# Can entropy of a network be written as a polynomial?

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 $$\log$$?)

• It seems clear that $E$ is not a polynomial, but the last (parenthetical) sentence in your question suggests that maybe you intended to ask instead about approximating $E$ by polynomials. Jun 20 '19 at 10:51
• What is $\mathbf{11}^{\mathcal T}$? (Are you using both $T$ and $\mathcal T$ to mean 'transpose'?) Jul 4 '19 at 0:57
• @LSpice That is a typo, should be $T$, transpose Jul 4 '19 at 1:00
• So I guess $\mathbf 1\mathbf 1^T$ is the all-$1$s matrix? Jul 4 '19 at 1:17
• @LSpice yes also this is a rank one matrix Jul 4 '19 at 2:31