M-Polynomial and Degree-Based Topological Indices
aa r X i v : . [ m a t h . C O ] J u l M -Polynomial and Degree-Based TopologicalIndices Emeric Deutsch
Polytechnic Institute of New York University, United States e-mail: [email protected]
Sandi Klavˇzar
Faculty of Mathematics and Physics, University of Ljubljana, SloveniaFaculty of Natural Sciences and Mathematics, University of Maribor, SloveniaInstitute of Mathematics, Physics and Mechanics, Ljubljana, Slovenia e-mail: [email protected] (Received *** July, 2014)
Abstract
Let G be a graph and let m ij ( G ), i, j ≥
1, be the number of edges uv of G such that { d v ( G ) , d u ( G ) } = { i, j } . The M -polynomial of G is introduced with M ( G ; x, y ) = X i ≤ j m ij ( G ) x i y j .It is shown that degree-based topological indices can be routinely computed from the polyno-mial, thus reducing the problem of their determination in each particular case to the singleproblem of determining the M -polynomial. The new approach is also illustrated with examples. Numerous graph polynomials were introduced in the literature, several of them turned outto be applicable in mathematical chemistry. For instance, the Hosoya polynomial [19], seelso [9, 11, 21], is the key polynomial in the area of distance-based topological indices. Inparticular, the Wiener index can be computed as the first derivative of the Hosoya polyno-mial, evaluated at 1, the hyper-Wiener index [5] and the Tratch-Stankevich-Zefirov index canbe obtained similarly [4, 16]. Additional chemically relevant polynomials are the matchingpolynomial [13, 12], the Zhang-Zhang polynomial (also known as the Clar covering polyno-mial) [25, 6, 24], the Schultz polynomial [18], and the Tutte polynomial [10], to name just a fewof them. In this paper we introduce a polynomial called the M -polynomial, and show that itsrole for degree-based invariants is parallel to the role of the Hosoya polynomial for distance-basedinvariants.In chemical graph theory (too) many topological indices were introduced, in particular (too)many degree-based topological indices. This fact is emphasized in the recent survey [15] whichcontains a uniform approach to the degree-based indices and a report on a comparative testfrom [17] how these indices are correlated with physico-chemical parameters of octane isomers.The test indicates that quite many of these indices are inadequate for any structure-propertycorrelation. But it could be that they are still applicable in a combination with other indices.In the literature one finds many papers that, for a given family of graphs, determine a closedformula for a given (degree-based) topological index. To overcome this particular approach in thearea of degree-based indices, in this paper we introduce the M -polynomial and demonstrate thatin numerous cases a degree-based topological index can be expressed as a certain derivative orintegral (or both) of the corresponding M -polynomial. This in particular implies that knowingthe M -polynomial of a given family of graphs, a closed formula for any such index can beobtained routinely. In the remaining cases when the function defining a given topological indexis such that it does not allow it to be (routinely) determined from the M -polynomial, one canuse equality (2) from the following section and try to get a closed formula from it. But inany case, knowing the M -polynomial is sufficient, hence the possible future research in the areashould concentrate on determining the M -polynomial of a relevant family of graphs instead ofcomputing the corresponding indices one by one. Moreover, it is our hope that a closer look tothe properties of the M -polynomial will bring new general insights.In the next section the M -polynomial is introduced and shown how degree-based indices canbe computed from it. In the last section it is discussed how the M -polynomial can be computedfor (chemical) graphs and typical examples for the use of the new polynomial are presented. The M -polynomial If G = ( V, E ) is a graph and v ∈ V , then d v ( G ) (or d v for short if G is clear from the context)denotes the degree of v . Let G be a graph and let m ij ( G ), i, j ≥
1, be the number of edges e = uv of G such that { d v ( G ) , d u ( G ) } = { i, j } . As far as we know, the quantities m ij were firstintroduced and applied in [14]. We now introduce the M -polynomial of G as M ( G ; x, y ) = X i ≤ j m ij ( G ) x i y j . For a graph G = ( V, E ), a degree-based topological index is a graph invariant of the form I ( G ) = X e = uv ∈ E f ( d u , d v ) , (1)where f = f ( x, y ) is a function appropriately selected for possible chemical applications [15].For instance, the generalized Randi´c index R α ( G ), α = 0, is defined with (1) by setting f ( x, y ) =( xy ) α [3]. Collecting edges with the same set of end-degrees we can rewrite (1) as I ( G ) = X i ≤ j m ij ( G ) f ( i, j ) , (2)cf. [7, 8, 15, 22].Using the operators D x and D y defined on differentiable functions in two variables by D x ( f ( x, y )) = x ∂f ( x, y ) ∂x , D y ( f ( x, y )) = y ∂f ( x, y ) ∂y , we have: Theorem 2.1
Let G = ( V, E ) be a graph and let I ( G ) = X e = uv ∈ E f ( d u , d v ) , where f ( x, y ) is apolynomial in x and y . Then I ( G ) = f ( D x , D y )( M ( G ; x, y )) (cid:12)(cid:12) x = y =1 . Proof.
