SHARP BOUNDS FOR THE GENERAL RANDIĆ INDEX R 1 OF A GRAPH

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1 ROCKY MOUNTAIN JOURNAL OF MATHEMATICS Volume 47, Number, 207 SHARP BOUNDS FOR THE GENERAL RANDIĆ INDEX R OF A GRAPH EI MILOVANOVIĆ, PM BEKAKOS, MP BEKAKOS AND IŽ MILOVANOVIĆ ABSTRACT Let G be an undirected simple, connected graph with n 3 vertices and m edges, with vertex degree sequence d d 2 d n The general Randić index is defined by R = d i d j (i,j) E Lower and upper bounds for R are obtained in this paper Introduction Let G = (V, E) be an undirected simple, connected graph with n 3 vertices and m edges, with vertex degree sequence d d 2 d n Denote by A the adjacency matrix of the graph G and by D the diagonal matrix of its vertex degrees Then L = I D /2 AD /2 is the normalized Laplacian matrix of G Its eigenvalues ρ ρ 2 ρ n = 0 are normalized Laplacian eigenvalues of graph G If ρ n 0, the graph G is connected, ie, it has only one component If ρ n k 0 and ρ n k+ = = ρ n = 0 for some k, k n, then the graph G has k connected components, see [2] The general Randić index R is defined [8, 9] by R = d i d j (i,j) E Here, d i and d j are the degrees of vertices i and j, respectively 200 AMS Mathematics subject classification Primary 05C50, 5A8 Keywords and phrases General Randić index, vertex degree sequence, normalized Laplacian spectrum (of graph) This research was supported by the Serbian Ministry of Education and Science Received by the editors on March 30, 205 DOI:026/RMJ Copyright c 207 Rocky Mountain Mathematics Consortium 259

2 260 MILOVANOVIĆ, BEKAKOS, BEKAKOS AND MILOVANOVIĆ The Randić index is an important molecular descriptor and has been closely related with many physico-chemical properties of alkanes, such as boiling points, surface areas, energy levels, etc For details on chemical applications of the general Randić index, see for example, [, 4, 5, 6, 3] For a survey of its mathematical properties and new results, see [3,, 2] Since the invariant R can be exactly determined for only a small number of graph classes, other methods for approximate calculation, asymptotic assessments, as well as inequalities that establish upper and lower bounds for R depending on other graph parameters are of interest In this paper, we are concerned with determining upper and lower bounds for R in terms of the number of vertices, number of edges, vertex degrees and extremal (the greatest and the smallest non-zero) normalized Laplacian eigenvalues 2 Preliminaries In what follows, we outline a few results of spectral graph theory and state a few analytical inequalities necessary for subsequent considerations In [3], Zumstein proved the following result: Lemma 2 ([3]) Let G be an undirected, simple graph of order n 2, with no isolated vertices Then (2) n ρ i = n and n ρ 2 i = n + 2R Lemma 22 ([6]) Let G be an undirected, simple graph of order n 2 with no isolated vertices Then (22) R n 2 d i Equality holds if and only if G is a k-regular graph, k n Lemma 23 ([5]) Let G be an undirected, simple graph of order n 2 with no isolated vertices Then n n (23) 2(n ) R 2

3 THE GENERAL RANDIĆ INDEX OF A GRAPH 26 with equality in the lower bound if G is a complete graph, and equality in the upper bound if and only if either (i) n is even and G is the disjoint union of n/2 paths of length, or (ii) n is odd and G is the disjoint union of (n 3)/2 paths of length and one path of length 2 In [0], also see [7], Rennie proved the following result: Lemma 24 ([0]) Let p, p 2,, p n be non-negative real numbers with the property p + p p n = Further, let a a 2 a n, be real numbers, and assume that there are r, R R such that Then (24) 0 < r a i R < +, for each i, i =, 2,, n n n p i p i a i + rr r + R a i Equality in equation (24) is obtained if and only if for some k, k n R = a = = a k a k+ = = a n = r Remark 25 Let us note that inequality (24) can be easily proved by induction or by maximizing the function on F (x, x 2,, x n ) = n n p i x i + rr {[x, x 2,, x n ] r x i R} p i x i 3 Main results We now obtain the lower bound for R in terms of the parameters n, ρ and ρ n

4 262 MILOVANOVIĆ, BEKAKOS, BEKAKOS AND MILOVANOVIĆ Theorem 3 Let G be an undirected, connected graph with n 2 vertices and m edges Then (3) R Equality holds if and only if G = K n n 2(n ) + 4 (ρ ρ n ) 2 Proof Let ρ ρ 2 ρ n > ρ n = 0 be the normalized Laplacian eigenvalues of the graph G Then (32) n ( n ) 2 (n ) ρ 2 i ρ i = i<j n (ρ i ρ j ) 2 n 2 ((ρ ρ i ) 2 + (ρ i ρ n ) 2 ) + (ρ ρ n ) 2 n 2 (ρ ρ n ) 2 + (ρ ρ n ) 2 2 = n (ρ ρ n ) 2 2 Bearing in mind Lemma 2 and the above inequality, we get n(n ) + 2(n )R n 2 n (ρ ρ n ) 2 2 By rearranging the above inequality we arrive at (3) Equality in (32) holds if and only if ρ = ρ 2 = = ρ n ; hence, the equality in (3) holds if and only if G = K n Remark 32 Since (ρ ρ n ) 2 0, inequality (3) is stronger than the left hand side inequality in (23)

