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(Γ₁) and v ∈ V(Γ₂) or adding an edge between u and v, then it is important to study the relation between Zagreb energies of Γ₁, Γ₂ 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.