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Document Type : Original Research Article

Author

Department of Mathematics, Open Educational College, Ministry of Education, Al-Qadisiya Centre, Iraq

Abstract

The most general algebraic polynomial to obtain a massive number of degree-based topological indices for a specific family of structures is the M-polynomial. In this paper, the M-polynomial of some graph operations was derived including join, corona product, strong product, tensor product, splice, and link of regular graphs. By using those expressions, numerous degree-based indices of the aforesaid operations were computed. The explicit expressions of the indices were also derived for some particular graphs.

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Introduction

The graph theory includes an important method called the topological index (or molecular descriptor) for correlating the physiochemical activity of molecular and networks. A molecular-graph is a simple connected graph in which nodes-edges are assumed to be atoms-chemical bonds of a compound. Topological index is nothing but a number obtained from molecular graphs which describes the molecular graph topology and is an invariant for isomorphic graphs. Such descriptors are considered to be fantastic tools in QSPR and QSAR modelling. Harold Wiener introduced the idea of topological index when he worked on boiling point alkanes in 1947 [1]. Several algebraic polynomials are spotted in the literary history to resolve the painstaking strategy of computing indices utilizing their customary understandings of a particular class of graphs [2,3,4,34-42]. The M-polynomial [15] plays an important role in computing a huge proportion of indices of a specific group of graphs in the scenario of degree-based topological-indices. If V(G) and E(G) represent the node and edge collection of a graph G, respectively, then degree of v V(G), written as dv, is defined as the number of members in E(G) incident on v. Let ni and  mi be the number of elements in V(Gi), E(Gi), respectively. If degree of all nodes of G is r, then it is called the r-regular graph. The join G1, G2 is generated by connecting each node belongs to G1 to each node of G2. Corona product [7] of G1 and G2 is formed by considering one copy of G1 and n1 copy of G2 and connecting the i-th node of G1 to all nodes belonging to i-th part of G2. The strong product [8] of two graphs contains node set V(G1)× V(G2) and (u1, v1) is adjacent with (u2, v2) iff u1=u2 with v1 v2 or v1=v2 and u1 u2. Tensor product contains the same node collection and (u1, v1) is adjacent with (u2, v2) if u1 u2 and v1 v2  [9,10,11,14]). Splice [12,13] of these two graphs is constructed by identifying the nodes v1, v2 in G1 G2. The link [14] of these two graphs is constructed by joining the nodes v1 and v2 with an edge in the union of G1 and G2.

Basavanagoud et al. [16] computed explicit expressions of algebraic polynomials for numerous operations. Mondal et al. [5] derived neighborhood Zagreb index of some graph operations. The present author obtained indices of various networks and dendrimer structures [17,18,19,20]. Khalifeh et al. [21] obtained Zagreb indices of different graph operations. The intention of the current report is to compute different degree-based topological descriptors of graph operations of regular graphs via M-polynomial.

Preliminaries

Lemma 2. If G is r-regular with n nodes and m edges, then the total number of edges is

Definition 1. The formulation of M-polynomial of G is defined as follows:

Where,  is the total count of connections u v for which {du, dv}={p, q}.

Gutman and Trinajstić presented Zagreb indices [23]. The first Zagreb-index is formulated as follows:

The second Zagreb descriptor is formed as:

For more details about this indices, see [5,6,22,32,33,21]. The second modified Zagreb index is defined as follows:

Bollobas and Erdos [24] and Amic et al. [25] initiated the concept of generalized Randić index and illustrated in a broad range in mathematical chemistry [26]. For more details, see [28,27]. Such index is formulated as follows:

The inverse Randić index is formulated as:

Symmetric division index is formulated as:

Harmonic index [29] is formulated as:

The inverse sum index [30] is formulated as:

The augmented Zagreb index [31] is formulated as follows:

The way by which indices based on degree are recovered from Mt-polynomial is presented in Table 1.

Main results

In this section, we discuss the M-polynomial of different graph operations namely join, corona product, strong product, tensor product, splice, link, and explore the topological indices for those operations. We utilize the following lemma to obtain the main results.