5 THE GENERAL RANDIĆ INDEX OF A GRAPH 263 Our next result is the upper bound for R in terms of n, m, d, and d n Theorem 33 Let G be an undirected, simple graph of order n 2, with m edges and with no isolated vertices Then (33) R n(d + d n ) 2m 2d d n Equality holds if and only if G is a k-regular graph, k n Proof For p i = n, a i = d i, i =, 2,, n, r = d n and R = d, inequality (24) becomes (34) Since n n inequality (34) becomes n d i + d d n n n n d i = 2m, d i d + d n n(d + d n ) 2m d i d d n According to the above inequality and inequality (22) we obtain the desired result Equality in (34) holds if and only if d = = d k and d k+ = = d n, for some k, k n, and in equation (22) if and only if d = d 2 = = d n Consequently, equality in (33) holds if and only if G is a k-regular graph, k n

6 264 MILOVANOVIĆ, BEKAKOS, BEKAKOS AND MILOVANOVIĆ Remark 34 Inequality (33) and the right term in inequality (23) are incomparable It is easy to see that inequality (33) is stronger than the right term in inequality (23) if G is a k-regular graph, k n, or if G = K,n Moreover, inequality (33) is stronger than the right term in inequality (23) if n is even However, if n is odd and G is the union of paths of length and a path of length 2, then the right term in inequality (23) is stronger than that of inequality (33) In the following theorem, we establish the upper bound for R depending on the parameters n, m, d 2 and d n Theorem 35 Let G be an undirected, simple graph of order n 3, with m edges and with no isolated vertices Then: (35) R 2d 2 + (n )(d 2 + d n ) (2m n + ) 2d 2 d n Equality holds if and only if G = K n Proof For p i = n, a i = d i, i = 2,, n, r = d n and R = d 2, the inequality transforms into (36) ie, (37) Since n n n n p i p i a i + rr r + R a i n d i + d 2d n n n d i d 2 + d n, + (n )(d 2 + d n ) (2m d ) d i d d 2 d n d d 2 and 2m d 2m n +,

7 THE GENERAL RANDIĆ INDEX OF A GRAPH 265 from inequality (37), it follows that (38) n + (n )(d 2 + d n ) (2m n + ) d i d 2 d 2 d n From inequality (38) and inequality (22) we obtain inequality (35) Equality in (36) holds if and only if d 2 = = d k and d k+ = = d n, for some k, k n, and in inequality (22) if and only if d = d 2 = = d n Equality 2m d = 2m n + holds if and only if d = n This means that equality in (35) holds if and only if G = K n Acknowledgments The authors would like to give sincere gratitude to Prof Margaret Bayer for a careful reading of the manuscript and for valuable comments, which greatly improved the quality of our paper REFERENCES M Cavers, S Fallat and S Kirkland, On the normalized Laplacian energy and general Randić index R of graphs, Linear Alg Appl 433 (200), FRK Chung, Spectral graph theory, American Mathematical Society, Providence, LH Clark and JW Moon, On the general Randić index for certain families of trees, Ars Combin 54 (2000), X Li and Y Yang, Best lower and upper bounds for the Randić index R of chemical trees, MATCH Comm Math Comp Chem 52 (2004), , Sharp bounds for the general Randić index, MATCH Comm Math Comp Chem 5 (2004), M Lu, H Liu and F Tian, The conectivity index, MATCH Comm Math Comp Chem 5 (2004), DS Mitrinović and PM Vasić, Analytic inequalities, Springer Verlag, Berlin, M Randić, On characterization of molecular branching, J Amer Chem Soc 97 (975), , On the history of the Randić index and energing hostility toward chemical graph theory, MATCH Comm Math Comp Chem 59 (2008), 5 24

8 266 MILOVANOVIĆ, BEKAKOS, BEKAKOS AND MILOVANOVIĆ 0 BC Rennie, On a class of inequalities, J Austral Math Soc 3 (963), L Shi, Bounds on Randić indices, Discr Math 309 (2009), G Yu and L Feng, Randić index and eigenvalues of graphs, Rocky Mountain J Math 40 (200), P Zumstein, Comparision of spectral methods through the adjacency matrix and the Laplacian of a graph, Masters thesis, ETH, Zürich, 2005 A Medvedeva 4, Faculty of Electronic Engineering, PO Box 73, 8000 Niš, Serbia address: ema@elfakniacrs Xanthi University of Technology, Department of Electrical & Computer Engineering, Xanthi 6700, Thrace, Greece address: anairetis@gmailcom Democritus University of Thrace, Department of Electrical & Computer Engineering, Xanthi 6700, Thrace, Greece address: anairetis@gmailcom A Medvedeva 4, Faculty of Electronic Engineering, PO Box 73, 8000 Niš, Serbia address: igor@elfakniacrs