In my research, I met a problem here.

Consider a weighted graph Laplacian matrix

$$\mathcal{L}_w(\mathcal{G}) = DWD^T,$$ where $D$ is a incidence matrix and $W$ is diagonal with each diagonal elements representing the weight of edge. Note that we know the graph Laplacian $\mathcal{L}(\mathcal{G}) = DD^T$.

Consider the entropy defined as $$E(w) = -\log \det(\mathcal{L_w(\mathcal{G})}+\gamma \mathbf{1}\mathbf{1}^T ),$$ where $\gamma$ is a small scalar.

Suppose the diagonal elements of $W$, i.e., $w$, are variables. Therefore, $E(w)$ is a function of $w_i$.

My question is can $E(w)$ be written as a polynomial?

(Should I choose the way of applying Taylor series for $\log$?)