Inequalities For The Intersection Numbers Of Distance-regular Graphs

In this paper we study a distance-regular graph Γ of diameter d≥3with intersection numbers c2>1and a1=0<a2. In the first part, we introduce the basic concepts and properties about distance-regular graphs. In the second part, we will show the necessary conditions for the existence of a strongly closed subgraph Δ of diameter m (2≤m≤d-1) in Γ, i.e. the inequalities for the intersection numbers of Γ. Using them we can prove there is no existence of a strongly closed subgraph Δ of diameter m in a given Γ. In the third part, let Y have classical parameters (d,b,α,β). Then we obtain some new inequalities for the intersection numbers of Γ. And Y is a Hermitian forms graph Her2(d) iff Ci· a2=ai+1-ai(1≤i≤m). In addition, we get a relation between c3and a2. In this case, Γ is1-homogeneous and its corresponding parameters can be given by b. In the forth part, we use a new way to prove an important inequality of Ahira Hiraki in [8].Main results:●Let Γ be a distance-regular graph of diameter d≥3such that c2>1and a1=0<a2. Assume there exists a strongly closed subgraph A of diameter m (2≤m≤d—1) in Γ. Let i be an integer with1≤i≤m. Then the following hold:(ⅰ) Ci·a2≤ai+1(7) and a2·(Ci-Ci-1)≤ai+1-ai.(8) Moreover, the quality holds in (7) if and only if the quality holds in (8).(ⅱ)(ai-ai-1)·a2≤di+1+Ci+1-ai-Ci(9) and ai·a2≤ai+1+Ci+1-1.(10)●Let Γ be a distance-regular graph of diameter d>3with classical parameters (d, b, α,β) such that c2>1and a1=0<a2. Let i be an integer with1≤i≤m. Then Ci·a2≤ai+1-ai,(11) In particular, the equality holds in (11) if and only if Γ is a Hermitian forms graph Her2(q).●Let Γ be a distance-regular graph of diameter d>3with classical parameters (d,b,α,β) such that c2>1and a1=0<a2. Then Γ is1-homogeneous and its corresponding parameters can be given by b.●Let Γ be a distance-regular graph of diameter d≥3with classical parameters (d, b,α,β) such that c2>1and a1=0<a2. Let θ0> θ1>…> θd be eigenvalues of Γ, and Y be a strongly closed regular subgraph of Γ of diameter two. Then the following hold.(i)θd<-1-b1/3+a2.(ii) c3=(2+(?))(2+a2-(?)).●Let Γ=(X, E) be a distance-regular graph of diameter d>2. Fix an integer (1≤h≤d), and suppose there exists a weak-geodetically closed subgraph Ω of F that has diameter h. Then the intersection numbers of Γ satisfy the following inequality Ci-1-Ci+1+(Ci-Ci-1)C2≤0(1≤i≤h).(12)...