Energy and spectral radius of Zagreb matrix of graph with applications

The $\mathcal{Z}$-matrix of a simple graph $\Gamma$ 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 $\Gamma$, and zero otherwise. The Zagreb eigenvalues of $\Gamma$ are the eigenvalues of its $\mathcal{Z}$-matrix and the Zagreb energy of $\Gamma$ 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 $\Gamma$ is a graph obtained by identifying $u\in\mathcal{V}(\Gamma_1)$ and $v\in\mathcal{V}(\Gamma_2)$ or adding an edge between $u$ and $v$, then it is important to study the relation between Zagreb energies of $\Gamma_1$, $\Gamma_2$ and $\Gamma$. 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 $\mathcal{Z}$-matrix. Also, the novel correlation of the density and refractive index of $n$-alkanes with spectral radius of the $\mathcal{Z}$-matrix has been observed.