Lemma 3. [16] For a a r-regular graph with n vertices and m edge connections, we have M(G;tx,y)=mxryr.

Where t is

Join

We compute the M-polynomial of join G1+G2 in the following theorem.

Theorem 1. If G1,tG2 are r1,tr2-regular, respectively, then for n1-n2 r1-r2, we find

Proof. Edge partition of G1+G2 is as follows:

|E{r1+n2,r2+n1}|=|{uvE(G1+G2):du=r1+n2,dv=r2+n1|=n1n2.

|E{r1+n2,r1+n2}|=|{uvE(G1+G2):du=r1t+n2,dv =r1t+n2}|=m1.

|E{r2+n1,r2+n1}|=|{uvE(G1+G2):du=r2+n1,dv=r2+n1|=m2.

The Mt-polynomial of Gt is derived as follows

Now employing such expression of Mt-polynomial, one easily finds degree-based topological index for G1+G2 in the following theorem.

Theorem 2. If G1+G2 be the join of graphs G1 and G2. Then, we have:

1.M1(G1+G2)=n1n2(r1t+tr2t+tn1t+tn2)+2m1(r1t+tn2)+2m2(r2t+tn1).

2.M2(G1+G2)=n1n2(r2+n1)(r1+n2)+m1(r1+n2)2+m2(r2+n1)2.

7.χα(G1+G2)=n1n2(r1t+tr2t+tn1t+tn2)α+2αm1(r1t+tn2)α+2αm2(r2t+tn1)α.

8.Mα(G1+G2)=n1n2(r1+n2)α-1+n1n2(r2+n1)α-1+2m1(r1t+tn2)α-1+2m2(r2t+tn1)α-1.

Proof. Consider tM((G1+G2);x,y)t=tf (x,y)=n1n2xr1+n2yr2+n1+m1xr1+n2yr1+n2+m2 xr2+n1 yr2+n1.

Then, we have:

The join of K¯and K¯yields the complete bipartite graph Kn,t. Now by using Theorem 1, we obtain the following corollary.

Corollary 1. The M-polynomial of complete bipartite graph Kn,t is given by  M(Kn,t)=ntxtyn.

By using Theorem 4, we obtain the following corollary.

Corollary 2. The topological indices of Kn,t are given by,

1. M1(Kn,t)=nt(n+t),
2. M2(Kn,t)=n2t2,
3. Mm(Kn,t)=1,
4. S D(Kn,t)=n2+t2,

Corona product

We compute the M-polynomial of the corona product of G1 and G2 in the following theorem.

Strong product

The strong product of two graphs G1 and G2 is denoted by G1G2. We compute the M-polynomial of the strong product of G1 and G2 in the following theorem.

Theorem 5. Let G1 and G2 be two regular graphs of degree r1 and r2, respectively. Then, we have:

Proof. The strong product of two regular graphs G1 and G2 with degree r1 and r2, respectively, is also a regular graph of degree n1n2-1 with n1n2 vertices. Consequently, the result follows from the Lemma 2 and Lemma 3.■

Now by using this M-polynomial, we calculate some degree based topological indices of the G1G2 in the following theorem.

Theorem 6. If G1G2 be the strong product. Then,

This completes the proof.

Tensor product

The tensor product of two graphs G1 and G2 is denoted by G1×G2. We compute the M-polynomial of the tensor product of G1 and G2 in the following theorem.

Theorem 7. Let G1 and G2 be two regular graphs of degree r1 and r2, respectively. Then, we have:

Proof. The tensor product of two regular graphs G1 and G2 with degree r1 and r2, respectively, is also a regular graph of degree n1n2-n1-n2+1 with n1n2 vertices. Consequently, the result follows from the Lemma 2 and Lemma 3.■

Splice

The splice of two graphs G1 and G2 is denoted by G1.G2. The M-polynomial of the splice of G1 and G2 is obtained in the following theorem.

