On the Geometric Arithmetic Index

Transcription

1 MATCH Communications in Mathematical and in Computer Chemistry MATCH Commun Math Comput Chem 74 (05) 03-0 ISSN On the Geometric Arithmetic Index José M Rodríguez a, José M Sigarreta b a Departamento de Matemáticas, Universidad Carlos III de Madrid, Avenida de la Universidad 30, 89 Leganés, Madrid, Spain jomaro@mathuc3mes b Facultad de Matemáticas, Universidad Autónoma de Guerrero, Carlos E Adame No54 Col Garita, Acalpulco Gro, México jsmathguerrero@gmailcom (Received September 30, 04) Abstract The concept of geometric-arithmetic index was introduced in the chemical graph theory recently, but it has shown to be useful The aim of this paper is to obtain new inequalities involving the geometric-arithmetic index GA and characterize graphs extremal with respect to them In particular, we improve some known inequalities and we relate GA to other well known topological indices Introduction A single number, representing a chemical structure in graph-theoretical terms via the molecular graph, is called a topological descriptor and if it in addition correlates with a molecular property it is called topological index; it is used to understand physicochemical properties of chemical compounds Topological indices are interesting since they capture some of the properties of a molecule in a single number Hundreds of topological indices have been introduced and studied, starting with the seminal work by Wiener [30] in which he used the sum of all shortest-path distances of a (molecular) graph for modeling physical properties of alkanes Topological indices based on end-vertex degrees of edges have been used over 40 years Among them, several indices are recognized to be useful tools in chemical researches

2 -04- Probably, the best know such descriptor is the Randić connectivity index (R) [5] There are more than thousand papers and a couple of books dealing with this index (see, eg, [3], [7], [8] and the references therein) During many years, scientists were trying to improve the predictive power of the Randić index This led to the introduction of a large number of new topological descriptors resembling the original Randić index The first geometric-arithmetic index GA, defined in [8] as GA (G) = (d u + d v ) where uv denotes the edge of the graph G connecting the vertices u and v, and d u is the degree of the vertex u, is one of the successors of the Randić index Although GA was introduced just five years ago, there are many papers dealing with this index There are other geometric-arithmetic indices, like Z p,q (Z 0, = GA ), but the results in [7, p598] show empirically that the GA index gathers the same information on observed molecules as other Z p,q indices The reason for introducing a new index is to gain prediction of some property of molecules somewhat better than obtained by already presented indices Therefore, a test study of predictive power of a new index must be done As a standard for testing new topological descriptors, the properties of octanes are commonly used We can find 6 physico-chemical properties of octanes at wwwmoleculardescriptorseu The GA index gives better correlation coefficients than Randić index for these properties, but the differences between them are not significant However, the predicting ability of the GA index compared with Randić index is reasonably better (see [7, Table ]) Although only about 000 benzenoid hydrocarbons are known, the number of possible benzenoid hydrocarbons is huge For instance, the number of possible benzenoid hydrocarbons with 35 benzene rings is [4] Therefore, the modeling of their physico-chemical properties is very important in order to predict properties of currently unknown species The graphic in [7, Fig7] (from [7, Table ], [7]) shows that there exists a good linear correlation between GA and the heat of formation of benzenoid hydrocarbons (the correlation coefficient is equal to 097) Furthermore, the improvement in prediction with GA index comparing to Randić index in the case of standard enthalpy of vaporization is more than 9% Hence, one can think that GA index should be considered in the QSPR/QSAR researches

3 -05- The aim of this paper is to obtain new inequalities involving the geometric-arithmetic index GA and characterize graphs extremal with respect to them In particular, we improve some known inequalities in Theorems 4, 37 and 30, and we relate GA to other well known topological indices in Section 3 Throughout this paper, G = (V (G), E(G)) denotes a (non-oriented) finite simple (without multiple edges and loops) connected graph with E(G) Note that the connectivity of G is not an important restriction, since if G has connected components G,, G r, then GA (G) = GA (G )+ +GA (G r ); furthermore, every molecular graph is connected Bounds for Geometric-Arithmetic Index We start with the following elementary result which allows to compute GA for many graphs Recall that a (, δ)-biregular graph is a bipartite graph for which any vertex in one side of the given bipartition has degree and any vertex in the other side of the bipartition has degree δ Proposition Let G be any graph Then the following statements hold: G is a regular graph if and only if GA (G) = m If G is a (, δ)-biregular graph, then GA (G) = m δ + δ If C n is the cycle graph with n vertices, then GA (C n ) = n If K n is the complete graph with n vertices, then GA (K n ) = ( n ) If Q n is the n-cube graph with n vertices, then GA (Q n ) = n n If K n,n is the complete bipartite graph with n, n vertices, then GA (K n,n ) = (n n ) 3/ n + n If S n is the star graph with n vertices, then GA (S n ) = (n )3/ n

