# Three Topological Indices of Two New Variants of Graph Products

Authors: Muhammad Bilal; Muhammad Kamran Jamil; Muhammad Waheed; Abdu Alameri
Journal: Mathematical Problems in Engineering
 (2021)

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## Abstract

Graph operations play an important role to constructing complex network structures from simple graphs, and these complex networks play vital roles in different fields such as computer science, chemistry, and social sciences. Computation of topological indices of these complex network structures via graph operation is an important task. In this study, we defined two new variants of graph products, namely, corona join and subdivision vertex join products and investigated exact expressions of the first and second Zagreb indices and first reformulated Zagreb index for these new products.


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## Body

## 1. Introduction

In mathematics, the graph theory is the study of graphs which are mathematical structures used to model pairwise connection between objects. The graph theory is applied in the various fields such as computer science, biology, chemistry, social sciences, and operation research [1, 2].LetG=VG,EG be a simple, connected graph with vertex set VG and edge set EG. The number of vertices and number of edges are called the order n and size m, respectively, of the graph G. A graph of order n and size m will be denoted by Gn,m. For any vertex v∈VG, the degree of vertex v is the number of edges incident on the vertex v, and it is written as dGv or simply dv. For a simple graph G, the subdivision of the graph G is denoted by SG and obtained by inserting a new vertex on every edge of G.Topological index is a numeric value which is associated with a chemical structure of a certain chemical compound. This numeric value can help to predict certain properties of that chemical compound. Hundreds of topological indices have been introduced, but few of them gain attention of the scientific community. Zagreb indices are among the oldest and useful topological indices. For a graphG, the first and second Zagreb indices are defined as(1)M1G=∑v∈VGdGv2=∑uv∈EGdGu+dGv,M2G=∑uv∈EGdGudGv.In 1972, these topological indices were applied for the first time to find the total electronπ-energy of molecular graphs [3]. Later, the Zagreb indices developed important applications in QSPR/QSAR studies, and a lot of research studies have been published on these [4–10].Milićević et al. in 2004 reformulated the Zagreb indices in terms of edge degree which is defined as(2)EM1G=∑e∈EGde2,EM2G=∑e∼fdedf,where de shows the degree of the edge e in G, which is defined as de=du+dv−2 with edge e=uv and e∼f shows that the edge e and f are adjacent [11].Graph operations, especially graph products, play a significant role not only in pure and applied mathematics but also in computer science, chemistry, electrical engineering, and pharmaceutics. For instance, the Cartesian product provides a significant model for connecting computers [12].LetG1n1,m1 and G2n2,m2 be two connected simple graphs. Corona product of graphs G1 and G2, denoted by G1°G2, is obtained by taking one copy of G1 and n1 copies of G2, and joining each vertex of ith copy of G2 to ith vertex of G1 [8]. In [13], authors introduced two variants of the corona product and discussed their spectral properties. The subdivision vertex variant of corona of G1 and G2 is attained from SG1 and n1 copies of G2 by joining the ith vertex of VG1 to every vertex in the ith copy of G2. Similarly, the subdivision-edge neighborhood corona is obtained by attaching the neighbors of the ith vertex of VG1 to every vertex in the ith copy of G2.The join graph ofG1 and G2 is obtained by joining each vertex of G1 to each vertex of G2, and it is denoted by G1+G2 [14].Khalifeh et al. [15] computed the first and second Zagreb indices of Cartesian product, composition, join, disjunction, and symmetric difference of graphs and applied the results on C4 tube, torus, and multiwalled polyhex nanotorus. Authors in [16] investigated the upper bounds on the multiplicative Zagreb indices of some product of graphs. Azari and Iranmanesh [17] discussed the rooted product of graphs and found the exact expression of first and second Zagreb indices for this product. Jamil and Tomescu [18] found the exact formulas of the first reformulated Zagreb index for Cartesian product, composition, join, corona product, splice, link, and chain of graphs. Some graph operations and their topological indices are presented in [13, 15–28].Now, we define variants of these graphs’ product.Definition 1.
LetG1n1,m1 and G2n2,m2 be simple connected graphs, and the corona join graph of G1 and G2 is obtained by taking one copy of G1, n1 copies of G2, and joining each vertex of the ith copy of G2 with all vertices of G1. The corona join product of G1 and G2 is denoted by G1⊕G2 and shown in Figure 1.Figure 1
Corona join productG1⊕G2.Definition 2.
ForG1n1,m1 and G2n2,m2, the subdivision vertex join is denoted by G1∔G2 and obtained by joining the each new vertex of SG1 to all vertices of G2. Figure 2 shows the illustration of subdivision vertex join for P3 and P2.Figure 2
Subdivision vertex join graphG1∔G2.


