Randić index, diameter and the average distance

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1 Radić idex, diameter ad the average distace arxiv: v1 [math.co] 9 Ju 009 Xueliag Li, Yogtag Shi Ceter for Combiatorics ad LPMC-TJKLC Nakai Uiversity, Tiaji , Chia lxl@akai.edu.c; shi@cfc.akai.edu.c Abstract The Radić idex of a graph G, deoted by R(G), is defied as the sum of 1/ d(u)d(v) over all edges uv of G, where d(u) deotes the degree of a vertex u i G. I this paper, we partially solve two cojectures o the Radić idex R(G) with relatios to the diameter D(G) ad the average distace µ(g) of a graph G. We prove that for ay coected graph G of order with miimum degree δ(g), if δ(g) 5, the R(G) D(G) +1 ; if δ(g) /5 ad 15, R(G) D(G) 3+ ad R(G) µ(g). Furthermore, for ay arbitrary real umber ε (0 < ε < 1), if δ(g) ε, the R(G) D(G) 3+ R(G) µ(g) hold for sufficietly large. Keywords: Radić idex; diameter; average distace; miimum degree AMS Subject Classificatio (000): 05C1, 05C35, 9E10. ad 1 Itroductio The Radić idex R(G) of a (molecular) graph G was itroduced by the chemist Mila Radić [8] i 1975 as the sum of 1/ d(u)d(v) over all edges uv of G, where d(u) deotes the degree of a vertex u i G, i.e., R(G) = 1. Recetly, may results o the extremal theory of the Radić d(u)d(v) uv E(G) idex have bee reported (see [6]). Supported by NSFC No , PCSIRT ad the 973 program. 1

2 Giveacoected, simpleadudirectedgraphg = (V,E)oforder. The distace betwee two vertices u ad v i G, deoted by d G (u,v) (or d(u,v) for short), is the legth of a shortest path coectig u ad v i G. The diameter D(G) of G is the maximum distace d(u,v) over all pairs of vertices u ad v of G. The average distace µ(g), a iterestig graph-theoretical ivariat, is defied as the average value of the distaces betwee all pairs of vertices of G, i.e., u,v V µ(g) = ( d(u,v). ) For termiology ad otatios ot give here, we refer to the book of Body ad Murty []. There are may results o the relatios betwee the Radć idex ad some other graph ivariats, such as the miimum degree, the chromatic umber, the radius, ad so o. I this paper, we will cosider the relatios of the Radić idex with the diameter ad the average distace. I [1], Aouchiche, Hase ad Zheg proposed the followig cojecture o the relatio betwee the Radić idex ad the diameter. Cojecture 1 ([1]) For ay coected graph of order 3 with Radić idex R(G) ad diameter D(G), R(G)D(G) +1 with equalities if ad oly if G = P. ad R(G) D(G) 3+, I [4], Fajtlowicz proposed the followig cojecture o the relatio betwee the Radić idex ad the average distace. Cojecture ([4]) For all coected graphs G, R(G) µ(g), where µ(g) deotes the average distace of G. I the followig, we will prove that for ay coected graph G of order with miimum degree δ(g), if δ(g) 5, the R(G) D(G) +1 ; if δ(g) /5 ad 15, R(G) 3+ D(G) ad R(G) µ(g). Furthermore, for ay arbitrary real umber ε (0 < ε < 1), if δ(g) ε, the R(G) 3+ D(G) ad R(G) µ(g) hold for sufficietly large.

3 Mai results At first, we recall some lemmas which will be used i the sequel. Lemma 1 (Erdös et al. [3]) Let G be a coected graph with vertices ad miimum degree δ(g). The D(G) 3 δ(g)+1 1. Lemma (Kouider ad Wikler [5]) If G is a graph with vertices ad miimum degree δ(g), the the average distace satisfyig µ(g) δ(g)+1 +. Lemma 3 (Li, Liu ad Liu [7]) Let G be a graph of order with miimum degree δ(g) = k. The R(G) k(k1) + k(k) if k (1) k(1) (p)(p1) (1) + p(p+k) k + p(p) if k > k(1) where p is a iteger give as follows: if 0 (mod 4) or if 1 (mod 4) ad k is eve if 1 (mod 4) ad k is odd p = or + if (mod 4) ad k is eve if (mod 4) ad k is odd or if 3 (mod 4) ad k is eve if 3 (mod 4) ad k is odd. It is easy to see from Lemma 3 that p is amog the umbers, 1,, +1 ad +. Lemma 4 Deote by g(,k) = k1 g(,k) k (1) k(1). The for 1 k /, Proof. If 3k, we ca directly obtai that g(,k) > 0. Now we assume that k < 3k. The ( ) k 1 3k g(,k) = = 1 k 1 1 ( 1 k 1 3 k + ). 1 1 k 3

