Yamada Polynomials Of A Class Of Spatial Connected Graphs

As an extension of knot theory,spatial graph theory plays an important role in the field of topology.Topology and graph theory are closely related.Modern spatial graph theory combines graph theory with topology research methods.Both polynomial invariants of graphs and polynomial invariants of knots have good topological properties.In the research of spatial graph theory,one of its core contents is to classify spatial graph equivalently.Since the 1980s,inspired by the method of studying polynomial invariants in knot theory,scholars from home and abroad have done a lot of research on spatial graph polynomial invariants,and have found many meaningful and valuable results.Yamada polynomial is a very important representation of spatial graph polynomial invariants.This paper mainly studies the properties of Yamada polynomials for a special class of spatial connected graphs.Firstly,for two graphs G and K,we construct a special class of graphs,1(n)and(m,n)by changing the number of edges connecting vertex pairs between these two graphs.Secondly,on the basis of the Yamada polynomial property of the two-point connected graph,shrinking and reducing the edges of graph,1(n)by using the property of the non-circular edge of the Yamada polynomial,we obtain the relationship between the Yamada polynomial of the graph,1(n)and the Yamada polynomial of the graph G and K.At the same time,using a similar approach,we obtain the relationship between the Yamada polynomial of the graph(m,n)and the Yamada polynomial of the graph G and K.Through further calculation,we extend the properties of Yamada polynomials of this kind of special two-point connected graph to general cases.Finally,according to the properties of the Yamada polynomial of the two-point connected graph,after further extension,we give the properties of a special class of Yamada polynomials of the three-point connected graph.Given two graphs₁G andG₂,we denote graphG₁(44)G ₂as a three-point connected graph of graph₁G andG₂,and use mathematical induction to describe the situations of the edge numberq(G)of graphG₁(44)G ₂,we find the relationship between the Yamada polynomials of this kind of special three-point connected graph and the Yamada polynomials of graph₁G andG₂ by combining the properties of the Yamada polynomials.