Zagreb indices and Zagreb energies of some chemical compounds

Introduction

In this article "graph" (molecular graph) referred as is simple finite undirected and connected graph. Let 𝐺 = (𝑉, 𝐸) is a graph with vertex set 𝑉 = (𝑣 1 , 𝑣 2 , 𝑣 3 , … … . . 𝑣 𝑛 ) and the edge set𝐸 = (𝑒 1 , 𝑒 2 , 𝑒 3 , … … . . 𝑒 𝑛 ). 𝑣 𝑖 𝑣 𝑗 Denotes an edge in 𝐺 if 𝑣 𝑖 𝑎𝑛𝑑 𝑣 𝑗 are adjacent in 𝐺. Topological indices [1,9] are the invariant by which the properties of chemical compound are studied and they are modeled as molecular graph based on the structure descriptors [2] . These descriptors help to represent he structural formula of a chemical compound. The vertices of a graph represent the atoms of chemical compound and edges represent the chemical bonds [3] . Recently the authors of [10] have obtained the wiener indices and Zagreb indices for some chemical compounds. By this motivation we have computed Zagreb indices of Glucose, Cholesterol, Vitamin C, Vitamin D, Vitamin K and ethanol. And Zagreb energies of Glucose and Vitamin C are obtained. We refer the following definitions to compute the required Zagreb indices and Zagreb energies for the said chemical compounds. Definition 1.1: Let 𝐺 = (𝑉, 𝐸) is a simple connected graph, and 𝑑 𝑖 = deg (𝑣 𝑖 ) and 𝑑 𝑗 = deg (𝑣 𝑗 ) then first and second Zagreb index [11] is defined as

These topological indices were introduced by M Ghorbani and N Azmi [8] .

Definition 1.2:

The hyper Zagreb index of simple connected graph was introduced by G H Shirdel, H Rezapour and A M Sayadi [4] . It is defined as

Definition 1.3: The first and second multiple Zagreb index [7] of simple connected graph were introduced by I Gutman [6] . It is defined as

Second multiple Zagreb index of 𝐺, 𝑃𝑀 2 (𝐺) = ∏ (𝑑 𝑖 𝑑 𝑗 )

Definition 1.4: The Zagreb matrices [5] for a graph 𝐺 are defined as follows First Zagreb matrix = 𝑍 (1) = 𝑧 𝑖𝑗 = { 𝑑 𝑖𝑖 2 𝑓𝑜𝑟 𝑣 𝑖 ∈ 𝑉(𝐺) 0 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒

Second Zagreb matrix = 𝑍 (2) = 𝑧 𝑖𝑗 = { 𝑑 𝑖 𝑑 𝑗 𝑓𝑜𝑟 𝑣 𝑖 𝑣 𝑗 ∈ 𝐸(𝐺) 0 𝑓𝑜𝑟 𝑖 = 𝑗 0 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒 The first Zagreb Matrix of a graph 𝐺 is 𝑍 (1)

Zagreb indices of Glucose:

Chemical structure of Glucose is (𝐶 6 𝐻 12 𝑂 6 ). Glucose is commonly known as "sugar". It is an important source of energy needed by the cells and organs. For human being energy is required for the normal functioning of the parts of the body. Brain and red blood cells completely depend on Glucose. Level of glucose in the blood is controlled by the insulin which is released by pancreas. Now we obtain the Zagreb indices of glucose. Where [ 16 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 16 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 16 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 16 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 16 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ]

References

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