Zero-Divisor Semigroups of Fan-Shaped Graph

Abstract:

      Let $P_{n}$ be a path graph with $n$ vertices, and let $F_n=P_n\cup\{c\}$, where $c$ is adjacent to all vertices of $P_{n}$. The resulting graph is called a fan-shaped graph. The corresponding zero-divisor semigroups have been completely determined by Tang et al. for $n=2,3,4$ and by Wu et al. for $n\geq 6$, respectively. In this paper, we study the case for $n=5$, and give all the corresponding zero-divisor semigroups of $F_n$.