Topological indices (TIs) are expressed by constant real numbers that reveal the structure of the graphs in QSAR/QSPR investigation. The reformulated second Zagreb index (RSZI) is such a novel TI having good correlations with various physical attributes, chemical reactivities, or biological activities/properties. The RSZI is defined as the sum of products of edge degrees of the adjacent edges, where the edge degree of an edge is taken to be the sum of vertex degrees of two end vertices of that edge with minus 2. In this study, the behaviour of RSZI under graph operations containing Cartesian product, join, composition, and corona product of two graphs has been established. We have also applied these results to compute RSZI for some important classes of molecular graphs and nanostructures.

1. Introduction

In the whole study, we only consider the molecular graph [1, 2], a graphical representation of molecular structure, in which every vertex corresponds to the atoms and the edges to the bonds between them. Assume J as a simple (molecular) graph with V_J vertex set and E_J edge set. The notations |V_J| and |E_J| represent the number of elements of J in V_J and E_J, respectively. Also, d_J(x) denotes the degree of a vertex (x) in J and is defined as the number of edges incident to x.

TIs can be expressed by real numbers related to graphs. There exist many applications as tools for modelling chemical and other properties of molecules for TIs. They determine the correlation between the specific properties of molecules and the biological activity with their configuration in the study of quantitative structure-activity relationships (QSARs) and quantitative structure-property relationships (QSPRs) [3]. To develop the scientific knowledge in 20th century, the concept of molecular structure plays an important role in chemical graph theory, a branch of mathematical chemistry which is closely related to chemical graph. The molecular structure descriptor, namely, topological index expresses the numerical value obtained from the molecular graph that represents its topology and is necessarily invariant under the automorphism of graphs.

The Zagreb indices, namely, first Zagreb index [4] and second Zagreb index [5] were introduced by Gutman et al. in 1972 and 1975, respectively. These two indices are, respectively, defined for molecular graph (J) as

(1)

(2)

Let x, y, and z be its three vertices of J forming a path of length two. If x and y are two adjacent vertices of J, we express as x ~ y and similarly for y and z as y ~ z. The edge connecting these vertices will be denoted by xy ~ yz. Then, the three auxiliary Zagreb-type indices are introduced by Basavanagoud et al. [6] in 2015. They are denoted as follows:

(3)

(4)

(5)

In 2004, Milicevic et al. [7] reformulated the Zagreb indices by replacing vertex degree with edge degree, and the edge degree of an edge is defined as d(e) = d(x) + d(y) − 2. The first and second reformulated Zagreb indices [8] of a graph J are defined as

(6)

(7)

where e ~ f means that the edges e and f share a common end vertex is, and e and f are adjacent. In 2015, Furtula and Gutman [9] introduced forgotten index (F-index) and is defined as

(8)

In mathematical chemistry, graph operations perform a significant role in the formation of new classes of graphs. By different graph operations on some general or particular graphs, some chemically interesting graphs can be obtained. In [10], Khalifeh et al. computed the first and second Zagreb indices under some graph operations. Some explicit formulae of Zagreb coindices under some graph operations were presented by Ashrafi et al. [11]. In [12], Das et al. derived multiplicative Zagreb indices of different graph operations. In [13], De et al. computed the reformulated first Zagreb index under some graph operations. Recently, the analytical expressions for various topological indices under some binary graph operations have been discussed in [14–16]. We also refer to [17–23] in this regard for interested readers.

2. Main Results

In the following, we study different binary graph operations such as join, Cartesian product, composition, and corona product of two molecular graphs and compute some exact formulae for RSZI with respect to those operations separately. Suppose J₁ and J₂ be two molecular graphs with the vertex sets , , such that , , and the edge sets , , such that , , respectively. We consider the notations P_n, C_n, and K_n for path, cycle, and complete graph with n vertices. To establish the main results, we also follow equations (1)–(8).

2.1. Join

The join [24] of two graphs J₁ and J₂, denoted by J₁ + J₂, contains the vertex set and the edge set .

In the following theorem, we compute RSZI for join of two graphs.

