On a Class of Distance Based Molecular Structure Descriptors

On a Class of Distance Based Molecular Structure Descriptors Franka Miriam Brückler a, Tomislav Došlić b, Ante Graovac c, Ivan Gutman d, a Department of Mathematics, University of Zagreb, Bijenička 30, 0000 Zagreb, Croatia bruckler@math.hr b Faculty of Civil Engineering, University of Zagreb, Kačić eva 26, 0000 Zagreb, Croatia doslic@master.grad.hr c NMR Center, The Rudjer Bošković Institute, Bijenička 54, 0000 Zagreb, Croatia, IMC, University of Dubrovnik, Branitelja Dubrovnika 29, 20000 Dubrovnik, Croatia, and Faculty of Science, University of Split, Nikole Tesle 2, 2000 Split, Croatia graovac@irb.hr d Faculty of Science, University of Kragujevac, P. O. Box 60, 34000 Kragujevac, Serbia gutman@kg.ac.rs Abstract A new class of distance based molecular structure descriptors is put forward, aimed at eliminating a general shortcoming of the Wiener and Wiener type indices, namely that the greatest contributions to their numerical values come from vertex pairs at greatest distance. The Q-indices, considered in this work, consist of contributions of vertex pairs that exponentially decrease with distance. It is shown that the Q-indices are equal to the Hosoya polynomial H(G, λ), evaluated at a pertinently selected value of the variable λ. AMS Classification: 92E0; 05C2; 05C90 Keywords: Distance (of vertices in graph); distance based molecular structure descriptor; distance based topological index; Wiener index; Harary index; Q-index

. Introduction In this work we are concerned with distance based molecular structure descriptors. If G is a molecular graph [, 2] and v, v 2,...,v n are its vertices, then the distance between two distinct vertices v i and v j, denoted by d(v i, v j G), is the length of (= number of edges in) a shortest path connecting v i and v j in G [3]. Molecular graphs are necessarily connected, and therefore the distance between any two of their vertices is a well defined (finite and non-zero) integer. In what follows it will be assumed that the graph G considered has n vertices and m edges. For brevity we shall say that G is an (n, m)-graph. By d(g, k) we denote the number of vertex pairs of the graph G that are at distance k. Evidently, for an (n, m)-graph, d(g, ) = m and ( ) n d(g, k) =. () 2 k Distance based molecular structure descriptors (also referred to as distance based topological indices) were extensively studied in chemical graph theory, see for instance the surveys [4, 7], the entry about distance matrix in [5] and the second chapter of [6]. The oldest and most thoroughly investigated of these descriptors is the Wiener index [8], W(G) = i<j d(v i, v j G) = k k d(g, k). (2) Its chemical applications and mathematical properties are well documented, and can be found in the reviews [9 2]. Other much studied Wiener-type indices are the hyper-wiener index W W and the Tratch Stankevich Zefirov index T SZ. The hyper-wiener index was originally defined [3] in terms other than distances, but for it the following relation was shown to hold [4]: WW(G) = i<j [ 2 d(v i, v j G) 2 + ] 2 d(v i, v j G) = ( 2 k2 + ) 2 k d(g, k). (3) k 2

The Tratch Stankevich Zefirov index is defined as [5] TSZ(G) = [ 6 d(v i, v j G) 3 + 2 d(v i, v j G) 2 + ] 3 d(v i, v j G) i<j = ( 6 k3 + 2 k2 + ) 3 k d(g, k). (4) k Some more Wiener type indices were considered elsewhere [6 8]. In [9] the multiplicative version of the Winer index was examined, defined as π(g) = i<j d(v i, v j G). Evidently, log π(g) = i<j log d(v i, v j G) = k (log k) d(g, k). All the mentioned distance based structure descriptors depend on vertex distances in such a way that the most distant vertex pairs have the greatest contribution. This is in direct contradiction with the generally accepted fact that any kind of interaction between physical objects (in particular, between atoms in a molecule) decrease with increasing distance. In view of this, although we are not intending to deny the importance of the Wiener and Wiener type descriptors in QSPR and QSAR researches, we point out the difficulty of giving a sound physical justification for their distance dependence. The above problem was recognized by other scholars as well, and as a way out of it, the Harary index was introduced. Initially, the Harary index was defined as [4] Har (G) = i<j d(v i, v j G) 2 = k d(g, k) k2 but soon the same authors changed their minds and re-defined the index as [20,2] Har(G) = i<j d(v i, v j G) = k d(g, k). (5) k The Harary index was also simultaneously and independently introduced in [22]. Another attempt to remedy the drawback of the Wiener index was made in a pair of references [7,8] which contain some ideas very close to the ones presented here. 3

