Computing Vertex PI, Omega and Sadhana Polynomials of F 12(2n+1) Fullerenes

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1 Iraia Joural of Mathematical Chemistry, Vol. 1, No. 1, April 010, pp IJMC Computig Vertex PI, Omega ad Sadhaa Polyomials of F 1(+1) Fullerees MODJTABA GHORBANI Departmet of Mathematics, Faculty of Sciece, Shahid Rajaee Teacher Traiig Uiversity, Tehra, , I R. Ira (Received Jauary 10, 010) ABSTRACT The topological idex of a graph G is a umeric quatity related to G which is ivariat uder automorphisms of G. The vertex PI polyomial is defied as PI v (G) euv u (e) v (e). The Omega polyomial (G,x) for coutig qoc strips i G is defied as (G,x) = c m(g,c)x c with m(g,c) beig the umber of strips of legth c. I this paper, a ew ifiite class of fullerees is costructed. The vertex PI, omega ad Sadhaa polyomials of this class of fullerees are computed for the first time. Keywords: Fulleree, vertex PI polyomial, Omega polyomial, Sadhaa polyomial. 1. INTRODUCTION Fullerees are molecules i the form of cage-like polyhedra, cosistig solely of carbo atoms. Fullerees F ca be draw for = 0 ad for all eve 4. They have carbo atoms, 3/ bods, 1 petagoal ad /-10 hexagoal faces. The most importat member of the family of fullerees is C 60 [1,]. Let be the class of fiite graphs. A topological idex is a fuctio Top from ito real umbers with this property that Top(G) = Top(H), if G ad H are isomorphic. Let G = (V,E) be a coected bipartite graph with the vertex set V = V(G) ad the edge set E = E(G), without loops ad multiple edges. The umber of vertices of G whose distace to the vertex u is smaller tha the distace to the vertex v is deoted by u (e). Aalogously, v (e) is the umber of vertices of G whose distace to the vertex v is smaller tha u. The vertex PI idex is a topological idex which is itroduced i [3]. It is defied as the sum of [ u (e) + v (e)], over all edges of a graph G. Let G be a arbitrary graph. Two edges e = uv ad f = xy of G are called codistat (briefly: e co f ) if they obey the

2 106 MODJTABA GHORBANI topologically parallel edges relatio. For some edges of a coected graph G there are the followig relatios satisfied [4,5]: e co e e co f f co e e co f, f co h e co h though the last relatio is ot always valid. Set C(e):= {f E(G) f co e}. If the relatio co is trasitive o C(e) the C(e) is called a orthogoal cut oc of the graph G. The graph G is called co-graph if ad oly if the edge set E(G) is the uio of disjoit orthogoal cuts. Let m(g,c) be the umber of qoc strips of legth c (i.e., the umber of cut-off edges) i the graph G, for the sake of simplicity, m(g,c) will hereafter be writte as m. Three coutig polyomials have bee defied [6-8] o the groud of qoc strips: c c ec (G, x) c m x, (G, x) c m c x ad (G, x) c m c x. (G, x) ad (G, x) polyomials cout equidistat edges i G while (G, x), o-equidistat edges. I a coutig polyomial, the first derivative (i x=1) defies the type of property which is couted; for the three polyomials they are: (G,1) c m.c E(G), (G,1) c m.c ad c (G,1) m.c.(e c). If G is bipartite, the a qoc starts ad eds out of G ad so (G, 1) = r /, i which r is the umber of edges i out of G. The Sadhaa idex Sd(G) for coutig qoc strips i G was defied by Khadikar et. al. [9,10] as Sd(G) cm(g,c)( E(G) c), where m(g,c) is the umber of strips of legth c. E c We ow defie the Sadhaa polyomial of a graph G as Sd(G, x) c m(g,c) x. By defiitio of Omega polyomial, oe ca obtai the Sadhaa polyomial by replacig x c with x E -c i omega polyomial. The the Sadhaa idex will be the first derivative of Sd(G, x) evaluated at x = 1. Herei, our otatio is stadard ad take from the stadard book of graph theory [11-17]. Example 1. Let C deotes the cycle of legth. x ( C, x ) x x ad Sd ( C, x ). 1 x Example. Suppose K deotes the complete graph o vertices. The we have:

3 Vertex PI, Omega ad Sadhaa Polyomials of F 1(+1) Fullerees 107 x x 1 ( ) ( K, x ) 1 x ad ( ) 1 ( Sd ( K, ) ) x x x. ( 1)( )/ x Example 3. Let T be a tree o vertices. We kow that E ( T ) 1. So, ( T, x ) ( T, x ) ( 1) x, Sd ( T, x ) ( T, x ) ( 1) x.. MAIN RESULTS AND DISCUSSION The aim of this sectio is to compute the coutig polyomials of equidistat (Omega, Sadhaa ad Theta polyomials) of a ifiite family F 1(+1) of fullerees with 1(+1) carbo atoms ad bods (the graph F 1(+1), Figure 1 is = 4). Theorem 4. The omega polyomial of fulleree graph F 1(+1) for is as follows: 1(+1) Ω(F, x ) 1x 1x 6x 3x. Proof. By figure 1, there are four distict cases of qoc strips. We deote the correspodig edges by f 1, f, f 3 ad f 4. By the table 1 proof is completed. Edge #Co distace Number of edges f f - 1 f f Table 1. The Number of Equidistat Edges. Corollary 5. The Sadhaa polyomial of fulleree graph F 1(+1) is as follows: 1(+1) Sd(F, x) 1x 1x 6x 3x. Now, we are ready to compute the vertex PI polyomial of fulleree graph F 1(+1). It is well-kow fact that a acyclic graph T does ot have cycles ad so u (e G) + v (e G) = V(T). Thus PI v (T) = V(T). E(T). Sice a fulleree graph F has 1 petagoal faces, PI v (F) < V(F). E(F). Let G be a coected graph. The PI v polyomials of G are defied as PI v (G;x) u (e G) v (e G) '.Obviously PI (G,1) PI (G) ad PI v (G,1) = euve(g) x v v

4 108 MODJTABA GHORBANI E(G). Defie N(e) = V ( u (e) + v (e)). The PI v (G) = [ V N(e)] V E N(e ad we have: e uv euv ) (e) (e) euve(g) euve(g) V(G) N(e) euve(g) PI (G, x) x u v x v x x. V(G) N(e) f f 1 f 3 f 4 Figure1.The graph of fulleree F 1(+1) for = 4. Example 6. Suppose F 30 deotes the fulleree graph o 30 vertices, see Figure. The PI v (F 30, x) = 10x x + 0x 6 + 5x 30 ad so PI v (F 30 ) = Figure. The Fulleree Graph F 30.

5 Vertex PI, Omega ad Sadhaa Polyomials of F 1(+1) Fullerees 109 Theorem 7. The vertex PI polyomial of fulleree graph F 1(+1) for is as follows: v 1(1) PI (F, x) 4x 1x 1x 6( - 3)x 4x 4x x 4x 4x 6(5 - )x. Proof. From Figures 3, oe ca see that there are te types of edges of fulleree graph F 1(+1). We deote the correspodig edges by e 1, e,,e 10. By table the proof is completed. Edge Number of vertex which are codistace from two eds of edges Num e 1 0 6(5-) e 1 e e e e e (-3) e e e Table. Computig N(e) for Differet Edges. e 5 e 10 e 1 e 4 e 6 e 4 e e 3 e 8 e 7 Figure 3. Types of Edges of Fulleree Graph F 1(+1).

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