4 Sice g(, k) for, we have ( 1 = k ) k k ( 1 < k k ) k < 0, k Therefore, the lemma follows. g(,k) > g(, ) = 1 0. Theorem 1 For ay coected graph G of order with miimum degree δ(g). (1) If δ(g) 5, the R(G)D(G) +1 ; () If δ(g) /5 ad 15, the R(G) 3+. Furthermore, for D(G) ay arbitrary real umber ε (0 < ε < 1), if δ(g) ε, the R(G) 3+ D(G) holds for sufficietly large. (3) If δ(g) /5 ad 15, the R(G) µ(g). Furthermore, for ay arbitrary real umber ε (0 < ε < 1), if δ(g) ε, the R(G) µ(g) holds for sufficietly large. Proof. Let G be a coected graph of order with miimum degree δ(g) = k. By Lemma 1, we have D(G) 3 k+1 1. (1) Suppose k 5, we will show R(G)D(G) +1. We cosider the followig two cases: Case 1. k. By Lemma 3, we oly eed to cosider the followig iequality, k(k 1) (1) + k(k) 3 k(1) k Let k(k 1) f(,k) = (1) + k(k) 3 k(1) k The by Lemma 4, f(,k) > k1 (1) + 3k k(1) f(,5) = (5) + 3 > (1) Case. < k 1. 4 > 0. If k 5, we have f(,k)

5 Let q(,p) = (p)(p1) + p(p+k) + p(p). I the followig, we will (1) k k(1) show that for every p {, 1,, +1, + }, I fact, if p =, deote by Notice that q(,p) 3 k h(,k) = q(, ) 3 k = (+) ()(k (+)) (1) 8k 4 k(1) 3 k h(, k) = 4 4 8k 8k k(1) + 3) (k +1) > 0, sice 8k 8k k(1), i.e., k 1. Thus, we have h(,k) > h(, ) = (+) 8(1) + 4 (1) > = By some calculatios, we have that > 0 for 8. For 4 7, it is easy to verify h(, ) > 0. I a similar way, we ca verify the iequality for each of the cases for p = 1,, +1 or +. The details are omitted. Let () Similarly, we cosider the followig two cases: Case 1. k. By Lemma 3, we oly eed to cosider the followig iequality, k(k 1) (1) + k(k) ( ) 3 3+ k(1) k f(,k) = k(k 1) (1) + k(k) ( ) 3 3+ k(1) k

6 The by Lemma 4, f(,k) > k1 (1) + 3k k(1) > 0. If k 5 ad 15, we have f(,k) f(, 5 ) = (5) (145)(3+ ) (1) 5 5(1) > 0. Actually, forayarbitrarypositiveumber ε(0 < ε < 1), (1)(+5) if k ε, the f(,k) > f(,ε) > ε(1ε) ( 3 > 0 for ε(1) sufficietly large. Case. < k 1. Let q(,p) = (p)(p1) (1) + p(p+k) k show that for every p {, 1,, +1, +}, ( ) 3 3+ q(,p) k ) 3+ ε+1 + k(1) p(p). I the followig, we will I fact, if p =, deote by h(,k) = q(, ( ) 3 3+ ) k +1 1 = (+) ()(k (+)) Notice that 8(1) ( 3 k +1 1 h(, k) 8k ) 3+. > 4 4 8k 8k k(1) > 0, sice 8k 8k k(1), i.e., k 1. Thus, we have 4 k(1) h(,k) > h(, ) = (+) 8(1) + 4 (1) ( ) > = (1). By some calculatios, we have that (1) (61) > 0 for 5. I a similar way, we ca verify the iequality for each of the cases for p = 1,, +1 or +. The details are omitted. 6

7 By the method similar to (), we ca obtai the result of (3). The proof is ow complete. Refereces [1] M. Aouchiche, P. Hase, M. Zheg, Variable eighborhood search for extremal graphs 19: Further cojectures ad results about the Radić idex, MATCH Commu. Math. Comput. Chem. 58 (007), [] J.A. Body, U.S.R. Murty, Graph Theory, Spriger, 008. [3] P. Erdös, J. Pach, R. Pollack, Z. Tuza, Radius, diameter, ad miimum degree, J. Combi. Theory Ser. B 47(1989), [4] S. Fajtlowicz, O cojectures of Graffiti, Discrete Math. 7(1988), [5] M. Kouider, P. Wikler, Mea distace ad miimum degree, J. Graph Theory 5 (1997) [6] X. Li, I. Gutma, Mathematical Aspects of Radić-Type Molecular Structure Descriptors, Mathematical Chemistry Moographs No.1, Kragujevac, 006. [7] X. Li, B. Liu, J. Liu, Complete solutio to a cojecture o Radić idex, Europea J. Operatioal Research (009), doi: /j.ejor [8] M. Radić, O characterizatio of molecular brachig, J. Amer. Chem. Soc. 97 (1975),