Energy and spectral radius of Zagreb matrix of graph with applications

The Z-matrix of a simple graph Γ is a square symmetric matrix, whose rows and columns correspond to the vertices of the graph and the ij th entry is equal to the sum of the degrees of i th and j th vertex, if the corresponding vertices are adjacent in Γ, and zero otherwise.The Zagreb eigenvalues of Γ are the eigenvalues of its Z-matrix and the Zagreb energy of Γ is the sum of absolute values of its Zagreb eigenvalues.We study the change in Zagreb energy of a graph when the edges of the graph are deleted or rotated.Suppose Γ is a graph obtained by identifying u ∈ V(Γ 1 ) and v ∈ V(Γ 2 ) or adding an edge between u and v, then it is important to study the relation between Zagreb energies of Γ 1 , Γ 2 and Γ.The highlight of the paper is that, the acentric factor of n-alkanes appear to have a strong positive correlation (where the correlation coefficient is 0.9989) with energy of the Z-matrix.Also, the novel correlation of the density and refractive index of n-alkanes with spectral radius of the Z-matrix has been observed.