Let r ≥ s ≥ D rx D sy ( M ( G ; x, y )) = D rx D sy X i ≤ j m ij ( G ) x i y j = D rx X i ≤ j m ij ( G ) j s x i y j = X i ≤ j m ij ( G ) i r j s x i y j . Suppose that f ( x, y ) = P r,s α r,s x r y s . Then f ( D x , D y )( M ( G ; x, y )) is equal to X r,s α r,s D rx D sy ( M ( G ; x, y )) = X r,s α r,s X i ≤ j m ij ( G ) i r j s x i y j = X i ≤ j m ij ( G ) X r,s α r,s i r j s x i y j . t follows that f ( D x , D y )( M ( G ; x, y )) (cid:12)(cid:12) x = y =1 = X i ≤ j m ij ( G ) X r,s α r,s i r j s = X i ≤ j m ij ( G ) f ( i, j ) . The result follows by (2). (cid:3)
In addition to the operators D x and D y we will also make use of the operators S x ( f ( x, y )) = Z x f ( t, y ) t dt, S y ( f ( x, y )) = Z y f ( x, t ) t dt . As in the proof of Theorem 2.1 we infer that the operators S rx S sy , S rx D sy , and D rx S sy appliedto the term x i y j of M ( G ; x, y ), and evaluated at x = y = 1, return i − r j − s , i − r j s , and i r j − s ,respectively. Hence, proceeding along the same lines as in the proof of Theorem 2.1, we canstate the following extension of Theorem 2.1. Theorem 2.2
Let G = ( V, E ) be a graph and let I ( G ) = X e = uv ∈ E f ( d u , d v ) , where f ( x, y ) = X i,j ∈ Z α ij x i y j . Then I ( G ) can be obtained from M ( G ; x, y ) using the operators D x , D y , S x , and S y . In the statement of Theorem 2.2, I ( G ) appears implicitly (contrary to the explicit formula ofTheorem 2.1). Hence in Table 1 explicit expressions for some frequent degree-based topologicalindices are listed. The general case for any function of the form f ( x, y ) = P i,j ∈ Z α ij x i y j shouldthen be clear. Note that for the symmetric division index we can apply Theorem 2.2 becauseits defining function f ( x, y ) can equivalently be written as x/y + y/x .topological index f ( x, y ) derivation from M ( G ; x, y )first Zagreb x + y ( D x + D y )( M ( G ; x, y )) (cid:12)(cid:12) x = y =1 second Zagreb xy ( D x D y )( M ( G ; x, y )) (cid:12)(cid:12) x = y =1 second modified Zagreb xy ( S x S y )( M ( G ; x, y )) (cid:12)(cid:12) x = y =1 general Randi´c ( α ∈ N ) ( xy ) α ( D αx D αy )( M ( G ; x, y )) (cid:12)(cid:12) x = y =1 general Randi´c ( α ∈ N ) xy ) α ( S αx S αy )( M ( G ; x, y )) (cid:12)(cid:12) x = y =1 symmetric division index x + y xy ( D x S y + D y S x )( M ( G ; x, y )) (cid:12)(cid:12) x = y =1 Table 1: Some standard degree based topological indices and the formulas how to compute themfrom the M -polynomialn order to handle additional indices that are not covered by Theorem 2.2, we introduce twoadditional operators as follows: J ( f ( x, y )) = f ( x, x ) , Q α ( f ( x, y )) = x α f ( x, y ) , α = 0 . With these two operators we have the following result whose proof again proceeds along thesame lines as the proof of Theorem 2.1:
Proposition 2.3
Let G = ( V, E ) be a graph and let I ( G ) = X e = uv ∈ E f ( d u , d v ) , where f ( x, y ) = x r y s ( x + y + α ) k , where r, s ≥ , , t ≥ , and α ∈ Z . Then I ( G ) = S kx Q α J D rx D sy ( M ( G ; x, y )) (cid:12)(cid:12) x =1 . In Table 2 three special cases of Proposition 2.3 from the literature are collected.topological index f ( x, y ) derivation from M ( G ; x, y )harmonic x + y S x J ( M ( G ; x, y )) (cid:12)(cid:12) x =1 inverse sum xyx + y S x J D x D y ( M ( G ; x, y )) (cid:12)(cid:12) x =1 augmented Zagreb (cid:16) xyx + y − (cid:17) S x Q − J D x D y ( M ( G ; x, y )) (cid:12)(cid:12) x =1 Table 2: Three more topological indices and the corresponding formulas M -polynomial As we have demonstrated in the previous section, the computation of the degree based topologicalindices can be reduced to the computation of the M -polynomial. In this section we first givesome general remarks how to compute this polynomial and then derive the polynomial for sometypical classes of graphs from the literature. In the general setting we follow an approach ofGutman [14] for which some definitions are needed first.As we are primarily interested in chemical applications, we will restrict us here to chemicalgraphs which are the connected graphs with maximum degree at most 4. (However, the state-ments can be generalized to arbitrary 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 = 0whenever G has at least three vertices (and is connected). For the other coefficients m ij of the -polynomial Gutman observed that the following relations hold: n + n + n + n = n (3) m + m + m = n (4) m + 2 m + m + m = 2 n (5) m + m + 2 m + m = 3 n (6) m + m + m + 2 m = 4 n (7) n + 2 n + 3 n + 4 n = 2 m . (8)These equalities are linearly independent. In some examples, all the quantities m ij (and conse-quently the M -polynomial) can be found directly. On the other hand, one can determine someof the m ij ’s first and then the remaining ones can be obtained from the above relations. Wenow demonstrate this on several examples. Let B ( n ; r ; ℓ , . . . , ℓ r ) be a polyomino chain with n squares, arranged in r segments of lengths ℓ , . . . , ℓ r , ℓ i ≥
