  We show that the (normalized) symmetric Laplacian of a simple graph can be
obtained from the partial trace over a pure bipartite quantum state that
resides in a bipartite Hilbert space (one part corresponding to the vertices,
the other corresponding to the edges). This suggests an interpretation of the
symmetric Laplacian's Von Neumann entropy as a measure of bipartite
entanglement present between the two parts of the state. We then study extreme
values for a connected graph's generalized R\'enyi-$p$ entropy. Specifically,
we show that
  (1) the complete graph achieves maximum entropy,
  (2) the $2$-regular graph: a) achieves minimum R\'enyi-$2$ entropy among all
$k$-regular graphs, b) is within $\log 4/3$ of the minimum R\'enyi-$2$ entropy
and $\log4\sqrt{2}/3$ of the minimum Von Neumann entropy among all connected
graphs, c) achieves a Von Neumann entropy less than the star graph.
  Point $(2)$ contrasts sharply with similar work applied to (normalized)
combinatorial Laplacians, where it has been shown that the star graph almost
always achieves minimum Von Neumann entropy. In this work we find that the star
graph achieves maximum entropy in the limit as the number of vertices grows
without bound.
  Keywords: Symmetric; Laplacian; Quantum; Entropy; Bounds; R\'enyi.
