ON MERRIFIELD – SIMMONS INDEX OF MOLECULAR GRAPHS

The Merrifield–Simmons index σ = σ(G) of a graph G is the number of independent vertex sets of G. This index can be calculated recursively and expressed in terms of Fibonacci numbers. We determine the molecular graphs for which σ can be recursively calculated in a single step.

by any contemporary scholar.The only surviving feature of this theory is a quantity that nowadays is referred to as the Merrifield-Simmons index .
Let G be a graph with vertex set V(G) = {v 1 , v 2 , . . ., v n }.An independent vertex set of G is a subset of V(G), such that no two vertices in it are adjacent.The number of distinct k-element independent vertex sets is denoted by n(G, k).By definition, n(G, 0) = 1 for all graphs, and n(G, 1) = n.
The Merrifield-Simmons index is then defined as i.e., it is just the total number of independent vertex sets of the underlying graph G [7].
The name "Merrifield-Simmons index" for the graph invariant σ was first time used by one of the present authors [8].Nowadays, in mathematical chemistry and mathematics this name is commonly accepted.For details of the theory of the Merrifield-Simmons index see the review [9], the recent papers [10][11][12][13][14], and the references cited therein.
For the present consideration we need the following recurrence relations [6,9].
Let v be a vertex of the graph G, and let N v be the set consisting of the vertex v and its first neighbors.Then The Fibonacci numbers F n , n ≥ 0 are defined recursively as When formula ( 2) is applied to the terminal vertex of the n-vertex path P n , we get a relation that has the same form as the recurrence relation for the Fibonacci numbers.

Calculating the Merrifield-Simmons index
Combining the recursion relations (1), (2), and the identity (4), it is possible to express the Merrifield-Simmons index of any (molecular) graph in terms of Fibonacci numbers.We illustrate this fact on the example of triphenylene.
The molecular graph of triphenylene G 0 is depicted in Fig. 1.The vertex to which relation (2) will be applied is indicated by a heavy dot.This yields The recurrence relation ( 2) needs now to be applied to the subgraphs G 1 and G 2 (again to the vertices indicated by heavy dots, see Fig. 1), resulting in: Fig. 1.The molecular graph of triphenylene (G 0 ) and its subgraphs needed for the calculation of the Merrifield-Simmons index σ(G 0 ).
The subgraph G 4 consists of two components, both being paths.The subgraph G 6 consists of three components, all three being paths.Therefore, applying (1) and (4), we get In order to compute σ(G 3 ) and σ(G 5 ), one needs to apply (1), ( 2) and (4) once again.Thus, = σ(P 10 ) σ(P 5 ) + σ(P 5 ) σ(P 5 ) σ(P 3 ) However, there exists large classes of molecular graphs in which the above described calculation can be accomplished in a single step.In the subsequent sections we describe these classes.
3 Simple calculation of the Merrifield-Simmons index of some acyclic molecular graphs Example 3.1.Consider the molecular graph T 0 of 3-ethyl-5-methyloctane, depicted in Fig. 2. When Eqs. ( 1), (2), and (4) are applied to its vertex labeled by v, then It can be easily recognized that T 0 is a special case of the molecular graph T 1 , in which the parameters a 1 , a 2 , b 1 , b 2 are non-negative integers.Thus, for T 0 , Bearing in mind that by applying Eqs. ( 1), (2), and (4) we get: Extending this argument, we arrive at the chemical trees T 2 and T 3 .By fully analogous calculation, we have: and .
The series T 1 , T 2 , T 3 cannot be continued because in the case of molecular graphs the vertex degree must not be greater than 4 (see [15]).Thus T 1 , T 2 , T 3 form a complete set of acyclic molecular graphs for which the recursive calculation of the Merrifield-Simmons index can be achieved in a single step.
4 Simple calculation of the Merrifield-Simmons index of some unicyclic molecular graphs The molecular graph U 0 is a special case of U 1 , in which the parameters a 1 , a 2 are non-negative integers whereas r is the size of the (unique) cycle, r ≥ 3.In particular, for U 0 , a 1 = a 2 = 1, r = 5.
Bearing in mind that by applying Eqs. ( 1), (2), and (4) we get: Note that two is the maximal number of branches that may be attached.Then, in full analogy to the previous case, we have: Another class of unicyclic molecular graphs with the required property is represented by U 3 , cf.Fig. 3.In this graph, two path fragments with x and y vertices are attached to the first neighbors of the vertex v. Recall that if x = y = 0, then U 3 is just the cycle of size r.For this molecular graph, which directly yields σ(U 3 ) = σ(P x+y+r−1 ) + σ(P x ) σ(P y ) σ(P r−3 ) .
6 Simple calculation of the Merrifield-Simmons index of some tricyclic molecular graphs From the diagram depicted in Fig. 5 we see that In the same way as in the acyclic, unicyclic, and bicyclic molecular graphs, the case

Concluding remarks
In view of the fact that the vertex degrees in molecular graphs (provided these represent organic compounds) must not exceed 4 [15], the acyclic graphs T 1 , T 2 , T 3 , uni- seems that tetracyclic and higher-cyclic molecular graphs of this kind do not exist.
Therefore, the graphs presented in this work appear to be the only possible of this kind.
It would be interesting to have a formal mathematical verification of the above claim.

Fig. 2 .
Fig. 2. Acyclic molecular graphs for which the recursive calculation of the Merrifield-Simmons index can be achieved in a single step.

Fig. 3 .
Fig. 3. Unicyclic molecular graphs for which the recursive calculation of the Merrifield-Simmons index can be achieved in a single step.
y, ≥ 0 and r ≥ 3 are the only unicyclic species for which the recursive calculation of the Merrifield-Simmons index can be achieved in a single step.5 Simple calculation of the Merrifield-Simmons index of some bicyclic molecular graphs Example 5.1.Consider the molecular graph B 0 of 1,8-diethyl-naphthalene, depicted in Fig. 4. When Eqs. (

Fig. 4 .
Fig. 4. Bicyclic molecular graphs for which the recursive calculation of the Merrifield-Simmons index can be achieved in a single step.

Example 6 . 1 .= F 10 + F 2 F 2 F 4 Fig. 5 .
Fig. 5. Tricyclic molecular graphs for which the recursive calculation of the Merrifield-Simmons index can be achieved in a single step.

cyclic graphs U 1 ,
U 2 , U 3 , U 4 , U 5 , bicyclic graphs B 1 , B 2 , B 3 , B 4 , and tricyclic graphsD 1 , D 2 (depicted in Figs.2-5) with parameters a 1 , a 2 , b 1 , b 2 , c 1 , c 2 , d 1 , d 2 ,x, y, z, w ≥ 0 and r, s, t ≥ 3 seem to be the only species for which the recursive calculation of the Merrifield-Simmons index can be achieved in a single step.For the same reason, it