Kite-free distance-regular graphs
Let \(G\) be a distance-regular graph with classical parameters \((d,b, \alpha, \beta)\). A kite of length \(i\) of \(G\) is a 4-tuple \((x, y, z, u)\) of vertices such that \(x\), \(y\) and \(z\) are mutually adjacent, and \(u\) is at distance \(d(u,x) = i\), and \(d(u,y) = d(u,z) = i - 1\), where \(2 \leq i \leq d\) and \(d\) is the diameter of \(G\). This paper shows that \(G\) contains no kite of length \(i\) for all \(2 \leq i \leq d\) if \(d \geq 3\) and \(b < -1\). Using this result the Hermitian forms graph \(\text{Her} (d,q)\) can be uniquely determined only by its intersection numbers, which improves a result of Ivanov and Shpectorov concerning the Hermitian forms graph \(\text{Her}(d,q)\).
scientific article; zbMATH DE number 790744
scientific article