Apr Vol.26 No.2. Pure and Applied Mathematics O157.5 A (2010) (d(u)d(v)) α, 1, (1969-),,.

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1 Pure and Applied Matheatics Apr Vol.26 No.2 Randić 1, 2 (1., ; 2., ) G Randić 0 R α (G) = v V (G) d(v)α, d(v) G v,α. R α,, R α. ; Randić ; O157.5 A (2010) G = (V (G), E(G)) V (G) E(G) G. α, G Randić ( ) R α = R α (G) = (d(u)d(v)) α, uv E(G) d(u) u, (d(u)d(v)) α uv., R 1 Randić 2, Randić 1975 [1],, [2,10-11]. 1977,Kier Hall [12] Randić, Randić, 0 R(G) = v V (G) d(v) 1 2., Li Zheng [13] Randić, 0 R α = 0 R(G) = v V (G) d(v)α, α. Randić,, [14-17]., 1 α < 0 0 < α < 1, [18] Randić. α, : α, Randić,. 2. G = (V (G), E(G)), V (G) E(G) G, N(v) d(v) v, P (n) n. M ( ), (JA08266), ( ). (1969-),,.

2 G, M G, M G, G. 1, 1, S 1,n 1 n, ( v n 1 ). T (n, ) n (n 2). F (i) k(k 2), S 1,k F ; (ii) T F, k(k 2), T S 1,k F, T S 1,k S 1,k T F., F (n, ) F n. F (n, ) S 1,k, S 1,k v k k( d(v k ) = k). T F (n, ),, n = V (T ) = (d(v k ) 1) = d(v k ) + 1 = k + 1, k = n 1, 0 R α (T ) =2 + (d(v k ) 2) + ( 1)2 α + =(n 1) + ( 1)(2 α 2) + =(n 1) + ( 1)(2 α 2) + d(v k ) α d(v k ) α (n 1)α α. T 0 (n, ) 1 S 1,n 1. T 0 (n, ) n., n = 2, T 0 (2, ) 2., 0 R α (T 0 (n, )) = (n ) + ( 1)2 α + (n ) α. 3 T T (n, ), (i) α = 0, 0 R α (T ) = d(v) α = n, n T ; (ii) α = 1, 0 R α (T ) = v V (T ) v V (T ) d(v) α = 2e, e T., α, α (0, 1) α (, 0) (1, + ) f(x) = x α (x 1) α (2 α 1), x 2, (i) f(x) 0, α (0, 1) ; (ii) f(x) 0, α (, 0) (1, + ); (i) (ii) x=2.

