M-Polynomial and Degree-Based Topological Indices

Transcription

1 Iranian Journal of Mathematical Chemistr, Vol. 6, No. 2, October 2015, pp IJMC M-Polnomial and Degree-Based Topological Indices EMERIC DEUTSCH 1 AND SANDI KLAVŽAR 2,3,4 1 Poltechnic Institute of New York Universit, United States 2 Facult of Mathematics and Phsics Universit of Lublana, Slovenia 3 Facult of Natural Sciences and Mathematics Universit of Maribor, Slovenia 4 Institute of Mathematics, Phsics and Mechanics, Lublana, Slovenia Correspondence should be addressed to sandi.klavzar@fmf.uni-l.si Received 14 Februar 2015; Accepted 27 Februar 2015 ACADEMIC EDITOR: ALI REZA ASHRAFI ABSTRACT Let G be a graph and let m i( G), i, 1, be the number of edges uv of G such that { dv ( G), d u ( G)} { i, } The M-polnomial of G is introduced with M ( G; x, i ( G) x i mi. It is shown that degree-based topological indices can be routinel computed from the polnomial, thus reducing the problem of their determination in each particular case to the single problem of determining the M-polnomial. The new approach is also illustrated with examples. KEYWORDS degree-based topological index Zagreb index Randić index graph polnomial 1. INTRODUCTION Numerous graph polnomials were introduced in the literature, several of them turned out to be applicable in mathematical chemistr. For instance, the Hosoa polnomial [19], see also [9, 11, 21], is the ke polnomial in the area of distance-based topological indices. In particular, the Wiener index can be computed as the first derivative of the Hosoa polnomial, evaluated at 1, the hper-wiener index [5] and the Tratch-Stankevich-Zefirov index can be obtained similarl [4, 16]. Additional chemicall relevant polnomials are the matching polnomial [13, 12], the Zhang-Zhang polnomial (also known as the Clar covering polnomial) [25, 6, 24], the Schultz polnomial [18], and the Tutte polnomial [10], to name ust a few of them. In this paper we introduce a polnomial called the M-polnomial, and

2 94 DEUTSCH AND KLAVŽAR show that its role for degree-based invariants is parallel to the role of the Hosoa polnomial for distance-based invariants. In chemical graph theor (too) man topological indices were introduced, in particular (too) man degree-based topological indices. This fact is emphasized in the recent surve [15] which contains a uniform approach to the degree-based indices and a report on a comparative test from [17] how these indices are correlated with phsico-chemical parameters of octane isomers. The test indicates that quite man of these indices are inadequate for an structure-propert correlation. But it could be that the are still applicable in a combination with other indices. In the literature one finds man papers that, for a given famil of graphs, determine a closed formula for a given (degree-based) topological index. To overcome this particular approach in the area of degree-based indices, in this paper we introduce the M-polnomial and demonstrate that in numerous cases a degree-based topological index can be expressed as a certain derivative or integral (or both) of the corresponding M-polnomial. This in particular implies that knowing the M-polnomial of a given famil of graphs, a closed formula for an such index can be obtained routinel. In the remaining cases when the function defining a given topological index is such that it does not allow it to be (routinel determined from the M-polnomial, one can use equalit (2) from the following section and tr to get a closed formula from it. But in an case, knowing the M-polnomial is sufficient, hence the possible future research in the area should concentrate on determining the M- polnomial of a relevant famil of graphs instead of computing the corresponding indices one b one. Moreover, it is our hope that a closer look to the properties of the M-polnomial will bring new general insights. In the next section the M-polnomial is introduced and shown how degree-based indices can be computed from it. In the last section it is discussed how the M-polnomial can be computed for (chemical) graphs and tpical examples for the use of the new polnomial are presented. 2. THE M-POLYNOMIAL If G ( V, E) is a graph and v V, then d v (G) (or d v fohort if G is clear from the context) denotes the degree of v. Let G be a graph and let m i ( G), i, 1, be the number of edges e = uv of G such that {d v (G), d u (G)} = {i, }. As far as we know, the quantities m i were first introduced and applied in [14]. We now introduce the M-polnomial of G as i M ( G; x, m ( G) x. For a graph G = (V, E), a degree-based topological index is a graph invariant of the form i i

