Antipodal Graphs And Bipartite Graphs With Q-polynomial

There are two parts in this thesis. In the first part, we study antipodal graph. First, by mean of combination we obtain some conditions of antipodal graph; then by studying the algebraical character of graph we find some new results about antipodal graph. In the second part, we study Q-polynomial bipartite distance-regular graphs with c₂ = 2,3,and obtain some relation about the intersection numbers ofΓby using a partition of vertice ofΓ. The following are our main results.? LetΓ= (X, E) be a distance-regular graphs with diameter d and valency k≥3. If b₁ = c_d-1, b₃ = 1, thenΓis antipodal.? LetΓ= (X, E) be a distance-regular graphs with valency k≥3 and diameter d = 2e + 1, while e∈N^*, e≥2. If b_i =c_d-i, i∈{1,2,…, e - 1, e + 1}, thenΓis antipodal.? LetΓ= (X, E) be a distance-regular graphs with valency k≥3 and diameter d = 2e, while e∈N^*, e≥2. If b_i = c_d-i,i∈{1,2,…, e -1}, then F is antipodal.? LetΓ= (X, E) denote a bipartite distance-regular graph with Q-polynomial structure, and with d≥3.1f b₁ = c_d-1, thenΓis antipodal 2-cover.? LetΓbe a antipodal graph with d≥3, k≥3. Suppose E is a nontrivial primitive idempotent ofΓ, andθ₀^*,θ₁^*,…,θ_d^* are dual eigenvalue sequence with respect to E. IfΓis Q-polynomial respect to E, thenmoreover the quotient is 1 or -1.? LetΓ= (X,E) denote a bipartite distance-regular graph with Q-polynomial structure, and with d≥4, k≥3. If c₂ = 2, then one of the following holds:(1)Γis 2-homogeneous.(2) c_i+1= c_i + 1 or c_i= c_i-1 + 1 can not hold simultaneously,where 2≤i≤d-1.? LetΓ= (X, E) denote a bipartite distance-regular graph with Q-polynomial structure, and with d≥4, k≥3. If c₂ = 3, then the following hold:(1) 2c_i = c_i+1 - 1 or c_i = c_i-1 + 1 can not hold simultaneously, where 2≤i≤d-1. (2) b_i = b_i+1 + 1 and 2b_i = b_i-1 - 1 can not hold simultaneously, where 2≤i≤d-1.