Research On Balance Of Several Kinds Of Graphs

Algebraic graph theory is an important part of graph theory,which is a relatively active research field in mathematical disciplines at present,and the research results can be widely applied to various disciplines.The adjacency spectrum theory of graphs is an important part of algebraic graph theory,which mainly examines the structure and properties of graphs through the study of the eigenvalues of the adjacency matrix of graphs.The difference between the positive and negative inertia indices of the adjacency matrix of a graph is called the signature of the graph.In recent years,a conjecture for the symbolic difference:-c₃（G）≤s（G）≤c₅（G）,has set off many scholars to study the symbolic difference,and many excellent research results have been achieved.The signature of a graph refers exclusively to the signature under its adjacency matrix,which reflects the characteristics of the adjacency matrix.A graph with zero signature is defined as an balanced graph.A graph is said to be inscribed as balanced if it is determined to be balanced,or if the conditions for its balance have been given.In this dissertation,based on the results of existing studies,the equilibrium of several types of graphs,such as quasi-paths,wheel graphs,and analogue wheel graphs,is inscribed using various methods such as algebraic analysis and simplification of graph structure based on the relationship between positive and negative inertia indices and rank and sign difference in simple graphs.A brief overview of the thesis in this dissertation is given below:First,we analyze the structural characteristics of the quasi-paths,wheel graphs,and analogue wheel graphs,which are obtained by adding a vertex v₀ to the path or circle,where the road and even circle are always balanced.However,the signature may either remain the same or increase or decrease when the vertex v₀ is deleted.Secondly,the graph structure is simplified by deleting circles or hanging trees and compressing internal paths,and the positive and negative inertia indices and signature of the simplified graph are equal to those of the original graph class.The relationship between the positive and negative inertia indices and the signature of the graphs is used to derive the range of values of the signature of these graphs,and then the change in rank after deleting the vertices v₀ is described by equating the adjacency matrices of these graphs or judging whether they are singular matrices,so as to analyze whether these graph classes are balanced.Finally,the accuracy of the research results is demonstrated by specific examples.Figure[14]Table[0]Reference[75]...