Topological indices of Kragujevac trees

Proyecciones Journal of Mathematics Vol. 33, No 4, pp. 471-482, December 2014. Universidad Católica del Norte Antofagasta - Chile Topological indices of Kragujevac trees Roberto Cruz Universidad de Antioquia, Colombia Iván Gutman University of Kragujevac, Serbia and Juan Rada Universidad de Antioquia, Colombia Received : May 2014. Accepted : November 2014 Abstract We find the extremal values of the energy, the Wiener index and several vertex-degree-based topological indices over the set of Kragujevac trees with the central vertex of fixed degree. 2010 Mathematics Subject Classification : 05C90, 05C35. Key Words : Topological indices; Kragujevac trees. 472 Roberto Cruz, Ivan Gutman and Juan Rada 1. Introduction The class of Kragujevac trees emerged in several studies addressed to solve the problem of characterizing the tree with minimal atom-bond connectivity index [6,11,12]. In this work, we are interested in finding the maximal and minimal value of several topological indices over a subset of Kragujevac trees of order n. Let G be a simple graph on n vertices. By uv we denote the edge connecting the vertices u and v. A vertex of degree one is referred to as a pendent vertex. An edge whose one end-vertex is pendent is referred to as a pendent edge. The formal definition of a Kragujevac tree was introduced in [15]: Let B1 , B2 , B3 , . . . be branches whose structure is depicted in Figure 1. A proper Kragujevac tree is a tree possessing a central vertex of degree at least 3, to which branches of the form B1 and/or B2 and/or B3 and/or . . . are attached. The set of all proper Kragujevac trees of order n will be denoted by Kgn . An improper Kragujevac tree is a tree obtained by inserting a new vertex (of degree 2) on a pendent edge of a proper Kragujevac tree. The set of all improper Kragujevac trees of order n will be denoted by Kg∗n . Figure 1 : The branches of Kragujevac trees Let K̃gn,m be the class of proper Kragujevac trees of order n with the central vertex of degree m, and no branches of type B1 . In this case, the values of n and m have different parity and are related by the formula P n−1 n=m+1+2 m i=1 ki , where 3 ≤ m ≤ 5 . The branches attached to the central vertex are of the form Bk1 , Bk2 , . . . Bkm where ki ≥ 2 for all i = 473 Topological indices of Kragujevac trees 1, . . . , m. An interesting property of this class is the fact that from any specific tree one can obtain any other tree of the family by moving pendent edges from one branch to another. Given two branches Bki and Bkj , the gap between them is defined as |ki − kj | and the gap of a tree T ∈ K̃gn,m is defined as g(T ) = max |ki − kj |. 1≤i<j≤m The tree with maximal gap is the tree, denoted by U , with m − 1 branches of type B2 and one branch of type Bh , where h = n+3−5m 2 Figure 2 : Trees of K̃gn,m with maximal and minimal gaps On the other hand, the tree with minimal gap is the tree, denoted by V , in which all the branches Bk1 , Bk2 , . . . , Bkm satisfy the condition of |ki − kj | ≤ 1. In this case g(V ) ≤ 1 and V has k l type Bmh1 and j m1 branches n−1−m , h2 = n−1−m and m − m1 branches of type Bh2 , where h1 = 2m 2m Figure (2). Here btc denotes the greatest integer less m1 = (3+2h1 )m−(n−1) 2 than t and dte denotes the lowest integer greater than t. Note that if a specific topological index is monotone with respect to the variation of gaps between branches, one extremal value of this index over K̃gn,m is attained in V and the other extremal value is attained in U . One of the indices which is monotone with respect to the variation of gaps in K̃gn,m is the energy. Given a graph G, its characteristic polynomial, denoted by ΦG , is defined as the characteristic polynomial of the adjacency matrix A of G, i.e., ΦG = det(x I − A), where I is the n × n identity matrix. The eigenvalues of G are the eigenvalues of A. Note that for a graph G, its 474 Roberto Cruz, Ivan