3 2 : Randić 341, x = 2, f(x) = 0. x 3,, ξ 1 (x 1, x), x α (x 1) α = αξ1 α 1, ξ 2 (1, 2), (2 α 1) = αξ2 α 1, 1 < ξ 2 < 2 (x 1) < ξ 1 < x.,f(x) = α(ξ1 α 1 ξ2 α 1 ), ξ (ξ 2, ξ 1 ), f(x) = α(α 1)ξ α 2 (ξ 1 ξ 2 ),,ξ α 2 (ξ 1 ξ 2 ),, α (0, 1), f(x) < 0 α (, 0) (1, + ), f(x) > 0., , 2, dg(x) dg(x) 1 g(x) = ( 1)(n x 1)α ( 1) α (i) g(x) 0, α (0, 1) ; (n 1)α α (ii) g(x) 0, α (, 0) (1, + ); (i) (ii) x = n 1, x = n 1 + x α, 0 < x < (n 1) n., g(x) = 0 g(x) α( 1)(n x 1)α 1 = ( 1) α + αx α 1 = α((x x)α 1 (n x 1) α 1 ) ( 1) α 1, = 0, x = n 1 n 1 n 1, (0, ) ( α (0, 1), 0 < x < n 1, g(x) n 1 dg(x) g(x) , n 1). dg(x), 0 < (x x) < (n x 1), > 0, < x < n 1, 0 < (n x 1) < (x x), < 0,, g(x), α (0, 1), 0 < x < (n 1), 3 [19] α (, 0) (1, + ),, 0 < x < (n 1), g(x) 0., 2. 4 [19] T n(n > 2), T, T n(n > 2) (n > 2), T - M v, M v. 5 φ(x) = x α (x 1) α (p α (p 1) α ), p 3, x 2 p, (i) φ(x) 0, α (0, 1) ; (ii) φ(x) 0, α (, 0) (1, + ); (i) (ii) x = p., x = p, φ(x) = 0. 2 x p 1,, ξ 1 (x 1, x), x α (x 1) α = αξ1 α 1, ξ 2 (p 1, p), (p α (p 1) α ) = αξ2 α 1, (x 1) < ξ 1 < x (p 1) < ξ 2 < p.,φ(x) = α(ξ1 α 1 ξ2 α 1 ), ξ (ξ 1, ξ 2 ), φ(x) = α(α 1)ξ α 2 (ξ 1 ξ 2 ), ξ α 2,(ξ 1 ξ 2 ),, α (0, 1), φ(x) > 0 α (, 0) (1, + ), φ(x) < 0., 5. 1 T T (n, ). α (0, 1), n ) + 2 α ( 1) + (n ) α 0 R α (T ) (n 1) + ( 1)(2 α 2) + (n 1)α α (1)

4 T = T 0 (n, ); T F (n, ) k = n 1 ; α (, 0) (1, ), (n 1) + ( 1)(2 α 2) + (n 1)α α 0 R α (T ) (n ) + 2 α ( 1) + (n ) α (2) T F (n, ) k = n 1 ; T = T 0 (n, ). (1). T T (n, ), = 1, T, (1). 2,. T T (n, ), u V (T ), N(u)={u 0, u 1, u 2,, u t 1 }(t 2),d(u 0 ) 2 d(u i ) = 1(i = 1, 2,, t 1)., T uu 0 = T T u, T T (n t, 1) V (T u ) = t., d(u 0 ) = d( 2). 0 < α < 1,,, 0 R α (T ) = 0 R α (T ) + t α + (t 1) + d α (d 1) α n t 1 + ( 2)(2 α 2) + = (n 1) + ( 1)(2 α 2) + t α + d α (d 1) α (2 α 1) (n 1) + ( 1)(2 α 2) + ( 1)(n t 1)α ( 1) α + t α + t 1 + d α (d 1) α (n 1)α ( 1)(n t 1)α (n 1)α α + ( 1) α α + (n 1)α α., T F (n t, 1) k = n t 1 1 ; 1 2, d = 2 t = n 1 n 1. t = k = n t 1 1,, k = n 1., (1) T F (n, ) k = n 1. (1)., T = T 0 (n, ),. T T (2, ), (1) T = T 0 (2, ). = 1, 2, T 0 (2, ) = P (2)., (1). 3,, T 2 ( T T (2, )), 3, T v 2 w, T u( v), uw E(T ), 2 d(u). T = T v w, T T (2( 1), 1). 0 < α < 1, d(u) = d,, 0 R α (T ) = 0 R α (T ) + 2 α d α (d 1) α ( 1) + 2 α ( 2) + ( 1) α + 2 α d α (d 1) α = + 2 α ( 1) + α α + ( 1) α + d α (d 1) α + 2 α ( 1) + α.

