The Normality Of 3,4-Valent Cayley Graphs Of Dihedral Groups Of Order 2~mp

A Cayley graph ?= Cay(G,S)of the group G with its subset S(?)1 is said to be normal if R(G),the group of right multiplications is normal in Aut(?);The graph ? is said Graph Regular Representation(GRR)if R(G)= Aut(?)and ?is undirected.In this paper,we study the related properties of the small valent Cayley graphs of dihedral groups by making use of some basic ideas and common methods of algebraic graph theory and relying on the knowledge of group theory.The first two chapters,as far as the related properties of Cayley graphs,main-ly state the research background and significance,analyse the current situation and development trend.Also,some basic concepts,properties and conclusions needed in this paper are introduced.In Chapter 3,we research the relative properties of 3-valent undirected Cay-ley graphs of dihedral groups of order 2m p,G =(a,b| a2m-1p?b2 = 1,ab = a-1>,where p is an odd prime and m>4.The 3-variable self-inverse generated sub-set S of G#is divided into four types under the action of Aut(G),then we solve completely the problem for the normality of 3-valent undirected Cayley graphs of the group G.In addition,we obtain many infinite families of 3-valent graphical regular representations and prove that none of such graphs are arc-transitive.In Chapter 4,we consider the 4-valent undirected Cayley graphs of dihedral groups G of order 2m p,G =(a,b|a2m-1p = b2 =1,ab = a-1),where p is an odd prime and m>2,and give the complete classification of 4-variable self-inverse generated subset S of G#G#under the action of Aut(G).Then we study the problems for the normality of the corresponding Cayley graphs and obtain many infinite families of 4-valent(non-)normal Cayley graphs and graphical regular representations for the group G.