The enhanced power graph P_(E)(S)of a semigroup S is a simple graph whose vertex set is S and two vertices a,y∈S are adjacent if and only if c,y∈(z)for some z∈S,where(z)is the subsemigroup generated by z.In this pa...The enhanced power graph P_(E)(S)of a semigroup S is a simple graph whose vertex set is S and two vertices a,y∈S are adjacent if and only if c,y∈(z)for some z∈S,where(z)is the subsemigroup generated by z.In this paper,we first describe the structure of P_(E)(S)for an arbitrary semigroup S,and then discuss the connectedness of P_(E)(S).Further,we characterize the semigroup S in the cases when P_(E)(S)is separately a complete,bipartite,regular,tree and null graph.The planarity,together with the minimum degree and independence number,of P_(E)(S)is also investigated.The chromatic number of a spanning subgraph,i.e.,the cyclic graph,of P_(E)(S)is proved to be countable.In the final part of this paper,we construct an example of a semigroup S such that the chromatic number of P_(E)(S)need not be countable.展开更多
基金the support of MATRICS Grant(MTR/2018/000779)funded by SERB,India.
文摘The enhanced power graph P_(E)(S)of a semigroup S is a simple graph whose vertex set is S and two vertices a,y∈S are adjacent if and only if c,y∈(z)for some z∈S,where(z)is the subsemigroup generated by z.In this paper,we first describe the structure of P_(E)(S)for an arbitrary semigroup S,and then discuss the connectedness of P_(E)(S).Further,we characterize the semigroup S in the cases when P_(E)(S)is separately a complete,bipartite,regular,tree and null graph.The planarity,together with the minimum degree and independence number,of P_(E)(S)is also investigated.The chromatic number of a spanning subgraph,i.e.,the cyclic graph,of P_(E)(S)is proved to be countable.In the final part of this paper,we construct an example of a semigroup S such that the chromatic number of P_(E)(S)need not be countable.