## 2. Main Results

In this section, we present the main results. The following lemmas are useful to obtain the exact expressions of topological indices of new variants of graph products. The proofs of the following two lemmas are directly from the definitions of corona join productG1⊕G2 and subdivision vertex join G1∔G2.Lemma 1.
LetG1=n1,m1 and G2=n2,m2 be two graphs; then, the degree behavior of vertices in the graph G1⊕G2 is(3)dG1⊕G2v=dG1v+n1n2,ifv∈VG1,dG2v+n1,ifv∈VG2.Lemma 2.
Let we have three simple connected graphsG1=n1,m1 and SG1=n1′,m1′; then, the degree behavior of vertices in the graph G1∔G2 is(4)dG1+G2v=dG1v,ifv∈VG1,2+n2,ifv∈VsG1,dG2v+m1,ifv∈VG2.
The following result gives the formula of the first Zagreb index forG1⊕G2.Theorem 1.
LetG1n1,m1 and G2n2,m2 be two simple graphs; then, the first Zagreb index of corona join product G1⊕G2 is given as(5)M1G1⊕G2=M1G1+n1M1G2+n13n2n2+1+4n1m1n2+m2n1.Proof.
From the definition of the first Zagreb index, we have(6)M1G1⊕G2=∑v∈VG1⊕G2dG1⊕G2v2.
Now, we apply Lemma1:(7)=∑v∈VG1dG1v+n1n22+∑v∈VG1∑v∈VG2dG2v+n12=∑v∈VG1dG1v2+n1n22+2dG1vn1n2+∑v∈VG1∑v∈VG2dG2v2+n12+2dG2vn1=∑v∈VG1dG1v2+n1n22∑v∈VG11+2n1n2∑v∈VG1dG1v+∑v∈VG11∑v∈VG2dG2v2+n12∑v∈VG11∑v∈VG21+2n1∑v∈VG11∑v∈VG2dG2v=M1G1+n13n22+4m1n1n2+n1M1G2+n13n2+4m2n12=M1G1+n1M1G2+n13n2n2+1+4n1m1n2+m2n1,which is our required result.
The next theorem is about the exact expression of the second Zagreb index forG1⊕G2.Theorem 2.
For simple graphsG1n1,m1 and G2n2,m2, the second Zagreb index of corona join product G1⊕G2 is given as(8)M2G1⊕G2=n1n2M1G1+n1M1G2+M2G1+n1M2G2+m2n1n12+4m1+n12n2m1n2+2m1+2m2n1+n12n2.Proof.
From the definition of the second Zagreb index, we have(9)M2G1⊕G2=∑uv∈EG1⊕G2dG1⊕G2u.dG1⊕G2v.
Now, we apply the Lemma1:(10)=∑uv∈EG1dG1u+n1n2dG1v+n1n2+∑v∈VG1∑uv∈EG2dG2u+n1dG2v+n1+∑v∈VG1∑uv∈EG1⊕G2u∈VG1v∈VG2dG1u+n1n2dG2v+n1=∑uv∈EG1dG1udG1v+n1n2∑uv∈EG1dG1u+dG1v+n1n22∑uv∈EG11+n1∑uv∈EG2dG2udG2v+n12∑uv∈EG2dG2u+dG2v+n13∑uv∈EG21+n1∑u∈VG1dG1u∑v∈VG2dG2v+n12∑u∈VG1dG1u∑v∈VG21+n12n2∑v∈VG2dG2v∑u∈VG11+n13n2∑u∈VG11∑v∈VG21=M2G1+n1n2M1G1+m1n12n22+n1M2G2+n12M1G2+m2n13+4m1m2n1+2n12n2m1+2n13n2m2+n14n22=n1n2M1G1+n1M1G2+M2G1+n1M2G2+m2n1n12+4m1+n12n2m1n2+2m1+2m2n1+n12n2,which is our required result.Theorem 3.