Theorem 9. Let G1 and G2 be two regular graphs of degree r1 and r2, respectively. Then, we have:

M(G1.G2)=(n1-1)xr1yr1+r2+(n2-1)xr2yr1+r2+

(m1-r1)xr1 yr1+(m2-r2)xr2 yr2.

Proof. The edge set of G1 · G2 has the following partitions.

|E{r1,r1+r2}|=|{uvE(G1·G2):du=r1, dv=r1+r2}|=n1-1.

|E{r2,r1+r2}|=|{uvE(G1·G2):du=r2, dv=r1+r2}|=n2-1.

|E{r1,r1}|=|{uvE(G1·G2):du=r1, dv=r1}|=m1-r1.

|E{r2,r2}|=|{uvE(G1·G2):du=r2, dv=r2}|=m2-r2.

Based on the definition, the M-polynomial of G is obtained as follows:

This completes the proof.■

Now by using this M-polynomial, we calculate some degree based topological indices of the G1·G2 in the following theorem.

Theorem 10. The degree based topological indices of G1·G2 are given by:

1.M1(G1·G2)=(n1-1)(2r1+r2)+(n2-1) (r1+2r2)+2r1(m1-r1)+2r2(m2-r2).

2.M2(G1·G2)=(r1+r2)(r1(n1-1)+r2(n2-1))+r2(m1-r1)+r2(m2-r2).

8.χα(G1·G2)=(n1-1)(2r1+r2)α+(n2-1) (r1+2r2)α+2αrα(m1-r1)+2αr2(m2-r2)α.

1. Mα(G1·G2)=(n1-1)rα-1(n2-1)rα-1

+2(m1-r1)rα-1+2(m2-r2)rα-1+(r1+r2)α-1(n1+n2-2).

The link of G1 and G2 is denoted by G1~G2. We compute the M-polynomial of the link of G1 and G2 in the following theorem.

Theorem 11. Let G1 and G2 be two regular graph of degree r1 and r2, respectively. Then, we have:

M(G1~G2)=xr2+1yr1+1+(n1-1)xr1 yr1+1+

(n2-1)xr2 yr1+1+(m1-r1)xr1 yr1+(m2-r2)xr2 yr2.

Proof. The edge set of G1~G2 is partitioned as follows:

|E{r2+1,r1+1}|=|{uvE(G1~G2):du=r2+1, dv=r1+1}|=1.

|E{r1,r1+1}|=|{uvE(G1~G2):du=r1, dv=r1+1}|=n1-1.

|E{r2,r2+1}|=|{uvE(G1~G2):du=r2, dv=r2+1}|=n2-1.

|E{r1,r1}|=|{uvE(G1~G2):du=r1, dv=r1}|=m1-r1.

}|=|{uvE(G1~G2):du=r2, dv=r2}|=m2-r2.

From the definition, the M-polynomial of G1 ~G2 is obtained as follows:

Theorem 12. The topological indices of G1~G2 are given by:

1. M1(G1~G2)=(r2+1)+(r1+1)+(n1-1)(2r1+1)

+(n2-1)(2r2+1)+2r1(m1-r1)+2r2(m2-r2).

1. M2(G1~G2)=(r2+1)(r1+1)+r1(r1+1)(n1-1)+r2(r2+1)(n2-1)+(m1-r1)r2+(m2-r2)r2.

Conclusion

In this paper, we obtained degree-based topological indices for different graph operations including join, corona product, strong product, tensor product, splice, and link of regular graphs. First, we computed M-polynomial of the aforesaid graph operations and later recovered many degree-based topological indices applying it.

Acknowledgments

The authors would like to thank the reviewers for their helpful suggestions and comments.

Conflict of Interest

The authors declare that there is no conflict of interests regarding the publication of this manuscript.

Orcid:

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How to cite this article: Raad Sehen Haoer.  Molecular descriptors of some graph operations through m-polynomial. Eurasian Chemical Communications, 2023, 5(2), 112-125. Link: http://www.echemcom.com/article_157141.html

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