The associated graded algebras of biserial fractional Brauer graph algebras

For any finite-dimensional algebra $A$, one can associate a graded algebra (denoted by $\mathrm{g}\mathrm{r}(A)$) with the radical filtration of $A$. An interesting question is to compare the representation theories of $A$ and $\mathrm{g}\mathrm{r}(A)$. When $A$ is a Brauer graph algebra with defining Brauer graph $\mathrm{\Gamma }$, we have described the representation type of $\mathrm{g}\mathrm{r}(A)$ in terms of the information of $\mathrm{\Gamma }$. In this paper, we describe the representation type of $\mathrm{g}\mathrm{r}(A)$ when $A$ is a biserial fractional Brauer graph algebra - a notion that properly generalizes a Brauer graph algebra - in terms of the information of the defining biserial fractional Brauer graph $\mathrm{\Gamma }$ of $A$.