LetG1=n1,m1 and G2=n2,m2 be two simple graphs; then, the first reformulated Zagreb index of corona join product G1⊕G2 is(11)EM1G1⊕G2=5n1n2M1G1+n1M1G2+EM1G1+n1EM1G2+8m1m2+4m1n1n2n1n2−2+4m2n12n1−2+n1+n1n2−24m1n1n2+4m2n12+n12n2n1+n1n2−2.Proof.
From the definition of the first reformulated Zagreb index and by Lemma1, we have(12)EM1G1⊕G2=∑uv∈EG1⊕G2dG1⊕G2u+dG1⊕G2v−22=∑uv∈EG1dG1u+n1n2+dG1v+n1n2−22+n1∑uv∈EG2dG2u+n1+dG2v+n1−22+n1∑uv∈EG1⊕G2u∈VG1v∈VG2dG1u+n1n2+dG2v+n1−22=∑uv∈EG1dG1u+dG1v−2+2n1n22+n1∑uv∈EG2dG2u+dG2v−2+2n12+n1∑uv∈EG1⊕G2u∈VG1v∈VG2dG1u+dG2v+n1+n1n2−22=∑uv∈EG1dG1u+dG1v−22+4n12n22+4n1n2dG1u+dG1v−2+n1∑uv∈EG2dG2u+dG2v−22+4n12+4n1dG2u+dG2v−2+n1∑uv∈EG1⊕G2u∈VG1v∈VG2dG1u+dG2v2+n1+n1n2−22+2n1+n1n2−2dG1u+dG2v=∑uv∈EG1dG1u+dG1v−22+4n12n22∑uv∈EG11+4n1n2∑uv∈EG1dG1u+dG1v−8n1n2∑uv∈EG11+n1∑uv∈EG2dG2u+dG2v−22+4n13∑uv∈EG21+4n12∑uv∈EG2dG2u+dG2v−8n12∑uv∈EG21+n1∑uv∈EG1⊕G2u∈VG1v∈VG2dG1u+dG2v2+n1n1+n1n2−22∑uv∈EG1⊕G2u∈VG1v∈VG21+2n1n1+n1n2−2∑uv∈EG1⊕G2u∈VG1v∈VG2dG1u+dG2v=∑uv∈EG1dG1u+dG1v−22+4n12n22∑uv∈EG11+4n1n2∑uv∈EG1dG1u+dG1v−8n1n2∑uv∈EG11+n1∑uv∈EG2dG2u+dG2v−22+4n13∑uv∈EG21+4n12∑uv∈EG2dG2u+dG2v−8n12∑uv∈EG21+n1∑u∈VG1dG1u2∑v∈VG21+n1∑v∈VG2dG2v2∑u∈VG11+2n1∑u∈VG1dG1u∑v∈VG2dG2v+n1n1+n1n2−22∑u∈VG11∑v∈VG21+2n1n1+n1n2−2∑u∈VG1dG1u∑v∈VG21+2n1n1+n1n2−2∑v∈VG2dG2v∑u∈VG11=EM1G1+4n12n22m1+4n1n2M1G1−8n1n2m1+n1EM1G1+4n13m2+4n12M1G2−8n12m2+n1n2M1G1+n12M1G2+8m1m2n1+n12n2n1+n1n2−22+4m1n1n2n1+n1n2−2+4m2n12n1+n1n2−2=5n1n2M1G1+n1M1G2+EM1G1+n1EM1G2+8m1m2+4m1n1n2n1n2−2+4m2n12n1−2+n1+n1n2−24m1n1n2+4m2n12+n12n2n1+n1n2−2,hence the required result.
In the following results, we discussed the Zagreb indices of the subdivision vertex join graph.Theorem 4.
For simple graphsG1=n1,m1 and G2=n2,m2, the first Zagreb index of subdivision vertex join G1∔G2 is given as(13)M1G1∔G2=M1G1+M1G2+n2n2+22+4m1m2+n2m12.Proof.
Applying Lemma2 in the definition of first Zagreb index, we have the following:(14)M1G1∔G2=∑v∈VG1∔G2dG1∔G2v2=∑v∈VG1dG1v2+∑v∈VsG12+n22+∑v∈VG2dG2v+m12=∑v∈VG1dG1v2+∑v∈VsG12+n22+∑v∈VG2dG2v2+m12+2m1dG2v=∑v∈VG1dG1v2+2+n22∑v∈VsG11+∑v∈VG2dG2v2+m12∑v∈VG21+2m1∑v∈VG2dG2v=M1G1+m1n2+22+M1G2+n2m12+4m1m2=M1G1+M1G2+m1n2+22+4m1m2+n2m12,which is our required result.Theorem 5.
LetG1=n1,m1 and G2=n2,m2 are two simple graphs and G1∔G2 be their subdivision vertex join graph; then, the second Zagreb index of G1∔G2 is given as(15)M2G1∔G2=m1M1G2+M2G2+m12m2+m12+n22m2+m1n2+2+n2∑uv∈ESG1dG1u.Proof.