Theorem 1. Let J₁ + J₂ = J be the join of J₁ and J₂ graphs. Then, RSZI of J is given by

(9)

Proof. Consider (d_J(x) + d_J(y) − 2)(d_J(y) + d_J(z) − 2) = X. Let J₁ and J₂ be two graphs with , vertices and , edges, respectively. Then, by Table 1, we obtain

(10)

First part:

(11)

Second part:

(12)

Third part:

(13)

Fourth part:

(14)

Fifth part:

(15)

Sixth part:

(16)

By adding , we get the desired result.

Table 1. The degree distribution of join J = J₁ + J₂.

2.2. Applications

The suspension of a graph H is the join or sum of H with a single vertex K₁.

Corollary 1. The RSZI of suspension of H that contain |V_J| = n and |E_J| = m can be expressed as EM₂(H + K₁) = EM₂(H) + 2nM₄(H) + F(H) + 2M₂(H) + (1/2)(4n − 9)M₁(H) + 2m² + 4m + 2m(n − 1)² + (n/2)(n − 1)³.

Example 1. The cone graph C_m,n is defined as C_m + K_n. Then, EM₂(C_m,n) = 2m(11n² + 5mn + 2) + (1/2)mn(m + n − 2)((m + n)² + 4(2n − 3)).

Example 2. The RSZI of suspension of graph such as are expressed in Table 2.

Table 2. The results are calculated using Corollary 1.

Suspension of	Namely	Join of	EM₂ values
C_n	Wheel graph (W_n)	C_n + K₁	(n/2)(n³ + n² + 47n − 15)
	Star graph (S_n)		(1/2)(n − 1)(n − 2)³
P_n	Fan graph (F_n)	P_n + K₁	(1/2)(n + 4)(n − 1)³ + 26n² − 46n − 17
mK₂	Flower graph	mK₂ + K₁	4m²(2m² − m + 2)

Example 3. The complete bipartite graph K_p,q is the join of K_p + K_q. Using Theorem 1, EM₂(K_p,q) = (1/2)pq(p + q − 2)³.

2.3. The Cartesian Product

The Cartesian product (CP) [25] of J₁ and J₂, denoted by J₁ × J₂, is a graph with and any if and only if or . Also, and .

Now, we obtain RSZI for Cartesian product of two graphs.

Theorem 2. If J₁ × J₂ = J be the CP of J₁ and J₂ graphs, then RSZI of J is

(17)

Proof. By definition of RSZI, from equation (7) and the degree distribution for CP of two graphs, we have

(18)

The notations U₁, U₂, and U₃ represent the sum of above terms in order.

Step 1:
(19)
Step 2:
(20)
Step 3:
(21)
By adding U₁, U₂, U₃, we get the required result.

2.4. Applications

Let P, Q, R, and S be the grids (P_n × P_m) = P, rook’s graph (K_n × K_m) = Q, C₄-nanotorus TC₄(n, m) = C_n × C_m = R, and C₄-nanotube TUC₄(n, m) = (P_n × C_m) = S. Then, by Theorem 2, we get the following results.

Example 4. The RSZI for P is given by EM₂(P) = 4[54mn − 89(m + n) + 132], for m, n ≥ 4.

Example 5. The RSZI for Q is given by EM₂(Q) = 2mn((m − 1)(m − 2)(2mn + 3m − 2n − 4 + (n − 1)(n − 2)(2mn + 3n − 2m − 4) + (n − 1)(m − 1)(6mn − 11m − 11n + 22)), for m, n ≥ 2.

Example 6. The RSZI for R is given by EM₂(R) = 216mn.

Example 7. The RSZI for S is given by EM₂(S) = 4m(54n − 89).

2.5. Lexicographic Product

The lexicographic product (LP) or composition [26] of two graphs J₁ and J₂ is denoted by J₁[J₂], and any two vertices (u₁, u₂) and (v₁, v₂) are adjacent if and only if or u₁ = v₁ and . The vertex set of J₁[J₂] is , and the degree of a vertex (a, b) ∈ J₁[J₂] is given by .

In the following theorem, we compute RSZI for composition of two graphs J₁ and J₂.

