The Separator Of K-circle Graphs

Graph spectral theory is an important research area of algebraic graph theory and combined matrix theory.Since the spectrum of graphs has a wide range of applications,the research on it has always been a hotspot on graph theory.In 2009,J.X.Li put forward the concept of separator and Laplacian separator of the graph.In 2011,Z.F.You gave the concept of signless Laplacian separator of the graph.The separator,the Laplacian separator and signless Laplacian separator of the graph can well reflect the structural characteristics of the graph.Therefore,the research on them has become one of the hot topics in the field of graph spectral theory in recent years.The studies of the separator,Laplace separator and signless Laplacian separator of the graph have been obtained some results.In 2009,Y.Z.Fan.was the first one to give the range of Laplacian separator of trees.In 2015,X.G.Jian gave the maximum signless Laplacian separator of unicyclic or bi-cyclic graphs.In 2017,G.D.Yu studied the maximum Laplacian separator of unicyclic or bicyclic graphs.This paper mainly studies the second maximum separator of the unicyclic graphs,the second maximum Laplacian separator of the unicyclic graphs,the second maximum signless Laplacian separator of the unicyclic graphs,the maximum Laplacian separator of the k-cyclic graphs,and characterizes the corresponding extremal graphs.The main contents are arranged as follows:In first chapter,it introduces the background and significance of this study,the current research situation,relevant symbols and definitions,and gives the structure of the full text.In second chapter,it mainly adopts the coalescence graph formula of the adjacency matrix.According to the maximum eigenvalue order of the unicyclic graph and the lower bound of the second largest eigenvalue,it gives the second largest separator of the unicyclic graph,and characterizes the corresponding graph.In third chapter,it mainly adopts the coalescence graph formula of the Laplacian matrix and the relevant conclusions of subgraph with a circle length of 3,it gives the second largest Laplacian separator of the unicyclic graph,and characterizes the corresponding graph.In fourth chapter,it mainly adopts the coalescence graph formula of the signless Laplacian matrix and the relevant conclusions of subgraph with a circle length of 3,it gives the second largest signless Laplacian separator of the unicyclic graph,and characterizes the corresponding graph.In fifth chapter,it mainly adopts the coalescence graph formula of the Laplacian matrix and the relevant conclusions of subgraph with a circle length of 3.According to the upper bound of the maximum eigenvalue of the con-nected graph,it gives the maximum Laplacian separator of the k-cyclic graph,and characterizes the corresponding graph.