From the definition of the second Zagreb index, we have(16)M2G1∔G2=∑uv∈EG1∔G2dG1∔G2u.dG1∔G2v.
Now, we apply Lemma2:(17)=∑uv∈ESG1u∈VG1v∈VsG1dG1u2+n2+∑uv∈EG2dG2u+m1dG2v+m1+∑uv∈EG1+G2u∈VsG1v∈VG22+n2dG2v+m1=2+n2∑uv∈ESG1u∈VG1v∈VsG1dG1u+∑uv∈EG2dG2udG2v+m1dG2u+dG2v+m12+2+n2∑u∈VsG11∑v∈VG2dG2v+m1=2+n2∑uv∈ESG1u∈VG1v∈VsG1dG1u+∑uv∈EG2dG2udG2v+m1∑uv∈EG2dG2u+dG2v+m12∑uv∈EG21+2+n2∑u∈VsG11∑v∈VG2dG2v+m1∑v∈VG21=2+n2∑uv∈ESG1dG1u+M2G2+m1M1G2+m12m2+m12+n22m2+m1n2=m1M1G2+M2G2+m12m2+m12+n22m2+m1n2+2+n2∑uv∈ESG1dG1u.Theorem 6.
LetG1=n1,m1 and G2=n2,m2 are two graphs; then, the first reformulated Zagreb index of subdivision vertex join G1+G2 is given as(18)EM1G1∔G2=5m1M1G2+EM1G2+m1n2n22+m12+n222m12+m1′+4m1m22m1+n2−2+∑uv∈ESG1dG1u2+2n2dG1u.Proof.
From the definition of the first reformulated Zagreb index, we have(19)EM1G1∔G2=∑uv∈EG1∔G2dG1∔G2u∔dG1∔G2v−22.
Now, we apply Lemma2:(20)=∑uv∈ESG1u∈VG1v∈VsG1dG1u+2+n2−22+∑uv∈EG2dG2u+m1+dG2v+m1−22+∑uv∈EG1+G2u∈VsG1v∈VG22+n2+dG2v+m1−22=∑uv∈ESG1u∈VG1v∈VsG1dG1u+n22+∑uv∈EG2dG2u+dG2v−2+2m12+∑uv∈EG1+G2u∈VsG1v∈VG2n2+dG2v+m12=∑uv∈ESG1u∈VG1v∈VsG1dG1u2+n22+2n2dG1u+∑uv∈EG2dG2u+dG2v−22+2m12+22m1dG2u+dG2v−2+∑uv∈EG1+G2u∈VsG1v∈VG2n22+dG2v+m12+2n2dG2v+m1=∑uv∈ESG1u∈VG1v∈VsG1dG1u2+n22∑uv∈ESG11+2n2∑uv∈ESG1u∈VG1v∈VsG1dG1u+∑uv∈EG2de2+4m12∑uv∈EG21+4m1∑uv∈EG2dG2u+dG2v−8m1∑uv∈EG21+n22∑u∈VsG11∑v∈VG21+∑u∈VsG11∑v∈VG2dG2v2+m12∑u∈VsG11∑v∈VG21+2m1∑u∈VsG11∑v∈VG2dG2v+2n2∑u∈VsG11∑v∈VG2dG2v+2m1n2∑u∈VsG11∑v∈VG21=∑uv∈ESG1u∈VG1v∈VsG1dG1u2+2n2dG1u+m1′n22+EM1G2+4m12m2+4m1M1G2−8m1m2+m1n23+m1M1G2+m13n2+4m12m2+4m1m2n2+2m12n22=5m1M1G2+EM1G2+m1n2n22+m12+n222m12+m1′+4m1m22m1+n2−2+∑uv∈ESG1dG1u2+2n2dG1u,which is our required result.


## 3. Conclusion

In this study, we proposed two new variants of special graph products and found their exact expressions for the first Zagreb index, second Zagreb index, and first reformulated Zagreb index. These new graph invariants can be used to construct come cellular networks, for example, the factor cellular network or some hybrid cellular networks. In future, other degree and distance-based topological indices of these graph operations can be found. The obtained results may help to construct and investigate the topological indices of complex networks structures.



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