Distance-regular Graphs With Strongly Closed Subgraphs

In this thesis, we consider the strongly closed subgraphs of distance-regular graphs. By means of intersection diagrams, properties of the distance-regular graphs and the known conclusionsof distance-regular graph, we prove the following conclusions :? LetΓdenote a 2-homogenous distance-regular graph with diameter D≥2 and c₂ > 1. Fix a integer m(1≤m < D), suppose there exists a strongly closed subgraphsΩwith diameter m ofΓ, thenγ_i = 1(2≤i≤m) and c_i satisfy c_i = c_i-1(c₂ -1) + 1(1≤i≤m + 1).? LetΓ= (X, E) denote a D-bounded distance-regular graph with diameter D≥2 and c₂ = 2, a_D = 0, then the following are equivalent:(i)Γis 2-homogenous.(ii) For any x, y, w satisfying (?)(x, y) = (?)(x, w) = i -1(2≤i≤D),(?)(w, y) = 2, we have? LetΓ= (X, E) denote a D-bounded 2-homogenous distance-regular graph. Assume c₂ = 2, a_D=0, thenΓis bipartite.? LetΓ= (X, E) denote a D-bounded distance-regular graph with diameter D≥2, and with the intersection number c₂ = 2, a_D = a_D-1 =0. ThenΓis D-bounded if and only ifΓis Hamming graph H(k, 2).? LetΓ= (X, E) denote a 2-homogenous D-bounded distance-regular graph with diameterD≥2, and with the intersection number c₂ = 2,a_D≠a_D-1 =0. Then the intersection array ofΓis {2D + 1,2D,…, D + 2; 1,2,…,D}.? LetΓ= (X, E) denote a distance-regular graph with classical parameters (D, b,α,β), and D≥4. Assume that a₂ > a₁ = 0 and c₂ > 1. For any x∈X, x₃∈Γ₃(x), if there exists a connected subgraph C' inΓ₃(x) such thatΩ= [x, C'] which is a strongly closed subgraph containing x, x₃ with diameter 3. ThenΓis 4-bounded.