4 -06- If W n is the wheel graph with n vertices, then GA (W n ) = n n + (n )3/ If P n is the path graph with n vertices, then GA (P ) =, GA (P n ) = n 3 + 4, if n 3 3 The double star graph S n,n is the graph consisting of the union of two star graphs S n+ and S n+ together with an edge joining their centers We have GA (S n,n ) = n n + n + + n n + n + + (n + )(n + ) n + n + If K n,,nk is the complete multipartite graph with n = n + + n k vertices, then k GA (K n,,nk ) = We will need the following result k i= j=i+ Lemma Let f be the function f(t) = n i n j (n ni )(n n j ) n n i n j t on the interval [0, ) Then f strictly +t increases in [0, ], strictly decreases in [, ), f(t) = if and only if t = and f(t) = f(t 0 ) if and only if either t = t 0 or t = t 0 Proof The statements follow from f (t) = ( t ) ( + t ) Corollary 3 Let g be the function g(x, y) = xy x+y ab g(x, y) a + b with 0 < a x, y b Then The equality in the lower bound is attained if and only if either x = a and y = b, or x = b and y = a, and the equality in the upper bound is attained if and only if x = y Besides, g(x, y) = g(x, y ) if and only if x/y is equal to either x /y or y /x Proof It suffices to apply Lemma, since g(x, y) = f(t) with t = x, and a t y b b a

5 -07- In [] and [8] (see also [7, p609-60]) appear the following inequalities: GA (G) (n )3/ n, GA (G) m n () Our next result provides a lower bound of GA (G) depending just on n and m, improving both inequalities in () Theorem 4 We have for any graph G GA (G) m n, n and the equality is attained if and only if G is a star graph Proof Recall that d u n for every u V (G) By Corollary 3, taking a = and b = n, we have GA (G) = d u d v (n ) n + = m n n By Corollary 3, the equality holds for G if and only if every edge joins a vertex of degree with a vertex of degree n, and this holds if and only if G is a star graph In [6] (see also [7, p609-60]) we find the bounds m δ + δ GA (G) m () In the papers [9] and [3] the authors obtain lower bounds of sum-connectivity and harmonic indices, respectively, depending just on n, for every graph with δ The following inequality provides a lower bound of GA (G) for every graph G with δ k, for any fixed k This result improves the first inequality in () Theorem 5 Consider any graph G with δ k () If n 0, then () If n, then GA (G) min GA (G) nk { nk } (k + ) k (n ) 3/, n + k

6 -08- Proof We have m = v V (G) We obtain from this inequality and () GA (G) m δ + δ where we consider the function for t [δ, n ] Since we have U (t) = 0 if and only if d v (n )δ + ( (n )δ + ) δ + δ = δ U( ), U(t) = ( (n )δ + t ) t t + δ = t3/ + (n )δt / t + δ U (t) = t + (4 n)δt + (n )δ t (t + δ), t = t ± = δ ( n 4 ± (n )(n 0) ) Hence, if n 0 then U (t) 0 for every t, and we conclude U(t) U(δ) for every t [δ, n ] Therefore, GA (G) δ U( ) δ U(δ) = nδ nk Assume now that n, then t, t + R and t < t + Since (n )(n 0) < n 6, we have δ = δ ( ) δ ( ) n 4 (n 6) < n 4 (n )(n 0) = t Furthermore, since n and δ, (n )(n 0) 3 and n δ We have two possibilities: ( ) δ ( ) n n 4 + (n )(n 0) = t+ (i) If t < n, then δ < t < n t +, U increases on [δ, t ] and decreases on [t, n ], and U(t) min { U(δ), U(n )} for every t [δ, n ] (ii) If n t, then δ n t, U increases on [δ, n ], and U(t) U(δ) = min { U(δ), U(n ) } for every t [δ, n ]

