Document Type : Original Research Article
Author
- Raad Sehen Haoer ^{} ^{}
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.
Graphical Abstract
Keywords
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,} G_{2} is generated by connecting each node belongs to G_{1} to each node of G_{2}. Corona product [7] of G_{1} and G_{2} is formed by considering one copy of G_{1} and n_{1} copy of G_{2} and connecting the i-th node of G_{1} to all nodes belonging to i-th part of G_{2}. The strong product [8] of two graphs contains node set V(G1)× V(G2) and (u_{1}, v_{1}) is adjacent with (u_{2}, v_{2}) iff u_{1}=u_{2} with v_{1} v_{2} or v_{1}=v_{2} and u_{1} u_{2}. Tensor product contains the same node collection and (u_{1}, v_{1}) is adjacent with (u_{2}, v_{2}) if u_{1} u_{2} and v_{1} v_{2 } [9,10,11,14]). Splice [12,13] of these two graphs is constructed by identifying the nodes v_{1,} v_{2} in G_{1} G_{2}. The link [14] of these two graphs is constructed by joining the nodes v_{1} and v_{2} with an edge in the union of G_{1} and G_{2}.
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^{r}y^{r}.
Where t is
Join
We compute the M-polynomial of join G_{1}+G_{2} in the following theorem.
Theorem 1. If G_{1},tG_{2} are r_{1,t}r_{2}-regular, respectively, then for n_{1}-n_{2} r_{1}-r_{2}, we find
Proof. Edge partition of G_{1}+G_{2} is as follows:
|E_{{r1+n2,r2+n1}}|=|{uv∈E(G_{1}+G_{2}):d_{u}=r_{1}+n_{2},d_{v}=r_{2}+n_{1}|=n_{1}n_{2}.
|E_{{r1+n2,r1+n2}}|=|{uv∈E(G_{1}+G_{2}):d_{u}=r_{1t}+n_{2},d_{v }=r_{1t}+n_{2}}|=m_{1}.
|E_{{r2+n1,r2+n1}}|=|{uv∈E(G_{1}+G_{2}):d_{u}=r_{2}+n_{1},d_{v}=r_{2}+n_{1}|=m_{2}.
The Mt-polynomial of Gt is derived as follows
Now employing such expression of Mt-polynomial, one easily finds degree-based topological index for G_{1}+G_{2} in the following theorem.
Theorem 2. If G_{1}+G_{2} be the join of graphs G_{1} and G_{2}. Then, we have:
1.M_{1}(G_{1}+G_{2})=n_{1}n_{2}(r_{1t}+tr_{2t}+tn_{1t}+tn_{2})+2m_{1}(r_{1t}+tn_{2})+2m_{2}(r_{2t}+tn_{1}).
2.M_{2}(G_{1}+G_{2})=n_{1}n_{2}(r_{2}+n_{1})(r_{1}+n_{2})+m_{1}(r_{1}+n_{2})^{2}+m_{2}(r_{2}+n_{1})^{2}.
7.χ_{α}(G_{1}+G_{2})=n_{1}n_{2}(r_{1t}+tr_{2t}+tn_{1t}+tn_{2})^{α+}2^{α}m_{1}(r_{1t}+tn_{2})^{α}+2^{α}m_{2}(r_{2t}+tn_{1})^{α}.
8.M^{α}(G_{1}+G_{2})=n_{1}n_{2}(r_{1}+n_{2})^{α-1+}n_{1}n_{2}(r_{2}+n_{1})^{α-1}+2m_{1}(r_{1t}+tn_{2})^{α-1}+2m_{2}(r_{2t}+tn_{1})α-1.
Proof. Consider tM((G_{1}+G_{2});x,y)t=tf (x,y)=n_{1}n_{2}x^{r}1+n2y^{r}2+n1+m_{1}x^{r}1+n2y^{r}1+n2+m2 xr_{2}+n_{1} yr_{2}+n_{1}.
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^{t}y^{n}.
By using Theorem 4, we obtain the following corollary.
Corollary 2. The topological indices of K_{n}_{,t} are given by,
- M_{1}(K_{n}_{,t})=nt(n+t),
- M_{2}(K_{n}_{,t})=n^{2}t^{2},
- M^{m}(K_{n}_{,t})=1,
- S _{D}(K_{n}_{,t})=n^{2}+t^{2},
Corona product
We compute the M-polynomial of the corona product of G_{1} and G_{2} in the following theorem.
Strong product
The strong product of two graphs G_{1} and G_{2} is denoted by G_{1}_{◙}G_{2}. We compute the M-polynomial of the strong product of G_{1} and G_{2} in the following theorem.
Theorem 5. Let G_{1} and G_{2} be two regular graphs of degree r_{1} and r_{2}, respectively. Then, we have:
Proof. The strong product of two regular graphs G_{1} and G_{2} with degree r_{1} and r_{2}, respectively, is also a regular graph of degree n_{1}n_{2-}1 with n_{1}n_{2} 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}_{◙}G_{2} in the following theorem.
Theorem 6. If G_{1}_{◙}G_{2} be the strong product. Then,
This completes the proof.■
Tensor product
The tensor product of two graphs G_{1} and G_{2} is denoted by G_{1}×G_{2}. We compute the M-polynomial of the tensor product of G_{1} and G_{2} in the following theorem.
