Distance Power And A_?-spectrum Of Cayley Graphs

Since the concept of Cayley graph was put forward,it has occupied an im-portant position in algebraic graph theory because of its symmetry and diversity.As the basic research direction of algebraic graph theory,spectral graph theory takes the spectrum of graph as the main object to study the related properties of graph.The distance power of Cayley graph is a further exploration of the spectral properties of Cayley graph,and a proposal of Semi-Cayley graph can also be used to better understand symmetric graphs.Based on these,this pa-per mainly studies the distance power of integral Cayley graphs on generalized dihedral groups and the A?-spectrum of Semi-Cayley graphs on Abel groups.Starting from integral Cayley graphs on generalized dihedral groups.In the second chapter,we study the sufficient conditions for the distance power of an integral Cayley graph on generalized dihedral group to be integral.According to the structure of the generalized dihedral group,we find its automorphism group and give the sufficient conditions for the distance power of a special subset on generalized dihedral group to be integral.Semi-Cayley graph is a natural generalization of Cayley graph.In the third chapter,according to the structure and adjacency matrix of Semi-Cayley graph on Abel group,we calculate the A?-spectrum expression of Semi-Cayley graph on Abel group.And apply it to special Semi-Cayley graph to obtain their A?-spectrum.At the same time,according to the isomorphism of graph,the formula of A?-spectrum of dihedral group and bicyclic group is obtained.