3 M-Polnomial and Degree-Based Topological Indices 95 I ( G) f ( d, d ), (1) u v euve where f = f(x, is a function appropriatel selected for possible chemical applications [15]. For instance, the generalized Randić index R ( G ), 0, is defined with (1) b setting f(x, = (x α [3]. Collecting edges with the same set of end degrees we can rewrite (1) as I ( G) i m i ( G) f ( i, ) (2) cf. [7, 8, 15, 22]. Using the operators D x and D defined on differentiable functions in two variables b we have: f ( x, Dx ( f ( x, ) x, x f ( x, D ( f ( x, ), Theorem 2.1. Let G = (V,E) be a graph and let I ( G) euve f ( d u, dv), where f(x, is a polnomial in x and. Then I ( G) f ( Dx, D )( M ( G; x, ) x 1. Proof. Let r 0 and s 0 be fixed. Then D ( ( ;, )) ( ) i x D M G x D x D i m i G x i m ( G) i r s x i i. D r x ( ) i m i G s x i It follows that f ( Dx, D )( M ( G; x, ) G x mi ( ) 1 r, si i r, s mi ( G) f ( i, ). i The result follows b (2). In addition to the operators Dx and D we will also make use of the operators x f ( t, f ( x, t) S x ( f ( x, ) 0 dt, S ( f ( x, ) dt t 0. t As in the proof of Theorem 2.1 we infer that the operators S, xs SxD, and D xs applied to the term x i of M(G; x,, and evaluated at x = = 1, return i r s, and i r s, respectivel. Hence, proceeding along the same lines as in the proof of Theorem 2.1, we can state the following extension of Theorem 2.1.

4 96 DEUTSCH AND KLAVŽAR Theorem 2.2. Let G = (V, E) be a graph and let i f ( x, x. i, Z I ( G) f ( d, d ), where i euve Then I(G) can be obtained from M(G; x, using the operators D x, D, S x, and S. In the statement of Theorem 2.2, I(G) appears implicitl (contrar to the explicit formula of Theorem 2.1). Hence in Table 1 explicit expressions foome frequent degreebased topological indices are listed. The general case for an function of the form i f ( x, i Z ix should then be clear. Note that for the smmetric division index we, can appl Theorem 2.2 because its defining function f (x, can equivalentl be written as x/ + /x. Topological index f ( x, Derivation from M(G;x, First Zagreb x+ (D x +D )(M(G;x,) x==1 Second Zagreb x (D x D )(M(G;x,) x==1 Second modified Zagreb 1 x (S x S )(M(G;x,) x==1 General Randić (αn) (x α (D α x D α )(M(G;x,) x==1 General Randić (αn) 1 ( x (S α x S α )(M(G;x,) x==1 u v Smmetric division index x 2 x 2 (D x S +D S x )(M(G; x,) x==1 Table 1. Some standard degree based topological indices and the formulas how to compute them from the M-polnomial. In order to handle additional indices that are not covered b Theorem 2.2, we introduce two additional operators as follows: J ( f ( x, ) f ( x, x), Q ( f ( x, ) x f ( x,, 0. With these two operators we have the following result whose proof again proceeds along the same lines as the proof of Theorem 2.1: Proposition 2.3. Let G = (V;E) be a graph and let I ( G) euve f ( d u, dv), where x f ( x,, k ( x ) and r, s 0, t 1, and Z then I k r ( G) S s xq JDxD( M ( G; x, ) x1.