7 -09- Hence, in both cases GA (G) δ U( ) δ min { U(δ), U(n ) } = min min { nk } (k + ) k (n ) 3/, n + k { nδ } (δ + ) δ (n ) 3/, n + δ Remark 6 One can check that if k =, then { } { min n, 3 (n ) 3/ n, if n =, = 3 (n ) n + 3/, if n, n+ and that if k = 3, then { 3n min, 4 } 3 (n ) 3/ = n + { 3n, if n =,, 4 3 (n ) 3/ n+, if n 3 The study of Gromov hyperbolic graphs is a subject of increasing interest, both in pure and applied mathematics (see, eg, [], [], [3], [4], [0] and the references cited therein) We say that a graph is t-hyperbolic (t 0) if any side of every geodesic triangle is contained in the t-neighborhood of the union of the other two sides We define the hyperbolicity constant δ(g) of a graph G as the infimum of the constants t 0 such that G is t-hyperbolic We consider that every edge has length The following inequality relates the geometric-arithmetic index with the hyperbolicity constant δ(g) Theorem 7 We have for any graph G that is not a tree GA (G) (4δ(G) )3/ 4δ(G) Proof It is well known that if G is not a tree then δ(g) > 0 We have that δ(g) is always an integer multiple of by [, Theorem 6] and that δ(g) / {, } by [0, Theorem ], 4 4 since G has not loops or multiple edges Hence, δ(g) 3 4 The function f(x) = (x )3/ x f (x) = for every x (, ) We know by () that is increasing in [, ), since (x )/ x ( x + ) > 0 GA (G) (n )3/ n

8 -0- Since δ(g) n by [0, Theorem 30], we have n 4δ(G) 3 and 4 GA (G) (n )3/ n (4δ(G) )3/ 4δ(G) One can think that perhaps it is possible to obtain an upper bound of GA (G) in terms of δ(g), ie, the inequality GA (G) Ψ ( δ(g) ), for every graph G and some function Ψ However, this is not possible, as the following example shows For each integer d 3 consider two copies A d and B d of the path graph with (ordered) vertices a,, a d and b,, b d, respectively Let G d be the graph obtained from A d and B d by connecting with an edge the vertices a i and b i for every i {,, d} One can check that δ(g) = 3 for every d 3 However, lim d GA (G d ) = 3 Relations between GA and other well known topological indices In order to obtain relations between GA and other well known topological indices we need the following classical result, which provides a converse of Cauchy-Schwarz inequality (see [5, p6]) Lemma 3 If 0 < n a j N and 0 < n b j N for j k, then ( k j= a j ) / ( k j= b j ) / ( ) ( k ) N N n n + a j b j n n N N j= We will denote by M (G) and M (G) the first and the second Zagreb indices of the graph G, respectively, defined in [4] as M (G) = u V (G) d u, M (G) = d u d v These indices have attracted growing interest, see eg, [5], [4], [9] (in particular, they are included in a number of programs used for the routine computation of topological indices)

9 -- In [8] (see also [7, p6]) we find the bound mm (G) GA (G) (33) δ The following result gives a lower bound for GA similar to (33) Proposition 3 We have for any graph G GA (G) δ mm (G) + δ, and the equality is attained if and only if G is a regular graph Proof Since Lemma 3 gives GA (G) = δ d u d v, d u d v δ ( M (G) ) ( / + δ (d u + d v ) δ, ( ) / ( d ud v ( ) + δ δ ) / δ mm (G) + δ ) / 4 (d u+d v) If the graph is regular (ie, δ = ), then the lower and upper bound are the same, and they are equal to GA (G) If we have the equality, then 4( ) = for every uv E(G); hence, d u = for every u V (G) and the graph is regular We will use the following particular case of Jensen s inequality Lemma 33 If f is a convex function in R + and x,, x m > 0, then [5] as f ( x + + x ) m ( f(x ) + + f(x m ) ) m m As we have said, we will denote by R(G) the Randić index of the graph G, defined in R(G) = We recall that probably R is the best know topological index (see, eg, [3], [7], [8], [6] and the references cited therein) The following result provides lower and upper bounds of GA involving the Randić index

