General Fifth M-Zagreb Indices and General Fifth M-Zagreb Polynomials of Dyck-56 Network

A chemical graph can be represented by polynomial, a numerical value or a matrix form. Topological indices are major source of the relationship between physical and biological properties and behavior of chemical structures. Topological indices are used in development quantitative structure activity relationships in which properties of molecules are correlated with their chemical structure. Let 𝐺(𝑉, 𝐸) be a graph with vertex set 𝑉 and edge set 𝐸, with |𝑉| = 𝑚 and |𝐸| = 𝑛. Let 𝑑𝑢 represent the degree of a vertex 𝑢 in graph 𝐺. 𝑆𝑢 stands for sum of degrees of the vertices incident with vertex 𝑢 in 𝐺. Milan Randic [1] introduced the first degree-based Topological index, i.e Randic Index, Defined as;

Furthermore, Bollabas and Erdos [2] introduced general Randic index as;

Gutman and Trinajstic [3] introduced first and second Zagreb indices as;

Recently in [4], Kulli introduced new indices that are general fifth 𝑀₁- Zagreb indices denoted by 𝑀₁ ^𝛼𝐺₅(𝐺) and general fifth 𝑀₂- Zagreb indices denoted by 𝑀₂ ^𝛼𝐺₅(𝐺) defined as;

The general fifth 𝑀₁- Zagreb polynomial denoted by 𝑀₁^𝛼𝐺₅(𝐺, 𝑥) and general fifth 𝑀₂- Zagreb polynomial denoted by 𝑀^𝛼₂𝐺₅(𝐺, 𝑥) is defined as;

Dyck-56_𝑛×𝑛 Networks were built up by unit representation of Dyck−56 (Dyck 1880) by identification procedure. Further detail about formation of Dyck Network can be found in [5]. Dyck graph is the 3-regular graph with 32 vertices and 48 edges, having chromatic number 2 and chromatic index 3 with diameter 5. Further details of formation of Dyck- 56𝑛×𝑛 network can be found in [5,6]. Selvan and Naranyakar in [7] calculated general Randic Index, First Zagreb index, ABC and GA indices for Dyck-x, ABC and GA indices for Dyck-56_𝑛×𝑛 network.

Figure 1 Dyck-56_𝑛×𝑛(𝐴) Network has 12𝑛² + 4𝑛 vertices and 18𝑛² − 2𝑛 edges. Based on the sum of the degrees of incident vertices to the end vertices of each edge, there are seven types of edges of Dyck-56_𝑛×𝑛(𝐴) Network. This edge partition is presented in (Table 1) [8-10].

Table 1: Edge Partition of Dyck-56_𝑛×𝑛(A) Network.

Theorem 1: Let 𝐺 be the graph Dyck-56_𝑛×𝑛(𝐴) Network, and the general fifth 𝑀₁- Zagreb index for 𝐺 is

Proof. Let 𝐺: = Dyck-56_𝑛×𝑛(𝐴) Network; the above result can be found by using Table 1 and equation 5.
So,

Which further reduces to

Theorem 2: Let 𝐺 be the graph Dyck-56_𝑛×𝑛(𝐴) Network, and the general fifth 𝑀₂- Zagreb index for 𝐺 is

Proof. Let 𝐺 be Dyck-56_𝑛×𝑛(𝐴) Network. The above result can be found using (Table 1) and equation 6.
So,

Which further reduces to

Theorem 3: Let 𝐺 be the graph Dyck-56_𝑛×𝑛(𝐴) Network, and the general fifth 𝑀₁- Zagreb polynomial for 𝐺 is [

Proof. Let 𝐺 be Dyck-56𝑛×𝑛(𝐴) Network. The above result can be found using (Table 1) and equation 7.
So,

Theorem 4: Let 𝐺 be the graph Dyck-56𝑛×𝑛(𝐴) Network, The general fifth 𝑀2- Zagreb polynomial for 𝐺 is

Proof. Let 𝐺 be Dyck-56_𝑛×𝑛(𝐴) Network. The above result can be found by using (Table 1) and equation 7.
So,

Figure 2 Dyck-56_𝑛×𝑛(𝐵) Network has 18𝑛² − 3𝑛 vertices and 24𝑛2 − 4𝑛 + 8 edges. On the base of sum of degrees of incident vertices to end vertices of each edge there are seven types of edges of Dyck-56_𝑛×𝑛(𝐵) Network. This edge partition is given in (Table 2) [11-13].

Theorem 5: Let 𝐺 be the graph Dyck-56𝑛×𝑛(𝐵) Network, The general fifth 𝑀1- Zegreb index for G is

Proof. Let 𝐺 be Dyck-56_𝑛×𝑛(𝐴) Network, The above result can be found by using (Table 1) and equation 5.
So,

Theorem 6: Let G be the graph Dyck-56_𝑛×𝑛(𝐵) Network. The general fifth 𝑀₂- Zegreb index for G is

Proof. Let 𝐺 denotes Dyck-56_𝑛×𝑛(𝐵) Network. The above result can be found by using (Table 2) and equation 6.

Table 2: Edge Partition of Dyck-56𝑛×𝑛(B) Network.

Theorem 7: Let 𝐺 be the graph Dyck-56_𝑛×𝑛(𝐵) Network, The general fifth 𝑀₁- Zagreb polynomial for 𝐺 is

Proof. Let 𝐺 is Dyck-56_𝑛×𝑛(𝐵) Network, The above result can be found by using (Table 2) and equation 7. So,

Theorem 8: Let G be the graph Dyck-56_𝑛×𝑛(𝐵) Network, The general fifth 𝑀₂- Zegreb polynomial for G is

Proof. Let 𝐺 be Dyck-56_𝑛×𝑛(𝐵) Network. The above result can be found by using (Table 2) and equation 8.

Degree based topological indices like General fifth M1 and General fifth M2 Zagreb indices for Dyck-56 Network are found in this paper. These indices are useful in study of QSAR/QSPR. Furthermore, General fifth M1 and General fifth M2-Zagreb polynomials for Dyck-56 Networks are found. These indices and polynomials are useful for the study to understand correlation between physical structures with chemical properties.

None.

No conflict of interest.

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