The Research On The Deletion Relation Of Groups And Some Semigroups

It is a heated topic of algebraic graph theory to study the relationship between groups and graphs.Researchers build a bridge between groups or semigroups and graphs by the way of researching automorphism groups and endomorphism monoids of graphs.The work is mainly to investigate the graph and it vertex-deleted subgraph which are fit the finite groups' deletion relation or semigroups' weakly deletion relation.First, the definition of deletion relation between two finite groups which is first given by Stephen G Hartke and Hannah Kolb is introduced. In addition, the theorems of two finite groups' deletion relation are proved completely by using the deduction proof method of the theory about finite group and algebraic graph theory in this paper. Cayley graphs and rigid trees will play a big part in the procedure of proofs.Next, the graphs which satisfy the deletion relation between two finite groups and have relatively less vertices are constructed by applying the definition of graph regular representation (GRR) of group G, which improves the construction method of graphs that is introduced by Stephen G Hartke and Hannah Kolb.Finally, as a generalization of the concept of deletion relation between two finite groups, the definition of semigroups' weakly deletion relation and theorems are determined by making use of the definition of the Cayley graph of semigroup and the quality of weakly vertex-transitive. In addition, one class of graphs which satisfy the definition of semigroups' weakly deletion relation is determined.