Left Multiplication Operators On The Riordan Group And Riordan Digraphs

The theory of Riordan arrays and Riordan group is an important research subject in combinatorics.It can be used to study combinatorial sequences,combinatorial identities,and various enumerative combinatorial problems.In this thesis,we introduce the left multiplication operators on the Riordan group,and carry out a systematic study on them.As a result,it can be found that the left multiplication operators provide a unified approach to numerous subgroups,operators and transformations of the Riordan group.Additionally,we also introduce the concept of Riordan digraphs,and study their properties.Thus,some special integer sequences are well related to digraphs,and some properties of integer sequences can be reflected to those of digraphs by using the theory of Riordan arrays.The main contents of this thesis are as follows:(1)By using the left multiplication operators,we obtain a family H of infinite Riordan subgroups.This family includes not only some well-known subgroups,but also the subgroup families introduced recently in literature.Fundamental properties of the left multiplication operators and the subgroup family H are presented.The Riordan involutions and pseudo-involutions,the stabilizer subgroups in family H,and the n-th commutator subgroup of any subgroup in family H are studied.(2)Some further applications of the left multiplication operators are also provided.In particular,we study a kind of left multiplication operators which are closely related to the diagonal translation operator and the[m]-complementary operator,and enrich the theory of complementary Riordan arrays.Moreover,we introduce and study the higher-level lifts of Riordan arrays as well as the vertical and horizontal 1/m Riordan arrays.(3)We define the Riordan digraphs on the basis of Riordan arrays,and give the conditions of the edge-sets of the Riordan digraphs by using the basic properties of Riordan arrays.Next,a sufficient condition for the existence of a Hamilton path in a Riordan digraph and a sufficient condition for a Riordan digraph to be primitive are given.Finally,a method to construct isomorphic Riordan digraphs is proposed by using diagonal translation operator on the Riordan group.