Some Studies On The Endomorphisms Of Graphs

The autmorphisms of graph connects group with graph theory,and becomes an important and effect way in the studying of the graph theory.Generalizing the way,the endomorphisms of graph tries to establish the relationship between graph theory and algebra theory of semigroup and to apply the theory of semigroups to graph theory.Many authors pay much attention to endomorphism monoids of graphs,regularities of graphs and Green's equivalences of endomorphism monoids.Such as Weimin Li gives a Green's equivalences of strong-endomorphisms in[16].Maeki posed a open question in[21]:the endomorphisms monoids of which graphs are regular? Moreover,M.Bottcher and U.Knauer in[5]introduced the endomorphism spectrum and the endomorphism type of a graph,and the following question was posed:For a graph X, under which conditions do the sets hEndX,1EndX,qEndX form monoids? However,it seems difficult to obtain a general answer to the above questions.So the strategy for solving these questions are finding various kinds of conditions for various kinds of graphs.In the paper,for the above question,we mainly take the n-prism,the split graph and the join of two split graphs as investigative objects and investigate some problems on the bases of these.Firstly,we character the endomorphisms of the n-prism prove that it is regular. Consequently,we give D Green's equivalences of it.Moreover,we character the quasi-strong endomorphisms of the n-prism when n is even and m≥2 and it was proved that it is not regular and the quasi-strong endomorphisms of n-prism forms monoids.At the same time, the corresponding endomorphism spectrum and the endomorphism type of the kind of graph were given.Secondly,we discuss the number of idempotent and the image of endomorphism of the End-regular split graphs.Accordingly,the number of L class and R class was given. At last,we character the local-strong endomorphisms of the join of two split graphs and prove that quasi-strong endomorphisms of it is local-strong.And we give a condition under which the join of two split graphs forms monoids.