5 M-Polnomial and Degree-Based Topological Indices 97 In Table 2 three special cases of Proposition 2.3 from the literature are collected. Topological index f ( x, Derivation from M(G;x, harmonic 2 x 2 S x J (M(G ; x,) x=1 inverse sum x x S x J D x D (M(G ; x,) x=1 augmented Zagreb 3 x 2 x Table 2. Three more topological indices and the corresponding formulas. S x 3 Q -2 J D x 3 D 3 (M(G ; x, ) x=1 3. HOW TO COMPUTE THE M-POLYNOMIAL As we have demonstrated in the previous section, the computation of the degree based topological indices can be reduced to the computation of the M-polnomial. In this section we first give some general remarks how to compute this polnomial and then derive the polnomial foome tpical classes of graphs from the literature. In the general setting we follow an approach of Gutman [14] for which some definitions are needed first. As we are primaril interested in chemical applications, we will restrict us here to chemical graphs which are the connected graphs with maximum degree at most 4. (However, the statements can be generalized to arbitrar graphs.) For a chemical graph G = (V, E), let n V ( G), m E( G), and let n i, 1 i 4, be the number of vertices of degree i. Note first that m 0 whenever G has at least three vertices (and is connected). For the other 11 coefficients m i of the M-polnomial Gutman observed that the following relations hold: n1 n2 n3 n4 n (3) m12 m13 m14 n1 (4) m 2m m m n (5) m23 2m33 m34 3n3 14 m24 m34 2m44 4n4 1 2n2 3n3 4n4 m m m (6) (7) n 2 (8) These equalities are linearl independent. In some examples, all the quantities m i (and consequentl the M-polnomial) can be found directl. On the other hand, one can determine

6 98 DEUTSCH AND KLAVŽAR some of the m i 's first and then the remaining ones can be obtained from the above relations. We now demonstrate this on several examples Polomino chains Let B n; r;,, ) be a polomino chain with n squares, arranged in egments of lengths ( 1 r 1, r, i 2, see Figure 1 for the particular case B(11;6;3,4,2,2,3,2). Figure 1. Polomino chain B(11; 6; 3; 4; 2; 2; 3; 2). Let a denote the number of segments of length 2 that occur at the extremities of the chain (a = 0, 1, or 2) and let b denote the number of non-extreme segments of length 2. In Figure 1 we have a = 1, b = 2. As we shall see, the knowledge of the parameters n, r, a, and b of a chain is sufficient to determine its M-polnomial. It is eas to see that V 2n 2 and E 3n 1. The numbers n i of vertices of degree i are given b n 2 = r + 3 (number of outer corners), n 4 = r 1 (number of inner corners), and n 3 = V n 2 n 4 = 2(n r). We have m 22 = 2, m 44 = b (the edges that halve the nonextreme segments of length 2), and m 24 = a + 2b (each extreme segment of length 2 contributes 1 and each nonextreme segment of length 2 contributes 2 24edges). Now, taking into account that there are no vertices of degree 1, from (5)(7) we obtain the expressions for m 23, m 33, and m 34. Setting B B n; r;,..., ) this ields n ( 1 r

7 M-Polnomial and Degree-Based Topological Indices 99 M ( B n ; x, 2x (2r a 2b 2) x ( a 2b) x (3n 6r 1 a 3b) x 3 3 (4r a 4b 4) x 3 4 bx 4 4. For example, for the first Zagreb index (corresponding to f(x, = x +, either from (2) or from the first entr of Table 1 we obtain 18n + 2r 4, in agreement with [23, Theorem 2.1]. Similarl, for the second Zagreb index (corresponding to f(x, = x we obtain 27n + 6r 19 a b, in agreement with [23, Theorem 2.4]. For the special case of the linear chain L n ( n 3), corresponding to r =1, a = b = 0, we obtain M(L n ; x, = 2x x (3n 5) x 3 3. For the special case of the zig-zag chain Z n, corresponding to r = n 1, a = 2, b = r 2 = n 3, we obtain M(Z n ; x, = 2x x (n 2)x x (n 3)x 4 4. We remark that for the first (second) Zagreb index we obtain 20n6 (resp. 32n24), in disagreement with the slightl mistaken expressions given in [23, Corollar 2.2(ii)] and [23, Corollar 2.5 (ii)] Starlike trees We next considetarlike trees, i.e. trees having exactl one vertex of degree greater than two [2]. If the degree of this vertex (called center) is n, then we denote a starlike tree b S(k 1,,k n ), where the k 's are positive integers representing the number of edges of the ras emanating from the center, see Figure 2 for an example. Figure 2. Starlike tree S(1, 2, 3, 4, 3, 3, 2, 1). Denote K n 1 k and let a be the number of ras having exactl one edge. The possible vertex degrees are 1, 2, and n and it is eas to see that m 11 = 0, m 12 = n a, m 1,n = a, m 2,n = n a, m n,n = 0. Since the total number of edges is K, we have m 2,2 = K (n a) a (n a) = K + a 2n, leading to M(S(k 1,, k n ); x, = (n a)x 2 + ax n + (K + a 2n) x (n a) x 2 n.

