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Levelness versus almost Gorensteinness of edge rings of complete multipartite graphs

Levelness and almost Gorensteinness are well-studied properties on graded rings as a generalized notion of Gorensteinness. In the present paper, we study those properties for the edge rings of the complete multipartite graphs, denoted by $\Bbbk[K_{r_1,\ldots,r_n}]$ with $1 \leq r_1 \leq \cdots \leq r_n$. We give the complete characterization of which $\Bbbk[K_{r_1,\ldots,r_n}]$ is level in terms of $n$ and $r_1,\ldots,r_n$. Similarly, we also give the complete characterization of which $\Bbbk[K_{r_1,\ldots,r_n}]$ is almost Gorenstein in terms of $n$ and $r_1,\ldots,r_n$.