Let M be a finitely generated free semimodule over a semiring S with identity having invariant basis number property with a basisα={α1,...,αk}.The complement■of the reduced non-zero component graph■of M,is the si...Let M be a finitely generated free semimodule over a semiring S with identity having invariant basis number property with a basisα={α1,...,αk}.The complement■of the reduced non-zero component graph■of M,is the simple undirected graph with■as the vertex set and such that there is an edge between two distinct vertices■and■if and only if there exists no i such that both ai,biare non-zero.In this paper,we show that the graph■is connected and find its domination number,clique number and chromatic number.In the case of finite semirings,we determine the degree of each vertex,order,size,vertex connectivity and girth of■.Also,we give a necessary and sufficient condition for■to be Eulerian or Hamiltonian or planar.展开更多
基金supported by CSIR Emeritus Scientist Scheme(21(1123)/20/EMR-II)of Council of Scientific and Industrial Researchthis research is also supported by Dr.M.G.R.Research Scholarship by Manonmaniam Sundaranar University。
文摘Let M be a finitely generated free semimodule over a semiring S with identity having invariant basis number property with a basisα={α1,...,αk}.The complement■of the reduced non-zero component graph■of M,is the simple undirected graph with■as the vertex set and such that there is an edge between two distinct vertices■and■if and only if there exists no i such that both ai,biare non-zero.In this paper,we show that the graph■is connected and find its domination number,clique number and chromatic number.In the case of finite semirings,we determine the degree of each vertex,order,size,vertex connectivity and girth of■.Also,we give a necessary and sufficient condition for■to be Eulerian or Hamiltonian or planar.
