Studies On The Properties Of Integral Invertible Graphs

A graph G is nonsingular if its adjacency matrix A（G） is nonsingular.A nonsingular graph G is said to have an inverse if A（G）^-1 is signature similar to a nonnegative matrix.Let H be the class of connected bipartite graphs with unique perfect matchings.In order to better describe the properties of inverse graph in H,Godsil proposed the concept of inverse graph in 1985,which was called a positive invertible graph and denoted as G⁺in this paper.And Pavl（?）kov（?） and （?）ev（?）ovi（?） presented the concept of negative invertible graph in 2017.This paper mainly studied the related properties of integral invertible graph（the inverse matrix of its adjacency matrix contains only integers）in H.The paper is divided into four chapters.In the first chapter,we first introduce the defi-nitions and notations used in the paper.Secondly,we present main results of this paper.In chapter 2,we study the bicyclic graph with unicyclic and bicyclic inverse graph,which is characterized by the path structure and the expansion of odd and even type,and give the result that bicyclic graph with unicyclic and bicyclic inverse.In chapter 3,we study the k-cyclic graph with c-cyclic inverse graph,which is charac-terized by the path structure,the expansion of odd and even type and simple minimal path,and give the result that k-cyclic graph with c-cyclic inverse.In chapter 4,we study the inverse of Kronecker product of graphs G and H,and judge the structure and positive and negative properties,based on the inverse matrix of the adja-cency matrix of graphs G and H and the mixed product property of the Kronecker product.