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Let $\Fth$ be a single vertex \textsf{k}-graph, and $\pi_\omega(\O_\theta)"$be the von Neumann algebra induced from the GNS representation of adistinguished state $\omega$ of its $\textsf{k}$-graph C*-algebra $\O_\theta$.In this paper, we prove the factoriality of $\pi_\omega(\O_\theta)"$ andfurther determine its type, when either $\Fth$ has the little pull-backproperty, or the intrinsic group of $\Fth$ has rank $0$. The key step toachieve this is to show that the fixed point algebra of the modular actioncorresponding to $\omega$ has a unique tracial state.

Factoriality and type classification of \textsf{k}-graph von Neumann algebras