The Inertia Of Distance Matrices Of Some Bicyclic Graphs And Their Line Graphs

Let G be a simple undirected graph with n vertices.The vertices of G are labeled as v₁,v₂…v_n,i.e.,|V?G?|= n.For v_i,vj_?V?G?,the distance between v_i and v_j is the length of a shortest path between them,and is denoted by dij.The distance matrix of G is defined as D?G?=?dij?n×n?.The inertia of the distance matrix of a graph is the triple of the integers?n₊?D?,n0?D?,n_-?D??,where n+?D?,n₀?D?,n_Ddenote the number of positive eigenvalues of D?G?,the multiplicity of zero eigenvalues of D?G?,and the number of negative eigenvalues of D?G?,respectively.In this thesis,we mainly give the inertia of the distance matrices of some bicyclic graphs and their line graphs.In Chapter 2,we mainly consider the bicyclic graphs which contain ?-cycde as their subgraphs.First,for the case that at least one cycle is even,by deleting the the vertices without changing the inertia of distance matrix of the graph,and then using the related conclusions of trees and unicyclic graphs,we get the inertia of distance matrices of these bicyclic graphs;Second,for the case that both cycles are odd,by doing some elementary transformations of matrices such that the distance matrix is congruent to a diagonal matrix,we get the inertia of the distance matrices of these bicyclic graphs.In Chapter 3,we mainly consider the line graphs of some bicyclic graphs,by using a similar method and combining some conclusions in Chapter 2,we get the inertia of the distance matrices of these graphs.