance between the given point and the two endpoints of that edge:5 dze(,)= min{(,),(,)} dzu dzv (3) Then, the edge euv= (,) is equidistant to f = (,xy ) if: dxe(,)= dye (,) (4) Or the edges euv= (,) and f = (,xy ) are equidistant if: dxe(,)== dye (,)and(, duf ) dvf (, ) (4`) (b) A second definition for equidistant edges joins the conditions for (topologically) parallel and perpendicular edges:9 dvx( , )=+=+= dvy ( , ) 1 dux ( , ) 1 duy ( , ), for edges (5) dux( , )=== duy ( , ) dvx ( , ) dvy ( , ), for ⊥ edges (6) Clearly, the two polynomials PI(,) G x (counted cf. version (a) - patterned by Ashrafi’s group) and Π(,)Gx (provided by version (b) - patterned by Diudea’s group) are different and their first derivatives in x=1 will give different PI-type indices. - 907 - Keeping in mind relation (2) and accounting that condition (4) is less strong than conditions (5) and (6) it is easily seen that: PIG′′(,1)≤Π (,1) G (7) The above results are exemplified for the graph G1 (Figure 1) in Table 1 (for each edge, the corresponding non-equidistant edges are listed). G1 Figure 1. Cuneane graph and its numbering Table 1. Counting polynomials for the structure G1 Version c Polynomial/index 1 PI(,) G x : non-equidistant edges (Ashrafi) (comments) 1-2 7-8 5-6 3-4 2-7 2-3 1-7 1-6 7 PI(,) G x7=+++ 2 x78 x 8 x 910 x 1-6 7-8 5-8 5-6 4-5 3-6 3-4 2-7 1-7 1-2 9 PI’(G,1) =104 1-7 7-8 5-8 5-6 4-8 3-6 2-7 2-3 1-2 1-6 9 2-3 7-8 5-6 4-8 4-5 3-6 3-4 2-7 1-2 1-7 9 2-7 7-8 5-8 4-8 3-6 3-4 1-2 1-6 1-7 2-3 9 3-4 7-8 5-8 4-8 4-5 3-6 1-2 1-6 2-3 2-7 9 Edge 2-7 is equidistant to 3-6 5-8 5-6 4-8 1-6 1-7 2-3 2-7 3-4 8 5-6 and 4-5, cf. (4) 4-5 7-8 5-8 5-6 4-8 1-6 2-3 3-4 7 e.g., 7 4-8 7-8 5-8 5-6 1-7 2-3 2-7 3-4 3-6 4-5 9 ⎧ dd[(6,5),2]== [(6,5)7] 2 ⎨ 5-6 7-8 5-8 1-2 1-6 1-7 2-3 3-6 4-5 4-8 9 ⎩dd[5,(2,7)]== (6,(2,7)] 2 5-8 7-8 1-6 1-7 2-7 3-4 3-6 4-5 4-8 5-6 9 7-8 1-2 1-6 1-7 2-3 2-7 3-4 4-5 4-8 5-6 5-8 10 - 908 - 2 Π(,)Gx : non-equidistant edges (Diudea) 1-2 7-8 5-8 5-6 4-8 3-4 2-7 2-3 1-7 1-6 9 Π=+++(,)Gx9 x 891011 2 x 5 x 4 x 1-6 7-8 5-8 5-6 4-8 4-5 3-6 3-4 2-7 1-7 1-2 10 Π=′(,1)120G 1-7 7-8 5-8 5-6 4-8 4-5 3-6 3-4 2-7 2-3 1-2 1-6 11 10 2-3 7-8 5-8 5-6 4-8 4-5 3-6 3-4 2-7 1-2 1-7 11 2-7 7-8 5-8 5-6 4-8 4-5 3-6 3-4 1-2 1-6 1-7 2-3 Edge 2-7 is equidistant to no any edge 10 3-4 7-8 5-8 4-8 4-5 3-6 1-2 1-6 1-7 2-3 2-7 cf. (5) and (6) 8 3-6 5-8 5-6 4-8 1-6 1-7 2-3 2-7 3-4 9 9 4-5 7-8 5-8 5-6 4-8 1-6 1-7 2-3 2-7 3-4 11 4-8 7-8 5-8 5-6 1-2 1-6 1-7 2-3 2-7 3-4 3-6 4-5 10 5-6 7-8 5-8 1-2 1-6 1-7 2-3 2-7 3-6 4-5 4-8 11 5-8 7-8 1-2 1-6 1-7 2-3 2-7 3-4 3-6 4-5 4-8 5-6 10 7-8 1-2 1-6 1-7 2-3 2-7 3-4 4-5 4-8 5-6 5-8 3 Θ(,)Gx : equidistant edges 1-2 4-5 3-6 3 Θ=+++(,)Gx 4 x 5 x234 2 x x 1-6 2-3 2 Θ′ 1-7 1 =24 2-3 1-6 2 Edge 2-7 is equidistant to no any edge 2-7 1 cf. (5) and (6) 3-4 5-6 2 3-6 1-2 4-5 7-8 4 Edge 3-6 is to (1-2) and (4-5) 4-5 1-2 3-6 3 and ⊥ to 7-8 4-8 1 which, cf. (5) and (6), are equidistant 5-6 3-4 2 5-8 1 Edge 7-8 is equidistant to 3-6 7-8 3-6 2 4 Ω(,)Gx: strips 1-2 3-6 4-5 3 Ω=++(,)Gx 5 x 2 x23 x 1-6 2-3 2 CI=122 1-7 1 2-7 1 3-4 5-6 2 4-8 1 5-8 1 7-8 1 - 909 - 2. Omega Polynomial Two edges e = (u,v) and f = (x,y) of G are called codistant (briefly: e co f ) if they obey the topologically parallel edges relation (5). For some edges of a connected graph G there are the following relations satisfied:8,10 e co e (8) ecof⇔ f coe (9) e co f &f co h ⇒ e co h (10) though the relation (10) is not always valid. Let C(e) := { f ∈ E(G); f co e} denote the set of edges in G, codistant to the edge e ∈ E(G) . If relation co is an equivalence relation (i.e., all the elements of C(e) satisfy the relations (8) to (10), then G is called a co-graph. Consequently, C(e) is called an orthogonal cut oc of G and E(G) is the union of disjoint orthogonal cuts: =∪∪∪ ∩ = ≠ = E(GCC )12 ..