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 d_v, is defined as the number of members in E(G) incident on v. Let n_i and  m_i 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 G_1, 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 v_1, 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 {d_u, d_v}={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)=mx^ry^r.

Where t is

Join

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

 Theorem 1. If G₁,tG₂ are r_1,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}}|=|{uv∈E(G₁+G₂):d_u=r₁+n₂,d_v=r₂+n₁|=n₁n₂.

|E_{{r1+n2,r1+n2}}|=|{uv∈E(G₁+G₂):d_u=r_1t+n₂,d_v =r_1t+n₂}|=m₁.

|E_{{r2+n1,r2+n1}}|=|{uv∈E(G₁+G₂):d_u=r₂+n₁,d_v=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(r_1t+tr_2t+tn_1t+tn₂)+2m₁(r_1t+tn₂)+2m₂(r_2t+tn₁).

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

7.χ_α(G₁+G₂)=n₁n₂(r_1t+tr_2t+tn_1t+tn₂)^α+2^αm₁(r_1t+tn₂)^α+2^αm₂(r_2t+tn₁)^α.

8.M^α(G₁+G₂)=n₁n₂(r₁+n₂)^α-1+n₁n₂(r₂+n₁)^α-1+2m₁(r_1t+tn₂)^α-1+2m₂(r_2t+tn₁)α-1.

Proof. Consider tM((G₁+G₂);x,y)t=tf (x,y)=n₁n₂x^r1+n2y^r2+n1+m₁x^r1+n2y^r1+n2+m2 xr₂+n₁ yr₂+n₁.

Then, we have:

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

Corollary 1. The M-polynomial of complete bipartite graph K_n,t is given by  M(K_n,t)=ntx^tyⁿ.

By using Theorem 4, we obtain the following corollary.

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

1. M₁(K_n,t)=nt(n+t),
2. M₂(K_n,t)=n²t²,
3. M^m(K_n,t)=1,
4. S _D(K_n,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 G_1◙G₂. 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 n₁n_2-1 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 G_1◙G₂ in the following theorem.

   Theorem 6. If G_1◙G₂ 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)x^r1y^r1+r2+(n₂-1)x^r2y^r1+r2+

   (m₁-r₁)x^r1 y^r1+(m₂-r₂)x^r2 y^r2.

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

   |E_{r1,r1+r2}|=|{uv∈E(G₁·G₂):d_u=r₁, d_v=r₁+r₂}|=n₁-1.

   |E_{r2,r1+r2}|=|{uv∈E(G₁·G₂):d_u=r₂, d_v=r₁+r₂}|=n₂-1.

   |E_{r1,r1}|=|{uv∈E(G₁·G₂):d_u=r₁, d_v=r₁}|=m₁-r₁.

   |E_{r2,r2}|=|{uv∈E(G₁·G₂):d_u=r₂, d_v=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.χ_α(G₁·G₂)=(n₁-1)(2r₁+r₂)^α+(n₂-1) (r₁+2r₂)^α+2^αr^α(m₁-r₁)+2^αr₂(m₂-r₂)^α.

   9. M^α(G₁·G₂)=(n₁-1)r^α-1(n₂-1)r^α-1

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

   Link

   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)=x^r2+1y^r1+1+(n₁-1)x^r1 y^r1+1+

   (n₂-1)x^r2 y^r1+1+(m₁-r₁)x^r1 y^r1+(m₂-r₂)x^r2 y^r2.

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

   |E_{r2+1,r1+1}|=|{uv∈E(G₁~G₂):d_u=r₂+1, d_v=r₁+1}|=1.

   |E_{r1,r1+1}|=|{uv∈E(G₁~G₂):d_u=r₁, d_v=r₁+1}|=n₁-1.

   |E_{r2,r2+1}|=|{uv∈E(G₁~G₂):d_u=r₂, d_v=r₂+1}|=n₂-1.

   |E_{r1,r1}|=|{uv∈E(G₁~G₂):d_u=r₁, d_v=r₁}|=m₁-r₁.

   }|=|{uv∈E(G₁~G₂):d_u=r₂, d_v=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:

   Raad Sehen Haoer: https://orcid.org/0000-0001-5396-7598‎

    

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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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Copyright © 2023 by SPC (Sami Publishing Company) + is an open access article distributed under the Creative Commons Attribution License(CC BY)  license  (https://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.