Gutman and Juan Rada adjacency matrix A is a real symmetric matrix and therefore the eigenvalues of G are real numbers λ1 , . . . , λn . The energy of G is denoted by E(G) and defined as E(G) = n P i=1 |λi |. This concept was introduced in [9] and it is related to the total π-electron energy in a molecule represented by a (molecular) graph. For the chemical applications and the mathematical properties of the energy of a graph we refer, respectively, to the review [10] and the recent book [16]. In Theorem 2.1 we find that the maximal value of the energy over K̃gn,m is attained in the tree V while the minimal value of the energy is attained in U . The Wiener index also appears to be monotone with respect to the variation of gaps in K̃gn,m . Let u and v be two vertices of the connected graph G. The distance between u and v is the length of a shortest path connecting u and v. The Wiener index of the graph G, denoted by W (G), is the sum of distances between all pairs of vertices of G. The Wiener index of trees has been extensively studied (see review [2]). In the case of K̃gn,m we find that the maximal value of the Wiener index K̃gn,m is attained in the tree V while the minimal value of the Wiener index is attained in U (see Theorem 3.1). In the last section of this work, we find the extremal values of several vertex-degree-based topological indices over the set K̃gn,m . We consider indices of the form ([5], [8], [14]) : T I = T I (G) = (1.1) X mij Ψ (i, j) 1≤i≤j≤n−1 where mij is the number of edges of the graph G connecting a vertex of degree i with a vertex of degree j. For instance, the following special cases of the function Ψ (x, y) have been extensively used: Ψ (x, y) = √1xy , for q the Randić index [17] ; Ψ (x, y) = x+y−2 xy , for the atom-bond connectivity (ABC) index [3] ; Ψ (x, y) = x+y, for the first Zagreb index [13] ; Ψ (x, y) = √ 2 xy xy, for the second Zagreb index [13] ; Ψ (x, y) = x+y , for the geometric1 , for the sum-connectivity index [21] arithmetic index [19] ; Ψ (x, y) = √x+y ; Ψ (x, y) = 2 x+y , (xy)3 , (x+y−2)3 for the augmented Zagreb index [4] ; and Ψ (x, y) = for the harmonic index [22]. 475 Topological indices of Kragujevac trees In Theorem 4.1 we find that the maximal value of the Randić index, the geometric-arithmetic index, the sum-connectivity index and the harmonic index over K̃gn,m is attained in V while the minimal value of these indices is attained in U . Conversely, the maximal value of the atom-bondconnectivity index, the first Zagreb index and the second Zagreb index is attained in U and the minimal value is attained in V. 2. Extremal values of the energy over K̃gn,m In order to determine the maximal and minimal value of the energy over K̃gn,m , we use the method to compare the energy of bipartite graphs consisting in defining a quasi-order relation which depends on the expression for the characteristic polynomial (see [7]). More precisely, G is a bipartite graph if and only if (2.1) n ΦG = x + n bX 2c (−1)k c2k (G) xn−2k k=1 where c2k (G) is a nonnegative integer for every k = 1, . . . , can define the relation (2.2) ¥n¦ 2 . Then we G1 ¹ G2 if and only if c2k (G1 ) ≤ c2k (G2 ) ¥ ¦ for every k = 1, . . . , n2 . If, in addition, there exists a k such that c2k (G1 ) < c2k (G2 ) then we write G1 ≺ G2 . Now, based on Coulson’s integral formula [1], it can be shown [7] that for a bipartite graph G, the energy is increasing with respect to the quasi-order relation. In other words, if G1 ≺ G2 then E(G1 ) < E(G2 ). Theorem 2.1. The maximal (resp. minimal) value of the energy over K̃gn,m is attained in the tree V (resp. U ). Proof. Let X ∈ K̃gn,m with branches Bk , Bl where k > l. Let Y ∈ K̃gn,m obtained from X replacing the branch Bk by Bk−1 and the branch Bl by Bl+1 (see Figure 3). 