Algebraic Properties Of Bipartite Distance-regular Graphs

This thesis mainly studies about an inequality involving the cosine sequences of the bipar-tite distance-regular graphs which are the special class, and the relationship between equality being attained and the properties of dual bipartite on Q-polynomial structures. Moreover, we investigate the properties of the eigenvalues and the pseudo cosine sequences, which are about the2-homogeneous bipartite distance-regular graphs. The thesis is divided into five sections.In section1, we introduce the basic concepts and properties.In section2, we show the inequality involving the cosine sequences of the bipartite distance-regular graphs. The conclusion is as follow.Let T=(X, R) denote a bipartite distance-regular graph with diameter d>3, and eigenvalues θ0>θ1>…> θd.Suppose E is a primitive idempotent of T,θ is the eigenvalue associated with E, and σ0, σ,…, σd is the cosine sequence of θ. Then, for all integers i (1≤i≤d-1), σ(σ-σiσi+1)-σ2(1-σi2)≥σi-1(σσi-σi+1).In section3, we obtain an necessary and sufficient condition of the above equality being attained. The conclusion is as follow.Let T=(X, R) denote a bipartite distance-regular graph with diameter d≥3, and eigenvalues θ0>θ1>…>θd. Suppose θ=θ1, and σ0, σ,…, σd is the cosine sequence of θ. Then the following are equivalent.(1)T is dual bipartite Q-polynomial with respect to θ.(2)σi≠1(1≤i<≤d), andσ(σ+σ3)-σ2(1+σ2)=0.In section4, we study the properties of eigenvalues which are about the2-homogeneous bipartite distance-regular graphs. The conclusions are as follow.Let T=(X, R) denote a2-homogeneous bipartite distance-regular graph with diameter d≥4and valency k≥3, eigenvalues θ0>θ1>…>θd. be the distinct eigenvalues of T, and θ1=k-2. Suppose E1is the primitive idempotent corresponding with θ1. Then there exist distinct primitive idempotents of T E and F such that E1(?)F∈Span(F,H). Moreover, let θf,θh be the eigenvalues associated withF, H. Then θf(θf-θh)=2k. .Let T=(X,R)denote a2-homogeneous bipartite distance-regular graph with diameter d≥3and valency k≥3,and let eigenvalues θ0>θ1>…>θd be the distinct eigenvalues of T,and θ1=k-2.Suppose E,F, H are the distinct nontrivial primitive idempotents of T such that E.F∈Span(F,H).(1)Suppose θf=θ1,then θe=k-4,θh=k-6.(2)Suppose θf=θd-1,then θe=k-4,θh=-(k-6).In section5,we study the properties of pseudo cosine sequences of2-homogeneous bipar-tite distance-regular graphs.The conclusion is as follow..Let T=(X,R)denote a2-homogeneous bipartite distance-regular graph with diameter d≥3and valency k≥3,and let eigenvalues θ0>θ1>…>θd denote the distinct eigenvalues of T,and θ1=k-2.For any θ,θ’∈R,suppose E,F,H are the nontrivial pseudo primitive idempotents associated with θ,θ1,θ’,and E(?)F=αF+βH,α,β∈R,αβ≠0.Then θ’=(k-4)θ-4/k-2We can conclude pairθ,θ1is taut.