Theorem 7. Let G_{1} and G_{2} be two regular graphs of degree r_{1} and r_{2}, respectively. Then, we have:
Proof. The tensor product of two regular graphs G_{1} and G_{2} with degree r_{1} and r_{2}, respectively, is also a regular graph of degree n_{1}n_{2}-n_{1}-n_{2}+1 with n_{1}n_{2} vertices. Consequently, the result follows from the Lemma 2 and Lemma 3.■
Splice
The splice of two graphs G_{1} and G_{2} is denoted by G_{1}.G_{2}. The M-polynomial of the splice of G_{1} and G_{2} is obtained in the following theorem.
Theorem 9. Let G_{1} and G_{2} be two regular graphs of degree r_{1} and r_{2}, respectively. Then, we have:
M(G_{1}.G_{2})=(n_{1}-1)x^{r}1y^{r}1+r2+(n_{2}-1)x^{r}2y^{r}1+r2+
(m_{1}-r_{1})x^{r}1 y^{r}1+(m_{2}-r_{2})x^{r}2 y^{r}2.
Proof. The edge set of G_{1} · G_{2} has the following partitions.
|E_{{r}1,r1+r2}|=|{uv∈E(G_{1}·G_{2}):d_{u}=r_{1}, d_{v}=r_{1}+r_{2}}|=n_{1}-1.
|E_{{r}2,r1+r2}|=|{uv∈E(G_{1}·G_{2}):d_{u}=r_{2}, d_{v}=r_{1}+r_{2}}|=n_{2}-1.
|E_{{r}1,r1}|=|{uv∈E(G_{1}·G_{2}):d_{u}=r_{1}, d_{v}=r_{1}}|=m_{1}-r_{1}.
|E_{{r}2,r2}|=|{uv∈E(G_{1}·G_{2}):d_{u}=r_{2}, d_{v}=r_{2}}|=m_{2}-r_{2}.
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 G_{1}·G_{2} in the following theorem.
Theorem 10. The degree based topological indices of G_{1}·G_{2} are given by:
1.M_{1}(G_{1}·G_{2})=(n_{1}-1)(2r_{1}+r_{2})+(n_{2}-1) (r_{1}+2r_{2})+2r_{1}(m_{1}-r_{1})+2r_{2}(m_{2}-r_{2}).
2.M_{2}(G_{1}·G_{2})=(r_{1}+r_{2})(r_{1}(n_{1}-1)+r_{2}(n_{2}-1))+r^{2}(m_{1}-r_{1})+r^{2}(m_{2}-r_{2}).
8.χ_{α}(G_{1}·G_{2})=(n_{1}-1)(2r_{1}+r_{2})^{α}+(n_{2}-1) (r_{1}+2r_{2})^{α}+2^{α}r^{α}(m_{1}-r_{1})+2^{α}r_{2}(m_{2}-r_{2})^{α}.
- M^{α}(G_{1}·G_{2})=(n_{1}-1)r^{α-1}(n_{2}-1)r^{α-1}
+2(m_{1}-r_{1})r^{α-1}+2(m_{2}-r_{2})r^{α-1}+(r_{1}+r_{2})^{α-1}(n_{1}+n_{2}-2).
Link
The link of G_{1} and G_{2} is denoted by G_{1}~G_{2}. We compute the M-polynomial of the link of G_{1} and G_{2} in the following theorem.
Theorem 11. Let G_{1} and G_{2} be two regular graph of degree r_{1} and r_{2}, respectively. Then, we have:
M(G_{1}~G_{2})=x^{r}2+1y^{r}1+1+(n_{1}-1)x^{r}1 y^{r}1+1+
(n_{2}-1)x^{r}2 y^{r}1+1+(m_{1}-r_{1})x^{r}1 y^{r}1+(m_{2}-r_{2})x^{r}2 y^{r}2.
Proof. The edge set of G_{1}~G_{2} is partitioned as follows:
|E_{{r}2+1,r1+1}|=|{uv∈E(G_{1}~G_{2}):d_{u}=r_{2}+1, d_{v}=r_{1}+1}|=1.
|E_{{r}1,r1+1}|=|{uv∈E(G_{1}~G_{2}):d_{u}=r_{1}, d_{v}=r_{1}+1}|=n_{1}-1.
|E_{{r}2,r2+1}|=|{uv∈E(G_{1}~G_{2}):d_{u}=r_{2}, d_{v}=r_{2}+1}|=n_{2}-1.
|E_{{r}1,r1}|=|{uv∈E(G_{1}~G_{2}):d_{u}=r_{1}, d_{v}=r_{1}}|=m_{1}-r_{1}.
}|=|{uv∈E(G_{1}~G_{2}):d_{u}=r_{2}, d_{v}=r_{2}}|=m_{2}-r_{2}.
From the definition, the M-polynomial of G_{1} ~G_{2} is obtained as follows:
Theorem 12. The topological indices of G_{1}~G_{2} are given by:
- M_{1}(G_{1}~G_{2})=(r_{2}+1)+(r_{1}+1)+(n_{1}-1)(2r_{1}+1)
+(n_{2}-1)(2r_{2}+1)+2r_{1}(m_{1}-r_{1})+2r_{2}(m_{2}-r_{2}).
- M_{2}(G_{1}~G_{2})=(r_{2}+1)(r_{1}+1)+r_{1}(r_{1}+1)(n_{1}-1)+r_{2}(r_{2}+1)(n_{2}-1)+(m_{1}-r_{1})r^{2}+(m_{2}-r_{2})r^{2}.
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
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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