THE FIRST AND SECOND ZAGREB INDICES TO MOLECULAR GRAPHS OF ALKANES, C_n H_((2n+2) ) - Muche, Tilahun; Martin, Sanaa (IJRDO -JOURNAL OF MATHEMATICS, 2023)
For a simple graph G=(V(G) ,E(G)) , let d(u), d(v) be the degree of the vertices u, and v in G. The first and second Zagreb indices of G are defined asM_1 (G) = ∑_(e=uv∈E(G))▒〖(d(u)+d(v))〗 and M_2 (G) = ∑_(e=uu∈E(G))▒〖d(u)d(v)〗, respectively. The first, generalized, and the second Multiplicative Zagreb indices of simple graph G are defined as 〖PM〗_1 (G)= ∏_(e=uv∈E(G))▒(d(u)+d(v)) and 〖PM〗_2 (G)= ∏_(e=uu∈E(G))▒(d(u)d(v)) respectively. The multiplicative Zagreb indices have been the focus of considerable research in computational chemistry dating back to Narumi and Katayama in 1980s.The general class of molecules known as the alkanes, or paraffins, is given by chemical formula C_n H_((2n+2) ) and it is natural to ask how many different molecules, number of atoms are there a with this formula. We derive a simple formula to find the first and second Zagreb indices for any positive integer n , show formulate a python approach to find a vertex distance matrix from a vertex incidence matrix, formulate Zagreb indices by (d(e)), the degree set of adjacent edges, and prove M_1 (G)= ∑_(e ∈E(G))▒〖d(e)〗 + ∑_(u ∈V(G))▒〖d(u)〗, ∑_(e=uv∈E(G))▒〖(d(u))^2+(d(v))^2 〗 = ∑_(u∈V(G))▒(d(u))^3 and the First and the second multiplicative Zagreb indices of a graph G〖PM〗_1 (G)=∏_(e=uv∈E(G))▒(d(e)+2) and 〖PM〗_2 (G)=∏_(u∈V(G))▒(d(u))^((d(u)) )