There are sure concoction exacerbates that are helpful for the survival of living things. Carbon, oxygen, hydrogen and nitrogen are the primary components that aides in the generation of cells in the living things. Carbon is a fundamental component for human life. It is helpful in the arrangements of proteins, sugars and nucleic acids. It is crucial for the development of plants as carbon dioxide. The carbon atoms can bond together in different ways, called allotropes of carbon. The outstanding structures are graphite and jewel. As of late, numerous new structures have been found including nanotubes, buckminster fullerene and sheets, precious stone cubic structure, and so forth. The utilizations of various allotropes of carbon are talked about in detail in [1].

A graph *G* is simply a collection of points and lines that connect the points or subset of points. The points are called vertices of *G* and lines are called edges of *G*. The vertices set and edges set of *G* are denoted as *V*(*G*) and *E*(*G*), respectively. If *e* is an edge of *G* that connects the vertices *u* and *v*, then we can write *e* = *uv*. A graph is called connected graph if there is a path between all pairs of vertices. The *degree of a vertex v* in the graph *G* is the number of edges which are incident to the vertex *v* and will be represented by *d*_{v}.

Let *Γ* be the family of finite graphs. A function *T* from *Γ* into set of real numbers having *T*(*G*) = *T*(*H*) property, for isomorphic *G* and *H*, is called a topological index. Someone can clearly notice that the vertices cardinality and the edges cardinality are topological indices. The earliest known topological index is Wiener index [2] and its based on distance, it is characterized as the sum of the half of distances between every pairs of vertices in a graph.

If *u*, *v* ∈ *V*(*G*), then the distance between the vertices *u* and *v* is given by the length of any arbitrary shortest path in *G* that connects *u* and *v*. Another well known and one of the earliest degree dependent index was due to Milan Randi´c [3] in 1975, characterized as the sum of the negative square root of the product of degree of the end vertices of each edge of the graph.

One can define the family of atom bond connectivity topological indices [4] consisting of elements(member) of the form $ABC(G)={\displaystyle \sum _{uv\in E(G)}\sqrt{\frac{{J}_{u}+{J}_{v}-2}{{J}_{u}{J}_{v}}}}$, where *J*_{u} is some number that in a uniquely way can be assigned with the vertex *u* of graph *G*. One of the element of *Γ* is the atom bond connectivity index introduced by Estrada et al. [5]:

$$ABC(G)={\displaystyle \sum _{uv\in E(G)}\sqrt{\frac{{d}_{u}+{d}_{v}-2}{{d}_{u}{d}_{v}}}}.$$(1)

Another well known member of *Γ* is the fourth version of atom bond connectivity denoted as *ABC*_{4} topological index of a graph *G*, introduced by Ghorbhani *et.al*. [6]:

$$AB{C}_{4}(G)={\displaystyle \sum _{uv\in E(G)}\sqrt{\frac{{S}_{u}+{S}_{v}-2}{{S}_{u}{S}_{v}}}}.$$(2)

where ${S}_{u}={\displaystyle \sum _{uv\in E(G)}{d}_{v}},{S}_{v}={\displaystyle \sum _{uv\in E(G)}{d}_{u}}$.

Here, we define a new member of this family *Γ*, namely multiple atom bond connectivity index and it is defined as follows:

$$AB{C}_{M}(G)={\displaystyle \sum _{uv\in E(G)}\sqrt{\frac{{M}_{u}+{M}_{v}-2}{{M}_{u}{M}_{v}}}}.$$(3)

where ${M}_{u}={\displaystyle \prod _{uv\in E(G)}{d}_{v}},{M}_{v}={\displaystyle \prod _{uv\in E(G)}{d}_{u}}$.

A family *Λ* of geometric arithmetic topological indices consisting of elements(member) of the form $GA(G)={\displaystyle \sum _{uv\in E(G)}\frac{2\sqrt{{J}_{u}{J}_{v}}}{{J}_{u}+{J}_{v}}}$, where *J*_{u} is some number that in a uniquely way can be assigned with the vertex *u* of *G*. One of the other member of *Λ* is the geometric arithmetic index *GA* of a graph *G* introduced by Vuki*č*evi*ć et.al*. [7]:

$$GA(G)={\displaystyle \sum _{uv\in E(G)}\frac{2\sqrt{{d}_{u}{d}_{v}}}{{d}_{u}+{d}_{v}}}.$$(4)

Another well known member of *Λ* is the fifth version of geometric arithmetic index and is denoted by *GA*_{5} topological index of a graph *G*, introduced by Graovoc *et.al*. [8]:

$$G{A}_{5}(G)={\displaystyle \sum _{uv\in E(G)}\frac{2\sqrt{{S}_{u}{S}_{v}}}{{S}_{u}+{S}_{v}}}.$$(5)

Here, we define a new member of *Λ* namely multiple geometric -arithmetic index and it is characterized as:

$$G{A}_{M}(G)={\displaystyle \sum _{uv\in E(G)}\frac{2\sqrt{{M}_{u}{M}_{v}}}{{M}_{u}+{M}_{v}}}.$$(6)

For more information and properties of topological indices, see [9 10 11 12 13 14 15].

Moreover this idea of computing the topological indices is helpful to discuss the concept of entropy. The entropy of Shannon, Rényi and Kolmogorov are analyzed and compared together with their main properties. The entropy of some particular antennas with a pre-fractal shape, also called fractal antennas, is studied [16]. Fractional derivative of the Riemann zeta function has been explicitly computed and the convergence of the real part and imaginary parts are studied with the help of topological indices [17, 18].

The aim of this paper is the introduction of the multiple atom bond connectivity index and multiple geometric arithmetic index. As an application we shall compute these new indices for the octagonal grid ${O}_{p}^{q}$, the hexagonal grid *H*(*p*, *q*) and the square grid *G*_{p}_{,q}. Also, we compared these results obtained with the ones obtained by other indices like fourth atom bond connectivity index and fifth geometric arithmetic index via their computation too. But first we shall see some examples.

#### Example 1

*Let G* = *K*_{n} be the complete graph, then for all u ∈ *V*(*K*_{n})*, the d*_{u} = *n* − 1*, so M*_{u} = (*n* − 1)_{n−1}*. Thus*

$$\begin{array}{l}AB{C}_{M}(G)=\frac{n\times \sqrt{{(n-1)}^{n-1}-1}}{\sqrt{2}{(n-1)}^{n-2}}\\ G{A}_{M}(G)=\frac{n(n-1)}{2}.\end{array}$$

#### Example 2

*If G* = *C*_{n} be the cycle graph, then for all u ∈ *V*(*C*_{n})*, then d*_{u} = 2*, so M*_{u} = 4*. Thus*

$$\begin{array}{l}AB{C}_{M}(G)=\frac{n}{4}\sqrt{6}\\ G{A}_{M}(G)=n.\end{array}$$

#### Example 3

*If G* = *P*_{n}, n ≥ 5 *be the path graph of length n, then for all u* ∈ *V*(*P*_{n})*, we can compute easily as*:

$$\begin{array}{l}AB{C}_{M}(G)=\frac{n-5}{4}\sqrt{6}+\frac{4}{\sqrt{2}}\\ G{A}_{M}(G)=n-1.\end{array}$$

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