Distance-Regular Graphs With Eigenvalues Satisfying Certain Conditions

The classification of distance-regular graphs is an important problem in algebraic combination.The method of eigenvalues of graphs is one of the important ones of studying distance-regular graphs.In this thesis,we study the classification and some properties of distance-regular graphs which eigenvalues satisfying certain conditions.The main results are as follows:1.Let ’ be a non-bipartite distance-regular graphs with diameter D and small eigenvalue θ_min≤-4/5k.When D=6 and D=7,we give the classification of Γ.（1）When D=6,Γ is one of the following graph:（a）the 13-gon with intersection array {2,1,1,1,1,1;1,1,1,1,1,1},（b）the Odd graph O₇ with intersection array {7,6,6,5,5,4;1,1,2,2,3,3},（c）the folded 13-cube with intersection array {13,12,11,10,9,8;1,2,3,4,5,6}.（2）When D=7 and c₄≤4,Γ is one of the following graph:（a）the 15-gon with intersection array {2,1,1,1,1,1,1;1,1,1,1,1,1,1};（b）the Odd graph O₈ with intersection array {8,7,7,6,6,5,5;1,1,2,2,3,3,4};（c）the folded 15-cube with intersection array {15,14,13,12,11,10,9;1,2,3,4,5,6,7}.2.Let s be an integer with s≥ 3,and let Γ be a non-complete distance-regular graph with a₁≠ 1.We prove that if k satisfies some certain conditions,then Γ is a geometric distance regular graph with smallest eigenvalue-s.3.Let Γ be a distance-regular graph with diameter D=4 and distinct eigenvaluesθ₀>θ₁>…>θ₄.When a₄ is an eigenvalue of Γ,we give the relations of the parameters of Γ and prove that a₄=θ₁ or a₄=θ₂.