8 100 DEUTSCH AND KLAVŽAR For example, for the first Zagreb index we obtain (using (2) or Table 1) n 2 3n + 4K. Similarl, for the second Zagreb index we obtain 2n 2 6n an + 2a + 4K. Both results are in agreement with those in [2, Corollar 3.7] obtained in a more complicated manner Triangulanes We consider the famil of triangulanes T n, defined in Figure 3, where the auxiliar triangulane G n is defined recursivel in the following manner (see also [1, 9, 20]). Let G 1 be a triangle and denote one of its vertices b 1. We define G n as the circuit of the graphs G n 1, G n 1, and K 1 (the 1-vertex graph) and denote b n the vertex where K 1 has been placed (see Figure 3 again). Figure 3. Graphs G 1, G 2, G n, and T n. The possible vertex degrees of G n are 2, 4 and, using the notation µ i instead of m i, it is eas to see that µ 22 = 2 n 1 and µ 24 = n n. We have µ 22 +µ 24 + µ 44 = E ( ) 3(2 1) since G n consists of 2 n 1 triangles. From here we obtain µ 44 = 3 2 n 1 5. Thus, as a bproduct, M(G n ; x, = 2 n 1 x (2 + 2 n )x (3 2 n 1 5)x 4 4. Now, in order to find the m i 's of T n from its defining picture, we note that at the attachment of each G n to the starting triangle, two 24-edges of G n turn into two 44-edges of T n. Consequentl, m 22 = 3µ 22 = 3 2 n 1 ; m 24 = 3(µ 24 2) = 3 2 n ; m 44 = 3(µ 44 +2) + 3 = 3(3 2 n 1 2) and G n

9 M-Polnomial and Degree-Based Topological Indices 101 ACKNOWLEDGMENTS M(T n ; x, = 3 2 n 1 x n x (3 2 n 1 2)x 4 4. Work supported in part b the Research Grant P10297 of the Ministr of Higher Education, Science and Technolog Slovenia. REFERENCES 1. A. R. Ashrafi, Wiener index of nanotubes, toroidal fullerenes and nanostars, in: F. Cataldo, A. Graovac, O. Ori (Eds.), The Mathematics and Topolog of Fullerenes, Springer, Dordrecht, 2011, pp M. Azari, A. Iranmanesh, Chemical graphs constructed from rooted product and their Zagreb indices, MATCH Commun. Math. Comput. Chem. 70 (2013) B. Bollobás, P. Erdös, Graphs with extremal weights, Ars Combin. 50 (1998) F. M. Brückler, T. Došlić, A. Graovac, I. Gutman, On a class of distancebased moleculatructure descriptors, Chem. Phs. Lett. 503 (2011) G. G. Cash, Relationship between the Hosoa polnomial and the hperwiener index, Appl. Math. Lett. 15 (2002) C.P. Chou, H. A. Witek, Closedform formulas for the ZhangZhang polnomials of benzenoid structures: chevrons and generalized chevrons, MATCH Commun. Math. Comput. Chem. 72 (2014) H. Deng, G. Huang, X. Jiang, A unified linear-programming modeling of some topological indices, to appear in J. Comb. Opt. DOI /s H. Deng, J. Yang, F. Xia, A general modeling of some vertex-degree based topological indices in benzenoid sstems and phenlenes, Comp. Math. Appl. 61 (2011) E. Deutsch, S. Klavžar, Computing Hosoa polnomials of graphs from primar subgraphs, MATCH Commun. Math. Comput. Chem. 70 (2013) T. Došlić, Planar polcclic graphs and their Tutte polnomials, J. Math. Chem. 51 (2013) M. Eliasi, A. Iranmanesh, Hosoa polnomial of hierarchical product of graphs, MATCH Commun. Math. Comput. Chem. 69 (2013) E. J. Farrell, An introduction to matching polnomials, J. Combin. Theor Ser. B 27 (1979)

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