In the subsequent section we offer an argument, suggesting that although in the Harary index the contribution of a pair of vertices attenuates with their distance, the speed of this attenuation is too slow. In other words, the contribution of distant vertices to the Harary type indices might be still too large. In Section 4 we offer a class of models in which the effect of far lying vertex pairs, although not fully neglected, is made reasonably small. 2. An excursion to Zagreb indices In an attempt to find a mathematical expression for the influence of branching of the molecular carbon atom skeleton on total π-electron energy of conjugated molecules [25] and, somewhat later, on physico chemical properties of saturated hydrocarbons [26], two simple structure descriptors were conceived, eventually named the first (M ) and the second (M 2 ) Zagreb indices: n M (G) = (d i ) 2 (6) i= M 2 (G) = (i,j) d i d j where d i is the degree (= number of first neighbors) of the vertex v i of the molecular graph G, and where indicates summation over all pairs of adjacent vertices v i, v j. (i,j) Another way to write the first Zagreb index is M (G) = (i,j)(d i + d j ). The Zagreb indices belong among the standard and most popular molecular structure descriptors [5,27], and are much applied for designing QSPR and QSAR models; for details see the reviews [28 30]. We now show how the Zagreb indices can be expressed in terms of distances. For the sake of simplicity we assume that the molecular graph G does not possess three and four-membered cycles. If so, then the number of pairs of vertices at distance two is equal to the sum over all vertices v i, of the number of vertex pairs attached to v i. This means, m(g, 2) = n i= ( ) [ n di = (d i ) 2 2 2 i= 4 ] n d i i=

which in view of Eq. (6) and the well known relation i= d i = 2m yields d(g, 2) = 2 M (G) m = 2 M (G) d(g, ). (7) In the absence of triangles and quadrangles, the number of pairs of vertices at distance three is equal to the sum over all edges (i, j) of the product of number of vertices adjacent to v i except v j, and the number of vertices adjacent to v j except v i. This means d(g, 3) = i )(d j ) (i,j)(d = (i,j) d i d j (i,j)(d i + d j ) + m = M 2 (G) M (G) + d(g, ). (8) Combining Eqs. (7) and (8) we obtain: M (G) = 2 d(g, ) + 2 d(g, 2) M 2 (G) = d(g, ) + 2 d(g, 2) + d(g, 3). Thus we arrive at a somewhat unexpected conclusion, namely that the Zagreb indices are determined by the number of vertex pairs at distance, 2, and 3. In other words, the Zagreb indices are also distance based molecular structure descriptors, containing only contributions coming from near lying vertex pairs, the distance of which is not greater than 3. The fact that the Zagreb indices are successful structure descriptors may be used as an argument for the design of the Q-indices, outlined in Section 4. As a preparation for this we recall the basic fact on the Hosoya polynomial. 3. Hosoya polynomial In his seminal work [3], Hosoya introduced a distance based graph polynomial whose coefficients are the numbers d(g, k): H(G, λ) = d(g, k) λ k. k It is immediate to see that the first derivative of this polynomial, at λ = is equal to the Wiener index, cf. Eq. (2). Because of this property, in [3] H(G, λ) was 5

