Distance-regular graphs with girth 3 or 4: I

AbstractWe study the structure of a distance-regular graph Γ with girth 3 or 4. First, we find some relationships among the intersection numbers of Γ when Γ contains a cycle {u1, u2, u3, u4} with ∂(u1, u3) = ∂(u2, u4) = 2. These relationships imply the diameter d, valency k, and intersection numbers a1 and cd of Γ are related by d ≤ (k + cd)(a1 + 2). Next, we show subgraphs induced by vertex neighbourhoods in distance-regular graphs where cycles mentioned above do not exist are related to certain strongly regular graphs