  Graph states play an important role in quantum information theory through
their connection to measurement-based computing and error correction. Prior
work has revealed elegant connections between the graph structure of these
states and their multipartite entanglement content. We continue this line of
investigation by identifying additional entanglement properties for certain
types of graph states. From the perspective of tensor theory, we tighten both
upper and lower bounds on the tensor rank of odd ring states
($|R_{2n+1}\rangle$) to read $2^n+1 \leq rank(|R_{2n+1}\rangle) \leq
3*2^{n-1}$. Next, we show that several multipartite extensions of bipartite
entanglement measures are dichotomous for graph states based on the
connectivity of the corresponding graph. Lastly, we give a simple graph rule
for computing the n-tangle $\tau_n$.
