Long Cycles And Spectral Of Graphs

Spectral graph theory is a very important field of study in algebraic graph theory.The main content it studies is the spectral properties of various algebraic representations of graphs,by studying the eigenspaces of graphs and relating the eigenvalues of graphs to the topology of graphs,especially with various structural parameters of graphs.It is an important part of algebraic graph theory by applying algebraic and geometric theory to inscribe the structure of the topological properties of graphs,and by using the topological structure of graphs to study problems related to algebra and geometry.At the same time,spectral graph theory has a wide range of applications in extreme value problems and computer science.So spectral graph theory has become an important and interesting topic in algebraic graph theory and combinatorial matrix theory today.When a graph contains a cycle containing all points,we call it a Hamiltonian.Determining whether a given graph is a Hamiltonian is an NP-complete problem.The edge and spectral conditions for graphs to be Hamiltonian have been well studied by many scholars,and good results have been obtained.In 2016,Bo Ning proposed edge and spectral conditions for graphs of minimal degree to be Hamiltonian,and in another paper,revealed edge and spectral radius conditions for graphs containing circles of length.In this paper,we focus on the Hamiltonian problem and the longest circle problem for graphs.In this paper,we study and give spectral sufficient conditions for graphs containing the longest circles.Firstly,the spectral characterizations of several special graphs are given.Secondly,the sufficient conditions when a simple connected graph contains a cycle of length n-2 are given.Finally,the longest cycle of a graph with a given toughness of 1 is at least Characterize the spectrum of n-1.The main contents are organized as follows:In Chapter 1,we firstly introduce background and significance of spectral graph theory,and related terms and concepts.Lastly,we introduce the research problem and development,and the main conclusions of this paper.In Chapter 2,we discuss the spectral inscription of the perimeter of a diagram;In Chapter 3,outlook for the latter part of the year.