5 2 : Randić 343, T = T 0 (2( 1), 1); 5, d = ( d(u) = ).,(1) T = T 0 (2, ). n > 2, n n. 4, T - M v, M v. uv E(T ), 2 d(u) n. T = T v, T T (n 1, ). 0 < α < 1, d(u) = d,, 0 R α (T ) = 0 R α (T ) d α (d 1) α (n 1) + 2 α ( 1) + (n 1) α d α (d 1) α = (n ) + 2 α ( 1) + (n ) α (n ) α + (n 1) α + d α (d 1) α (n ) + 2 α ( 1) + (n ) α., T = T 0 (n 1, ); 5, d = (n ) ( d(u) = n )., n > 2, (1) T = T 0 (n, )., T T (n, )(n 2), (1) T = T 0 (n, )., 1 (1). 1 (1) 1 2, (2), (2) T F (n, ) k = n 1. 1 (1) 3, 4 5, (2), (2) T = T 0 (n, )., 1. [1] Randić M. On characterization of olecular branching[j]. J. Aer. Che. Soc., 1975,97: [2] Hansen P, Mélot H. Variable neighborhood search for extreal graphs 6: Analyzing bounds for the connectivity index[j]. J. Che. Inf. Coput. Sci., 2003,43:1-14. [3] Caporossi G, Gutan I, Hansen P, Pavlović L. Graphs with axiu connectivity index[j]. Coput. Biol. Che., 2003,27: [4] Caporossi G, Gutan I, Hansen P. Variable neighborhood search for extreal graphs IV: Cheical trees with extreal connectivity index[j]. Coput. Che., 1999,23: [5] Gutan I, Miljković O, Caporossi G, et al. Alkanes with sall and large Randić connectivity indices[j]. Che. Phys. Lett., 1999,306: [6] Araujo O, de la Peña J. A. Soe bounds for the connectivity index of a cheical graph[j]. J. Che. Inf. Coput. Sci., 1998,38: [7] Bollobás B, Erdös P. Graphs of extreal weights[j]. Ars. Cobin., 1998,50:

6 [8] Liu H, Pan X, Xu J. On the Randić index of unicyclic conjugated olecules[j]. J. Math. Che., 2006,40: [9] Zhang L, Lu M, Tian F., Maxiu Randić index on trees with K-pendant vertices[j]. J. Math. Che., 2007,41: [10] Chen X D, Qian J G,Conjugated trees with iniu general Randić index[j], Discrete Applied Matheatics. 2009,157: [11] Li X, Gutan I. Matheatical Aspects of Randić-Type Molecular Structure Descriptors[M]. Kragujevac, [12] Kier L B, Hall L H. The nature of structure-activity relationships and their relation to olecular connectivity[j]. Europ. J. Med. Che., 1977,12: [13] Li X, Zheng J. An unified approach to the extreal trees for different indices[j]. Math. Coput. Che., 2005,51: [14] Pavlović L. Maxial value of the zeroth-order Randić index[j]. Discr. Appl. Math., 2003,127: [15] Hu Y, Li X, Shi Y, Gutan I. On olecular graphs with sallest and greatest zeroth-order general Randić index[j]. MATCH Coun. Math. Coput. Che., 2005,54: [16] Zhang S, Zhang H. Unicyclic graphs with the first three sallest and largest first general Zagreb index[j]. Match. Coun. Math. Coput. Che., 2006,55: [17] Hua H, Lin M, Wang H. On zeroth-order general Randić index of conjugated unicyclic graphs[j]. J. Math. Che., 2007,43: [18],. Randić [J]. :, 2008,47: [19] Hou Y, Li J. Bounds on the largest eigenvalues of trees with a given size of athing[j]. Lin. Algebra Appl., 2002,342: Extreal trees with repected to zeroth-order general Randić index LIN Qi-fa 1, QIAN Jian-guo 2 (1.Departent of Matheatics, Ningde Teachers College, Ningde 2.School of Matheatical Sciences, Xiaen University, Xiaen , China; , China) Abstract: The zeroth-order general Randić index of a graph G is defined by 0 R α(g) = v V (G) d(v)α, where α is a real nuber and d(v) is the degree of v. In this paper, an upper bound and a lower bound for the trees with given size of the axiu atching are deterined, respectively. Further, the corresponding extreal trees are characterized. Keywords: tree, zeroth-order general Randić index, axiu atching 2000MSC: 05C70