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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 $\log$?)

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    $\begingroup$ 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. $\endgroup$ Jun 20 '19 at 10:51
  • $\begingroup$ What is $\mathbf{11}^{\mathcal T}$? (Are you using both $T$ and $\mathcal T$ to mean 'transpose'?) $\endgroup$
    – LSpice
    Jul 4 '19 at 0:57
  • $\begingroup$ @LSpice That is a typo, should be $T$, transpose $\endgroup$ Jul 4 '19 at 1:00
  • $\begingroup$ So I guess $\mathbf 1\mathbf 1^T$ is the all-$1$s matrix? $\endgroup$
    – LSpice
    Jul 4 '19 at 1:17
  • $\begingroup$ @LSpice yes also this is a rank one matrix $\endgroup$ Jul 4 '19 at 2:31

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