10 -- Theorem 34 We have for any graph G m R(G) GA (G) R(G), and the equality in each inequality holds if and only if G is regular Proof Since f(x) = /x is a convex function in R +, Lemma 33 gives m dud v (du+dv) m m GA (G) R(G) m (d u + d v ) m, If the equality holds, then ( ) = for every uv E(G) and we conclude d u = for every u V (G) In order to prove the upper bound note that GA (G) = (d u + d v ) (d u + d v ) = R(G) If the equality holds, then ( ) = for every uv E(G) and we conclude d u = for every u V (G) Reciprocally, if G is regular, then R(G) = m Hence, the lower and upper bound are the same, and they are equal to m = GA (G) Remark 35 If we replace the function f(x) = /x by the convex function f(x) = x in the proof of Theorem 34, we obtain the known inequality (33) We will need also the following lemma Lemma 36 We have for any graph G m M (G) Furthermore, the equality is attained if only if G is regular or biregular Proof Note that in the sum ( ) each term d u appears exactly d u times, since u is the endpoint of precisely d u edges Hence, ( ) = u V (G) d u = M (G),

11 -3- and Cauchy-Schwarz inequality gives m = du + d v du + d v = M (G) ( ) Furthermore, by Cauchy-Schwarz inequality, the inequality is attained if only if there exists a constant µ such that, for every uv E(G), du + d v = µ, = µ (34) If uv, uw E(G), then µ = = d u + d w, d w = d v, and we conclude that (34) is equivalent to the following: for each vertex u V (G), every neighbor of u has the same degree Since G is connected, this holds if and only if G is regular or biregular In [] (see also [7, p60]) appears the inequality GA (G) M (G) (35) Our next result improves this inequality and also gives a lower bound of GA involving the first Zagreb index Theorem 37 We have for any graph G δm M (G) GA (G) δ M (G) Furthermore, the equality in each inequality is attained if and only if G is regular Proof First of all we have GA (G) m = u V (G) d u u V (G) d u δ = δ M (G) If we have the equality, then GA (G) = m and G is regular If the graph is regular, then M (G) = nδ = mδ, GA (G) = m and the equality holds

12 -4- Lemma 36 gives GA (G) = d u d v δ δm M (G) If the equality holds, then d u d v = δ for every uv E(G); hence, d u = δ for every u V (G) and the graph is regular If G is regular, then M (G) = nδ = mδ, GA (G) = m and the equality holds M The following result gives a lower bound for GA involving the Zagreb indices M and Theorem 38 We have for any graph G GA (G) δm + δ M (G), M (G) and the equality is attained if and only if G is a regular graph Proof Since Lemmas 3 and 36 give GA (G) = δ d u d v, d u d v δ ( M (G) ) ( / δ, ( ) / ( d ud v ( ) + δ δ ) / d u+d v δm + δ + δ ) / 4 (d u+d v) M (G) M (G) If the equality holds, then (d u+d v ) = for every uv E(G); hence, d u = for every u V (G) If G is regular, then M (G) = n = m, M (G) = m, GA (G) = m and we have the equality We deal now with two additional topological descriptors, called harmonic and sumconnectivity index, defined respectively as H(G) =, S(G) = du + d v These indices have attracted a great interest in the lasts years (see, eg, [9], [0], [6], [9], [3], [3], [33] and [34]) Next, we relate them with the geometric-arithmetic index

13 -5- Proposition 39 We have for any graph G δh(g) GA (G) H(G), and the equality in each inequality is attained if and only if G is regular Proof We have δ, for every uv E(G) Hence, we obtain the inequalities by summing in uv E(G) The equality in the first (respectively, second) inequality is attained if and only if = δ for every uv E(G), ie, d u = δ (respectively, d u = ) for every u V (G) Reciprocally, if G is regular, then both bounds have the same value, and they are equal to GA (G) Next, we obtain inequalities relating the geometric-arithmetic index with the second Zagreb index Note that, since nδ m by the handshaking lemma, the upper bound improves the known inequality (33) Theorem 30 We have for any graph G δmm (G) nm (G) GA (G), + δ δ and the equality in each inequality is attained if and only if G is a regular graph Proof Since δ (d u + d v ), Lemma 3 gives ( GA (G) = ) / ( 4d ud v ) / (d u + d v ) (d u+d v) ( ) + δ δ δm / + δ d ud v = δmm (G) + δ In order to prove the upper bound, fix any function h Note that in the sum ( h(du )+ h(d v ) ) each term h(d u ) appears exactly d u times, since u is the endpoint of precisely d u edges Hence, ( h(du ) + h(d v ) ) = u V (G) d u h(d u ), ( d u + d v ) = u V (G) d u d u = n

