Yamada Polynomial Of Spatial Graphs And The Density Of Their Zeros

Spatial graph theory developed in the 1980s since topologists began using the methods of knot theory on intrinsic knotting and linking of graphs in R3 or S3.Spatial graph theory arises as a natural generalization of the knot theory.In particular,in 1983,Conway and Gordon proved that any embedding of the complete graph K7 in R3 contains a knotted cycle and any embedding of the complete graph K6 in R3 contains a pair of linked cycles.Later,this area came to be known as spatial graph theory to distinguish it from the study of abstract graphs.It is natural that the modern theory of spatial graphs combines topological and graph-theoretical methods.The power of polynomial invariants of knot as well as polynomial invariants for graphs was a natural motivation for investigation of polynomial invariants of spatial graphs initiated by L.H.Kauffman.In 1989,S.Yamada introduced Yamada polynomial of spatial graphs in R3.It is a concise and useful invariant for spatial graphs.Being motivated by problems on knotting and linking of DNA and chemical compounds,the study of spatial graphs is in the center of interest for last decades.There are many interesting results on Yamada polynomial and its generalizations.In this paper,we mainly focus on the Yamada polynomial of spatial graphs and the distri-bution of their zeros.The main contents are as follows:1.We study properties of the Yamada polynomial of spatial graphs and obtain the relations of Yamada polynomial of spatial graphs with the chain polynomial of labeled graphs.2.We give a method to construct spatial graphs via labeled graphs,and we study properties of the Yamada polynomial of spatial graphs we constructed.In particular,we discuss formulae for computing the Yamada polynomial of spatial graphs obtained by replacing edges of cycle graphs,theta-graphs,or bouquet graphs by spatial parts.3.A certain class spatial graphs are considered.By calculating the Yamada polynomial,we study the zeros distribution of Yamada polynomial for them.We prove that zeros of the Yamada polynomial of spatial graphs are dense in the complex plane.