476 Roberto Cruz, Ivan Gutman and Juan Rada Figure 3 : Kragujevac trees used in the proof of Theorem 2.1 Denote by ΦX the characteristic polynomial of the tree X. Removing the edge ab using Schwenk’s theorem [18] we have ΦX = ΦX−ab −ΦX−a−b = (x2 − 1)ΦZ − x(x2 − 1)k−1 ΦW , where Z is the tree obtained from X replacing the branch Bk by Bk−1 and W is the tree obtained from X removing branch Bk . Now, removing the edge cd from W we obtain (2.3) h i ΦX = (x2 − 1)ΦZ − x(x2 − 1)k−1 ΦT ΦBl − (x2 − 1)l ΦT 0 , where T is the tree obtained from X removing the branches Bk , Bl and T 0 is obtained from T removing the central vertex c. If we apply the same procedure to the tree Y , removing first the edge 0 0 a b we have ΦY = (x2 − 1)ΦZ − x(x2 − 1)l ΦW 0 , where W 0 is the tree obtained from Y removing branch Bl . Now, removing the edge c0 d0 from W 0 we obtain the following expression for the characteristic polynomial of Y : h i (2.4) ΦY = (x2 − 1)ΦZ − x(x2 − 1)l ΦT ΦBk−1 − (x2 − 1)k−1 ΦT 0 . It is easy to see that the characteristic polynomial of a branch Bs is ΦBs = ¡ ¢s−1 ¡ 2 ¢ x − s − 1 ; then replacing the characteristic polynomial of x x2 − 1 Bl in (2.3), the characteristic polynomial of Bk−1 in (2.4) and taking the difference ΦX − ΦY we obtain ΦX − ΦY = (l − k + 1)x2 (x2 − 1)k+l−2 ΦT . Since T is a tree, i.e. bipartite graph, the polynomial ΦT is of the form bip. It implies that the polynomial in the previous equation is also of the form bip and its first coefficient is negative because l < k − 1. Then, using the quasiorder relation qo we have that X ≺ Y , and E(X) < E(Y ). Topological indices of Kragujevac trees 477 It means that when we have a tree in K̃gn,m with two branches with gap greater that one, we can construct another tree in the same class with greater value of the energy, by reducing the gap between branches, and with less value of the energy, by increasing the gap between branches. Then, the tree with maximal energy is V while the tree with minimal energy is U . 2 3. Extremal values of the Wiener index over K̃gn,m Theorem 3.1. The maximal (respectively minimal) value of the Wiener index over K̃gn,m is attained in the tree V (respectively U ). Proof. Let X and Y be the trees in K̃gn,m depicted in Figure 3. To find the difference W (X) − W (Y ) we use the following expression for a Wiener index of a tree [20] : P W(T)= n1 (e)n2 (e), where T is a tree, summation goes for all edges of e T and n1 (e) and n2 (e) are the numbers of vertices lying on the two sides of the edge e. Hence, in the difference W (X) − W (Y ) we have only to consider the edges ac and cd of X and the edges d0 c0 and c0 a0 of Y (see Figure 3): W (X) − W (Y ) = (2k + 1)(n − 2k − 1) + (2l + 1)(n − 2l − 1) −(2k − 1)(n − 2k + 1) − (2l + 3)(n − 2l − 3) = 8l − 8k + 8 = 8(l − k + 1) < 0. It means that when we have a tree in K̃gn,m with two branches with gap greater that one, we can construct another tree in the same class with greater value of the Wiener index, by reducing the gap between branches, and with less value of the Wiener index, by increasing the gap between branches. Then, the tree with maximal Wiener index is V while the tree with minimal Wiener index is U . 2 4. Extremal values of some vertex-degree-based topological indices over K̃gn,m Theorem 4.1. The maximal (respectively minimal) value of the Randić index, the geometric-arithmetic index, the sum-connectivity index and the harmonic index over K̃gn,m is attained in the tree V (respectively U ). 478 Roberto Cruz, Ivan Gutman and Juan Rada The maximal (respectively minimal) value of the atom-bond-connectivity index, the first Zagreb index and the second Zagreb index over K̃gn,m is attained in the tree U (respectively V ). Proof. Let X and Y be the trees in K̃gn,m depicted in Figure 3. Using the expresion gen for the general vertex-degree-based topological index we have: ∆ = T I(X) − T I(Y ) = [Ψ(k + 1, m) + Ψ(l + 1, m) + kΨ(k + 1, 2) + lΨ(l + 1, 2)] − [Ψ(k, m) + Ψ(l + 2, m) + (l + 1)Ψ(l + 2, 2) + (k − 1)Ψ(k, 2)] = [Ψ(k + 1, m) − Ψ(k, m) + kΨ(k + 1, 2) − (k − 1)Ψ(k, 2)] − [Ψ(l + 2, m) − Ψ(l + 1, m) + (l + 