Some Studies On The Maximal Eigenvalue In Cacti And The Least Eigenvalue In Non-bipartite Graphs Of A_?-matrix

Through the careful study of the known literature,we found a large number of unresolved question about A?-matrix.For example,when 0???1 and the number of cut edges and match edges are fixed,the structure of cactus with largest eigenvalue and its corresponding characteristic equation are unknown.When 1/2???1,the structure of non-bipartite graphs with minimum A?-eigenvalue is also undiscovered.We take these problems as the starting point to look for a way to solve them.With the help of the tutor and on the basis of reading the literature,we carefully study and analyze the achievements of others,and get inspiration from them,so as to determine the solutions and ideas to solve the problems.The key to solve the problem is to first determine the number and length of cycles in the graph,and seek out the number and length of paths.Finally,the relationship between cycle and cycle,path and path,and cycle and path is described.The study of problems mainly uses classification and comparison,analysis and synthesis,abstraction and generalization.In the process of proof,we mainly use the method of synthesis and counter-certification.Ultimately,we get that the structure of cactus with maximum A?-eigenvalue is related to its order n.If n is odd,the extremal graph contains only n-1/2 triangles,and these triangles have one common vertex;If n is even,the extremal graph contains n-2/2 triangles,and these triangles have one common vertex,but there is a suspending edge at this common vertex.The non-bipartite with minimum A?-eigenvalue has a connection with the value of ?.If?=1/2,the extremal graph is a path length n-3 attaching to a triangle;If 1/2<?<1.,the extremal graph is a star order n-2 attaching to a triangle.