named Wiener polynomial. Later, for obvious reasons, the name was changed into Hosoya polynomial [32], which is nowadays used by the majority of mathematical chemists. Some elementary properties of the Hosoya index are: H(G, 0) = 0 (9) ( ) n H(G, ) = 2 (0) H (G, ) = W(G) () 2 H (G, ) + H (G, ) = WW(G). (2) Formulas (0), (), and (2) are immediate consequences of the definition (), (2), and (3), respectively. Some less well known properties of the Hosoya polynomial are: W(G) = H(G, λ) λ (3) λ= WW(G) = 2 λ H(G, λ) 2 λ2 (4) λ= TSZ(G) = 3 3! λ 3 λ2 H(G, λ) (5) λ= where TSZ is the Tratch Stankevich Zefirov index, Eq. (4). Formula (3) is, of course, same as Eq. (). It may be that the formulas (4) and (5) are reported here for the first time. The above three expressions suggest that also for k > 3, the following invariants may be of some relevance in mathematical chemistry: k! k λ k λk H(G, λ). (6) λ= At this point we also state two integral relations, involving the Hosoya polynomial: 0 H(G, λ) dλ = k d(g, k) k + 0 H(G, λ) λ dλ = Har(G) (7) 6

where Har is the Harary index, Eq. (5). 4. The Q-model We consider a class of invariants for the molecular graph G whose vertex set is V (G): Q = γ(u, v). {u,v} V (G) Thus Q is an additive function of increments associated with pairs of vertices of G. We further require that γ(u, v) depends solely on the distance d(u, v) between the vertices u and v. Thus, γ(u, v) = f(d(u, v)) where f is a function that needs to chosen. This implies Q = k 0 f(k) d(g, k). It may be convenient to choose f so that f(0) = 0, in which case Q = k f(k) d(g, k). Note that choice f(k) = results in ( Q = n ) 2 choice f(k) = k results in Q = W(G) choice f(k) = k 2 /2 + k/2 results in Q = WW(G) choice f(k) = k 3 /6 + k 2 /2 + k/3 results in Q = TSZ(G) choice f(k) = /k 2 results in Q = Har (G) choice f(k) = /k results in Q = Har(G) choice f(k) = λ k results in Q = H(G, λ). The Q-index is conceived as follows. Consider a molecular graph G and let v be one of its vertices. Assume that the intensity of the interactions between this vertex and its first neighbors are given by some parameter λ (for each first neighbor). Let then the intensity of the interaction of v with its second neighbors be λ 2 (for 7

each second neighbor), with its third neighbors be λ 3 (for each third neighbor), etc. Evidently, if λ < then according to our model the interaction between vertex v and the other vertices decreases exponentially with the distance between v and the other vertices. Summing all these interaction for vertex v, and then summing over all vertices of G, we get a distance based structure descriptor in which the attenuation of the interaction between two vertices rapidly (exponentially) decreases, and by adjusting the parameter λ may become arbitrarily fast. We refer to this new structure descriptor as the Q-index with parameter λ, and denote it by Q(λ). Bearing in mind the considerations in the previous parts of this paper, especially in Section 3, we immediately recognize that the Q-index is intimately related with the Hosoya polynomial, namely that Q(λ) = 2 H(G, λ). (8) The multiplier 2 comes from the fact that each pairwise interaction has been counted twice. As already mentioned, the parameter λ in Eq. (8) must be less than unity. As a first guess, we might set λ = /2. We see that the class of Q-indices is conceived by setting 0 < λ < instead of λ = in one of the expressions involving the Hosoya polynomial. We may think of doing the same in all above stated such formulas, in particular in Eqs. () (7). Then each such formula would yield a new invariant, maybe worth our attention. The study of the QSPR/QSAR applications of both the Q-indices and the other, above specified, options is left for a forthcoming communication. References [] N. Trinajstić, Chemical Graph Theory, CRC Press, Boca Raton, 983; 2nd revised ed. 992. [2] I. Gutman, O. E. Polansky, Mathematical Concepts in Organic Chemistry, Springer Verlag, Berlin, 986. 8

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