Distance-regular Graphs With Kite-free Or Triangle-free

In this thesis we study two kinds of distance-regular graphs. (1) Distance-regular graphs without any kites of length 2. If the eigenvalues ofΓsatisfy (?), obtain the equivalentconditions of the multiplicity ofθsatisfying mult(θ) = a₁ + 1by means of algebraical method. (2) Triangle-free distance-regular graphs. In this part, we get the relationship of some intersection numbers of T by using combinatory method and the circle chasing technique.We obtain the following conclusions:·LetΓ= (X, E) be a distance-regular graph with diameter d≥3 and valency k≥3.Suppose there are not any kites of length 2 in T and a₁≠0. Letθbe a nontrivial eigenvalue ofΓ, and E = |X|^(-1)(?) which is primitive idempotent with respect toθ. Then the following(1) - (3) hold:(1)θ=-(?).(2) For all vertices x,y∈X such that (?)(x, y) = 1, we have(3) There exist vertices x, y∈X such that (?)(x, y) = 1, and such that·LetΓ= (X, E) be a distance-regular graph with diameter d≥3 and valency k≥3. Suppose there are not any kites of length 2 in T and a₁≠0. Letθbe a nontrivial eigenvalueofΓ, and E = |X|^(-1) (?) which is primitive idempotent with respect toθ. If the conditions(1) - (3) hold in the theorem above, then the following (1) - (3) are equivalent:(1) mult(θ) = a₁ + 1.(2) For all x, y∈X with (?)(x, y) = 1, form a basis for EV.(3)There exist x,y∈X with (?)(x, y) = 1, such that·LetΓ= (X, E) be a distance-regular graph with diameter d≥5, valency k≥4 and a₁ = 0. For any i∈{2, 3,…, d - 3},γ_i exists, butγ_d-2 does not exist. If c₂ = 1, thenγ_i = 1, where i∈{2,3,…, d - 3}, and b_d-3≥2.·Let = (X, E) be a distance-regular graph with diameter d≥5, valency k≥4 and a₁ = 0. For any i∈{2,3,…, d - 3},γ_i exists, butγ_d-2 does not exist. If c₂ = 1 and a_d-4 = 1, then a_d-3 = 1≤a_d-2, where d≡2(mod3), and the intersection array ofΓis:·LetΓ= (X, E) be a distance-regular graph with diameter d≥5, valency k≥4 and a₁ = 0. For any i∈{2,3,…, d - 3},γ_i exists, butγ_d-2 does not exist. Ifγ_d-1 exists and c₂ = 1, thenγ_d-1 = 0 and b_d-2 = 1 hold.·LetΓ= (X, E) be a distance-regular graph with diameter d≥5, valency k≥4, a₁ = 0 and c₂ = 1. For any i∈{2,3,…, d - 3},γ_i exists, butγ_d-2 does not exist. IfΓis also a distance-transitive graph or a symmetric distance-regular graph, we have d = 5 or 6.