14 -6- Therefore, Lemma 33 with f(x) = x gives Cauchy-Schwarz inequality gives GA (G) = d u d v (M (G)) / δ ( + ) = n d u d v d u d v / / nm (G) δ 4 ( ) If the graph is regular, then the lower and upper bound are the same, and they are equal to GA (G) If the equality holds in the lower bound, then 4( ) = for every uv E(G); hence, d u = for every u V (G) and the graph is regular If the equality is attained in the upper bound, then ( ) = δ for every uv E(G) and we conclude d u = δ for every u V (G) Theorem 3 We have for any graph G δs(g) GA (G) S(G) m and the equality in each inequality holds if and only if G is regular Proof Since f(x) = x is a convex function in R +, Lemma 33 gives δs(g) = δ ( ) / m m du + d v m m d u d v = m GA (G) If the equality holds, then d u d v = δ for every uv E(G) and we conclude d u = δ for every u V (G) In order to prove the upper bound note that GA (G) = d u d v d u + d v =, du + d v du + d v d u d v = d u d v du + d v du + d v du + d v = S(G) /

15 -7- If the equality holds, then = for every uv E(G) and we conclude d u = for every u V (G) Reciprocally, if G is regular, then S(G) = are the same, and they are equal to m = GA (G) Theorem 3 We have for any graph G GA (G) S(G) R(G), and the equality holds if and only if G is regular or biregular Proof Using the Cauchy-Schwarz inequality, we obtain S(G) = du + d v m Hence, the lower and upper bound = d u d v d u d v (d u d v ) /4 = GA (G)R(G) Hence, the equality is attained if and only if there exists a constant c such that for every uv E(G) d u d v If uv, uw E(G), then = c (d u d v ), d ud /4 v = c ( ), c = + = +, d w = d v, d u d v d u d w c = d u + d v (36) and we conclude that (36) is equivalent to the following: for each vertex u V (G), every neighbor of u has the same degree Since G is connected, this holds if and only if G is regular or biregular The modified Narumi-Katayama index NK = NK (G) = u V (G) d du u = is introduced in [], inspired in the Narumi-Katayama index defined in [3] (see also [], []) Next, we prove an inequality relating the modified Narumi-Katayama index with the geometric-arithmetic index d u d v

16 -8- Theorem 33 We have for any graph G GA (G) m NK (G) /(m), and the equality holds if and only if G is regular Proof Using the fact that the geometric mean is at most the arithmetic mean, we obtain m GA (G) = m m d u d v /m d u d v /m = NK (G) /(m) If the equality holds, then ( ) = for every uv E(G); hence, d u = for every u V (G) and the graph is regular If the graph is regular, then NK (G) = m and the equality holds Acknowledgement: Supported in part by a grant from Ministerio de Economía y Competititvidad (MTM ), Spain, and by a grant from CONACYT (CONACYT-UAG I00/6/0), México References [] S Bermudo, J M Rodríguez, J M Sigarreta, Computing the hyperbolicity constant, Comput Math Appl 6 (0) [] S Bermudo, J M Rodríguez, J M Sigarreta, E Tourís, Hyperbolicity and complement of graphs, Appl Math Lett 4 (0) [3] S Bermudo, J M Rodríguez, J M Sigarreta, J M Vilaire, Gromov hyperbolic graphs, Discr Math 33 (03) [4] G Brinkmann, J Koolen, V Moulton, On the hyperbolicity of chordal graphs, Ann Comb 5 (00) 6 69 [5] K C Das, On comparing Zagreb indices of graphs, MATCH Commun Math Comput Chem 63 (00) [6] K C Das, On geometric arithmetic index of graphs, MATCH Commun Math Comput Chem 64 (00)

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