Theorem 3. Let J₁[J₂] = J be the composition of J₁ and J₂. The RSZI of J is given by

(22)

Proof. By using the definition of RSZI and from the equation (7), we have

(23)

Now,

(24)

Next,

(25)

For

(26)

= Part(A) − Part(B) (say), where

(27)

Therefore, we get

(28)

Lastly,

(29)

By taking the summation of the five cases T₁, T₂, T₃, T₄, and T₅ and after simplification, we get the desired result.

2.6. Applications

The fence graph is the composition of P_n and P₂.

Example 8. From Theorem 3, RSZI of (P_n[P₂]) is calculated as EM₂(P_n[P₂]) = 160(8n − 17), where n > 3.

The closed fence graph is the composition of C_n and P₂.

Example 9. The RSZI of C_n[P₂] is given by EM₂C_n[P₂] = 1280n.

2.7. Corona Product

For the corona product (COP) [27] of J₁ and J₂, denoted by J₁∘J₂, the degree of a vertex r ∈ J₁∘J₂ is given in Table 3.

Table 3. The degree distribution of COP for J and J_2,i is the i^th copy of the graph J₂.

Now, we obtain the explicit expression of RSZI for corona product of two graphs.

Theorem 4. The RSZI of J₁∘J₂ is given by

(30)

Proof. From definition of RSZI and from equation (7) and Table 3, we have

(31)

First,

(32)

Second,

(33)

Third,

(34)

Fourth,

(35)

Last,

(36)

By simplifying the sum , we get the required result.

2.8. Applications

The t-thorny graph of a graph J, denoted as J^t, is obtained by joining t-number of thorns (pendent edges) to each vertex of J. It is defined as the corona product of J and complement of complete graph K_t. To know more about the thorn graphs, it may be followed in [28]. By using Theorem 4, we have the following results.

Example 10. The RSZI of J^t is given by EM₂(J^t) = EM₂(J) + tEM₁(J) + 2tM₄(J) + (2t(3t − 2) + (^tC₂))M₁(J) + 2mt(t − 1)(3t − 5) + (n/2)t(t − 1)³.

Example 11. The RSZI of t-thorny of P_n is given by .

Example 12. The RSZI of C_n is given by .

The bottleneck graph of a graph J is defined as the corona product of K₂ and J.

Example 13. The RSZI is given by EM₂(K₂∘J) = 2M₅(J) + 2M₃(J) + 2F(J) + (4n − 1)M₁(J) + 4M₂(J) + (n² + 3n + 2m)(n² + 2m) − 2mn.

3. Conclusion

In this study, we have executed the explicit expressions for RSZI under several graph operations such as join, Cartesian product, lexicographic product, and corona product. By applying these results, RSZI is also computed for some classes of graphs by specializing the components of graph operations. As a future work, we want to generalize the above theorems for n graphs. These results will also be helpful for further development using remaining graph operations.

Conflicts of Interest

The authors declare that there are no conflicts of interest.

Acknowledgments

The authors would like to express their sincere gratitude to the anonymous referees for valuable suggestions, which led to great deal of improvement of the original manuscript. The second author acknowledges the support of DST-FIST, New Delhi (India) (SR/FST/MS- I/2018/21) for carrying out this work.

Open Research

Data Availability

No data were used to support this study.

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[Retracted] On the Reformulated Second Zagreb Index of Graph Operations

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Retracted: On the Reformulated Second Zagreb Index of Graph Operations

Abstract

1. Introduction

2. Main Results

2.1. Join

2.2. Applications

2.3. The Cartesian Product

2.4. Applications

2.5. Lexicographic Product

2.6. Applications

2.7. Corona Product

2.8. Applications

3. Conclusion

Conflicts of Interest

Acknowledgments

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[Retracted] On the Reformulated Second Zagreb Index of Graph Operations

Retraction(s) for this article

Retracted: On the Reformulated Second Zagreb Index of Graph Operations

Abstract

1. Introduction

2. Main Results

2.1. Join

2.2. Applications

2.3. The Cartesian Product

2.4. Applications

2.5. Lexicographic Product

2.6. Applications

2.7. Corona Product

2.8. Applications

3. Conclusion

Conflicts of Interest

Acknowledgments

Open Research

Data Availability

References

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