Kragujevac J. Sci. 33 (2011) UDC : ZAGREB POLYNOMIALS OF THORN GRAPHS. Shuxian Li

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1 Kragujevac J. Sci. 33 (2011) UDC : ZAGREB POLYNOMIALS OF THORN GRAPHS Shuxian Li Department of Mathematics, South China Normal University, Guangzhou , China (Received May 5, 2010) ABSTRACT. We give the expressions for the two Zagreb polynomials of the thorn graphs, which are then generalized to general form. 1 Introduction Let G be a simple graph with vertex set V (G) and edge set E(G). For u V (G), d G (u) or d u denotes the degree of u in G. The first Zagreb index and the second Zagreb index, introduced in [1] and elaborated in [2], are defined respectively as: ZG 1 (G) d 2 u ZG 2 (G) d u d v. Note that ZG 1 (G) may be written as ZG 1 (G) (d u + d v ). The Zagreb indices reflect the extent of branching of the molecular carbon-atom skeleton, and can thus be viewed as molecular structure descriptors [3,4]. Mathematical properties for these two graph invariants may be found in [5 12].

2 34 Recently, Fath Tabar [13] put forward the first and the second Zagreb polynomials of the graph G, defined respectively as ZG 1 (G, x) x du+dv ZG 2 (G, x) x d ud v where x is a dummy variable. For the graph G, we define ZG 0 p,q(g, x) x pd ud v +q(d u +d v ). Obviously, ZG 1 (G, x) ZG 0 0,1(G, x) and ZG 2 (G, x) ZG 0 1,0(G, x). Another polynomial related to the first Zagreb index is ZG 1(G, x) d u x d u. Define a polynomial as ZG 0 (G, x) x d u. For a graph G with vertex set {v 1,..., v n }, the thorn graph G (p 1,..., p n ) of G is obtained by attaching p i thorns (pendent edges) to vertex v i V (G) for i 1,..., n. We write G a,b G (p 1,..., p n ) if p i ad vi + b, where a and b are integers such that p i 1 for v i V (G). Let V 1 (E 1, respectively) be the set of pendent vertices (edges, respectively) of G a,b but not in G. Obviously, V 1 E 1 i V (G) (ad i + b) 2am + bn. These kinds of thorn graphs are well known in chemistry [14]. In this paper we give the expressions for the two Zagreb polynomials of the thorn graphs, and then generalize them. 2 Zagreb polynomials of thorn graphs Theorem 1. Let G be a graph with n vertices and m edges. Then ZG 1 (G a,b, x) x 2b ZG 1 (G, x a+1 ) + ax b+1 ZG v (G, x a+1 ) + bx b+1 ZG 0 (G, x a+1 ) ZG 1(G a,b, x) (a + 1)x b ZG v (G, x a+1 ) + bx b ZG 0 (G, x a+1 ) + (2am + bn)x ZG 2 (G a,b, x) x b2 ZG 0 a+1,b(g, x a+1 ) + ax b ZG v (G, x a+1 ) + bx b ZG 0 (G, x a+1 ).

3 35 Proof. For u V (G a,b ), d u denotes the degree of u in G a,b. If u V (G) V (G a,b ), then d u (a + 1)d u + b. For ZG 1 (G a,b, x), ZG 1 (G a,b) + x d u+d v + x d u+d v x [(a+1)d u+b]+[(a+1)d v +b],,v V 1 x (a+1)du+b+1 x (a+1)(d u+d v )+2b + (ad u + b)x (a+1)du+b+1 x 2b + ax b+1 + bx b+1 x (a+1)(du+dv) d u x (a+1)d u x (a+1)d u x 2b ZG 1 (G, x a+1 ) + ax b+1 ZG v (G, x a+1 ) + bx b+1 ZG 0 (G, x a+1 ). For ZG v (G a,b, x), ZG v (G a,b, x) d ux d u + + u V 1 1 x 1 [(a + 1)d u + b]x (a+1)d u+b (ad u + b)x (a + 1)x b d u x (a+1)d u + bx b x (a+1)du + (2am + bn)x (a + 1)x b ZG v (G, x a+1 ) + bx b ZG 0 (G, x a+1 ) + (2am + bn)x.

4 36 For ZG 2 (G a,b, x), ZG 2 (G a,b, x) + x d u d v + x d u d v x [(a+1)du+b][(a+1)dv+b],,v V 1 x [(a+1)dv+b] 1 x (a+1)2 d u d v +b(a+1)(d u +d v )+b 2 + (ad u + b)x (a+1)d u+b x b2 + ax b x (a+1)[(a+1)d ud v +b(d u +d v )] d u x (a+1)du + bx b x (a+1)du x b2 ZG 0 a+1,b(g, x a+1 ) + ax b ZG v (G, x a+1 ) + bx b ZG 0 (G, x a+1 ). The proof is completed. Similarly, ZG 0 (G a,b, x) x d u + x b u V 1 x 1 x (a+1)d u+b + (ad u + b)x x (a+1)d u + (2am + bn)x x b ZG 0 (G, x a+1 ) + (2am + bn)x. Theorem 2. Let G be a graph with n vertices and m edges. Then ZG 0 p,q(g a,b, x) x pb2 +2qb ZG 0 p(a+1),pb+q(g, x a+1 ) + ax pb+q(b+1) ZG v (G, x (a+1)(p+q) ) + bx pb+q(b+1) ZG 0 (G, x (a+1)(p+q) ). Proof. It is easily seen that ZG 0 p,q(g a,b, x) x pd ud v+q(d u+d v) + x pd ud v+q(d u+d v)

5 37 + x p[(a+1)2 d u d v +b(a+1)(d u +d v )+b 2 ]+q[(a+1)(d u +d v )+2b] x p[(a+1)du+b]+q[(a+1)du+b+1] x p(a+1)2 d u d v +(pb+q)(a+1)(d u +d v )+pb 2 +2qb + (ad u + b)x (p+q)(a+1)d u+pb+q(b+1) x pb2 +2qb x (a+1)[p(a+1)dudv+(pb+q)(du+dv)] + x pb+q(b+1) (ad u + b)x (p+q)(a+1)du x pb2 +2qb ZG 0 p(a+1),pb+q(g, x a+1 ) + ax pb+q(b+1) ZG v (G, x (a+1)(p+q) ) + bx pb+q(b+1) ZG 0 (G, x (a+1)(p+q) ) as desired. REFERENCES [1] I. Gutman, N. Trinajstić, Graph theory and molecular orbitals. Total π-electron energy of alternant hydrocarbons, Chem. Phys. Lett. 17 (1972) [2] I. Gutman, B. Ruščić, N. Trinajstić, C. F. Wilcox, Graph theory and molecular orbitals. XII. Acyclic polyenes, J. Chem. Phys. 62 (1975) [3] A. T. Balaban, I. Motoc, D. Bonchev, O. Mekenyan, Topological indices for structure-activity corrections, Topics Curr. Chem. 114 (1983) [4] R. Todeschini, V. Consonni, Handbook of Molecular Descriptors, Wiley VCH, Weinheim, [5] S. Nikolić, G. Kovačević, A. Miličević, N. Trinajstić, The Zagreb indices 30 years after, Croat. Chem. Acta 76 (2003) [6] I. Gutman, K. C. Das, The first Zagreb index 30 years after, MATCH Commun. Math. Comput. Chem. 50 (2004)

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