1)Ψ(l + 2, 2) − lΨ(l + 1, 2)] = F (k) − F (l + 1) where F(x)=Ψ(x + 1, m) − Ψ(x, m) + xΨ(x + 1, 2) − (x − 1)Ψ(x, 2).For k = l + 1, T I(X) − T I(Y ) = 0. Assume that k > l + 1. Note that for each of the indices considered in this theorem, for a fixed y the function Ψ is continuous and differentiable as a function of x, for x > 0. It means that the function F (x) is continuous in [l + 1, k] and differentiable in (l + 1, k). By the mean value theorem there exists ξ ∈ (l + 1, k) such that ∆ = F (k) − F (l + 1) = F 0 (ξ)(k − l − 1) ¸ ∙ ∂Ψ ∂Ψ (ξ + 1, m) + Ψ(ξ + 1, 2) + ξ (ξ + 1, 2) (k − l − 1) = ∂x ∂x ∙ ¸ ∂Ψ ∂Ψ (ξ, m) + Ψ(ξ, 2) + (ξ − 1) (ξ, 2) (k − l − 1) − ∂x ∂x = [G(ξ + 1) − G(ξ)] (k − l − 1) where ∂Ψ Now, since for all the considered inG(x)= ∂x(x,m)+Ψ(x,2)+(x−1) ∂Ψ (x,2). ∂x dices, ∂Ψ ∂x is also continuous and differentiable as a function of x, for x > 0, the function G is continuous in the closed interval [ξ, ξ + 1] and differentiable in the open interval (ξ, ξ + 1). By the mean value theorem there exists z ∈ (ξ, ξ + 1) such that ∆ = G0 (z)(k − l − 1) = " # ∂2Ψ ∂2Ψ ∂Ψ (z, 2) + (z − 1) (z, m) + 2 (z, 2) (k − l − 1). ∂x2 ∂x ∂x2 479 Topological indices of Kragujevac trees If for a specific vertex-degree-based topological index, T (x) = ∂Ψ ∂2Ψ ∂2Ψ (x, m) + 2 (x, 2) + (x − 1) (x, 2) ≤ 0 ∂x2 ∂x ∂x2 for all x ≥ 1, then ∆ = T I(X) − T I(Y ) ≤ 0 for every value of k > lIt means that the tree with maximal index is V while the tree with minimal index is U . Conversely, if for a specific vertex-degree-based topological index T (x) ≥ 0 for all x ≥ 1, then the tree with maximal index is U while the tree with minimal index is V . In the next table we show that for Randić index, the geometric-arithmetic index, the sum-connectivity index and the harmonic index, T (x) ≤ 0, while for the atom-bond-connectivity index, the first Zagreb index and the second Zagreb index, T (x) ≥ 0. Index T (x) Randić Geometric-arithmetic √ m(m2 +6mx−3x2 ) − 2x√x(m+x)3 − Sum-connectivity Harmonic − ³ − √ √ √ −x m−3( √ m− 2) 4x2 2mx <0 √ 2(x3 +15x2 −24x−4) √ 2x x(2+x)3 <0 5 5 (m+x) 2 (11+x)−3(x+2) 2 5 5 4(x+2) 2 (m+x) 2 ´ <0 4[(3m3 +2x3 −8)+3x2 (3m−2)+3x(3m2 −4)] (x+2)3 (m+x)3 <0 (m−2)(4x+3m−6) √ 4x2 (x+m−2) xm(x+m−2) >0 Atom-bond connectivity First Zagreb 2>0 Second Zagreb 4>0 2 References [1] C. A. Coulson, On the calculation of the energy in unsaturated hydrocarbon molecules, Proc. Cambridge Phil. Soc., 36, pp. 201-203, (1940). 480 Roberto Cruz, Ivan Gutman and Juan Rada [2] A. A. Dobrynin, R. Entriguer and I. Gutman, Wiener index of trees: Theory and applications, Acta Appl. Math, 66, pp. 211-249, (2001). [3] E. Estrada, L. Torres, L. Rodrı́guez and I. Gutman, An atom-bond connectivity index: Modelling the enthalpy of formation of alkanes, Indian J. Chem., 37A, pp. 849-855, (1998). [4] B. Furtula, A. Graovac and D. Vukičević, Augmented Zagreb index, J. Math. 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Schwenk, Computing the characteristic polynomial of a graph, in: Lectures Notes in Mathematics, vol 406, Springer-Verlag, Berlin, Heidelberg, pp. 153-172, (1974). [19] D. Vukičević and B. Furtula, Topological index based on the ratios of geometrical and arithmetical means of end-vertex degrees of edges, J. Math. Chem., 46, pp. 1369-1376, (2009). [20] H. Wiener, Structural determination of paraffin boiling points, J. Amer. Chem. Soc., 69, pp. 17-20, (1947). [21] B. Zhou and N. Trinajstić, On a novel connectivity index, J. Math. Chem., 46, pp. 1252-1270, (2009). [22] L. Zhong, The harmonic index for graphs, Appl. Math. Lett., 25, pp. 561-566, (2012). Roberto Cruz Instituto de Matemáticas, Universidad de Antioquia Medellı́n, Colombia e-mail : [email protected] Ivan Gutman Faculty of Science, University of Kragujevac Kragujevac, Serbia e-mail : [email protected] and 482 Roberto Cruz, Ivan Gutman and Juan Rada Juan Rada Instituto de Matemáticas, Universidad de Antioquia Medellı́n, Colombia e-mail : [email protected]