2, see Fig. 1 for the particular case B (11; 6; 3 , , , , , B (11; 6; 3 , , , , , 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 Fig. 1 wehave a = 1, b = 2. As we shall see, the knowledge of the parameters n , r , a , and b of a chain issufficient to determine its M -polynomial. It is easy to see that | V | = 2 n + 2 and | E | = 3 n + 1.The numbers n i of vertices of degree i are given by n = r + 3 (number of outer corners), n = r − n = | V | − n − n = 2( n − r ). We have m = 2, = b (the edges that halve the non-extreme segments of length 2), and m = a + 2 b (eachextreme segment of length 2 contributes 1 and each non-extreme segment of length 2 contributes2 24-edges). Now, taking into account that there are no vertices of degree 1, from (5)-(7) weobtain the expressions for m , m , and m . Setting B n = B ( n ; r ; ℓ , . . . , ℓ r ) this yields M ( B n ; x, y ) = 2 x y + (2 r − a − b + 2) x y + (4 r − a − b − x y + bx y . For example, for the first Zagreb index (corresponding to f ( x, y ) = x + y ), either from (2) or fromthe first entry of Table 1 we obtain 18 n + 2 r −
4, in agreement with [23, Theorem 2.1]. Similarly,for the second Zagreb index (corresponding to f ( x, y ) = xy ) we obtain 27 n + 6 r − − a − b ,in agreement with [23, Theorem 2.4]. For the special case of the linear chain L n ( n ≥ r = 1, a = b = 0, we obtain M ( L n ; x, y ) = 2 x y + 4 x y + (3 n − x y . Forthe special case of the zig-zag chain Z n , corresponding to r = n − a = 2, b = r − n − M ( Z n ; x, y ) = 2 x y + 4 x y + 2( n − x y + 2 x y + ( n − x y . We remark thatfor the first (second) Zagreb index we obtain 20 n − n − We next consider starlike trees, i.e. trees having exactly one vertex of degree greater than two [2].If the degree of this vertex (called center) is n , then we denote a starlike tree by S ( k , . . . , k n ),where the k j ’s are positive integers representing the number of edges of the rays emanating fromthe center, see Fig. 2 for an example.Figure 2: Starlike tree S (1 , , , , , , , K = P nj =1 k j and let a be the number of rays having exactly one edge. The possiblevertex degrees are 1, 2, and n and it is easy to see that m = 0, m = n − a , m ,n = a , ,n = n − a , m n,n = 0. Since the total number of edges is K , we have m , = K − ( n − a ) − a − ( n − a ) = K + a − n , leading to M ( S ( k , . . . , k n ); x, y ) = ( n − a ) xy + axy n + ( K + a − n ) x y + ( n − a ) x y n . For example, for the first Zagreb index we obtain (using (2) or Table 1) n − n + 4 K . Similarly,for the second Zagreb index we obtain 2 n − n − an + 2 a + 4 K . Both results are in agreementwith those in [2, Corollary 3.7] obtained in a more complicated manner. We consider the family of triangulanes T n , defined in Fig. 3, where the auxiliary triangulane G n is defined recursively in the following manner (see also [1, 9, 20]). Let G be a triangle anddenote one of its vertices by y . We define G n as the circuit of the graphs G n − , G n − , and K (the 1-vertex graph) and denote by y n the vertex where K has been placed (see Fig. 3 again). y G y G G n G n − G n − y n − y n − G n G n G n y n y n y n T n Figure 3: Graphs G , G , G n , and T n The possible vertex degrees of G n are 2, 4 and, using the notation µ ij instead of m ij , it iseasy to see that µ = 2 n − and µ = 2 + 2 n . We have µ + µ + µ = | E ( G n ) | = 3(2 n − G n consists of 2 n − µ = 3 · n − −
5. Thus, as aby-product, M ( G n ; x, y ) = 2 n − x y + (2 + 2 n ) x y + (3 · n − − x y . Now, in order to find the m ij ’s of T n from its defining picture, we note that at the attachment ofeach G n to the starting triangle, two 24-edges of G n turn into two 44-edges of T n . Consequently, m = 3 µ = 3 · n − , m = 3( µ −
2) = 3 · n , m = 3( µ + 2) + 3 = 3(3 · n − − M ( T n ; x, y ) = 3 · n − x y + 3 · n x y + 3(3 · n − − x y . cknowledgments Work supported in part by the Research Grant P1-0297 of the Ministry of Higher Education,Science and Technology Slovenia.
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