1, the significance of the topic, the main research content and the key problems to be solved.

2. The main contents of this paper are: the existence of cayley graphs, hamilton cycles and paths of groups, and some special and commonly used groups are summarized.

3. Significance of the topic: 1. The highly abstract group is concretized into a visible model corresponding to the group structure. 2. This paper establishes the relationship between two important modern disciplines "Group Theory" and "Graph Theory". 3. This paper also makes us have a further understanding of some "old friends" cyclic groups, dihedral groups, direct products of groups, generators and their operational relations.

4. Key problems to be solved: The graphical representation of some special groups and the existence of Hamilton cycles and paths are summarized, and we try to prove the familiar theorems from the graphs and derive some results. For Hamilton Road and cayley({(a, 0), (b, 0), (e,1)}: existence of Hamilton cycle in Q4+zm), the existence of Hamilton cycle in cayley graph ({(a, 0), (b, 0), (e) is proved. An undirected cayley graph composed of two generators and its existence are summarized.

5. Basis of demonstration and innovation of research: This paper introduces the concept of cayley graph of groups, and studies and summarizes some commonly used groups. Studying cayley graphs of groups will give us an intuitive understanding of abstract groups and observe the excellent properties of cayley graphs of some special groups. Learning this topic will not only further understand and review cyclic groups, dihedral groups, direct products of groups, generators and their operational relations, but also be very interesting.

The innovation of research is to express cayley graphs of some special groups, and observe the relationship between groups (such as the direct product of groups) through graphs, so as to prove and generalize the existence of hamilton cycles and paths of some special groups, such as the existence of hamilton groups, cayley graphs of q4+zm, q8+zm, s6 and the existence of hamilton cycles.

6, take an examination of literature directory

1 Jiang changhao, graph theory and network flow, Beijing, China forestry publishing house, XX.7

2 i.grossman w.magnus, Groups and Their Graphs

3 igor pak and rados radoicic, hamilton Path in cayley Graph

7. The overall arrangement and concrete progress of the inspection work.

At the beginning of February and the end of February, I will study the information given to me by Teacher Lin.

Consult relevant information in early March and mid-March.

In late March, the direction of the paper was determined and the final draft began.

The first draft was finalized in early April, and was revised and corrected under the guidance of Teacher Lin.

The paper was completed in early May.

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