On a characterization of bilinear forms graphs
The following distance regular graphs have been characterized by their parameters: the Hamming graphs \(H(d,n)\) with parameters \((d,1,0,n-1)\) for \(d\geq 2\) and \(n\neq 4\); the Johnson graphs \(H(n,d)\) with parameters \((d,1,1,n-d)\) for \((d,n)\neq (2,8)\); the Grassmann graphs \(J_q(n,d)\) with parameters \((d,q,q,\sum_{k=1}^{n-d}q^k)\) for \(d\geq 3\) and \(n\neq 2d\), \(2d+1\) (with some exceptions if \(q=2\) or \(q=3\)); the bilinear forms graphs \(H_q(n,d)\) with parameters \((d,q,q-1,q^n-1)\) for \(n\geq 2d\geq 6\) and \(q\neq 4\). In this paper the bound \(n\geq 2d\) for the bilinear forms graphs is improved. Theorem 1.1. Let \(\Gamma\) be a distance regular graph with classical parameters \((d,q,q-1,\beta)\) and \(d\geq 3\). Suppose that either \(q=2\) and \(\beta \geq q^{d+4}-1\) or \(q\geq 3\) and \(\beta \geq q^{d+3}-1\). Then \(q\) is a prime power, \(\beta=q^n\) for some integer \(n\), and \(\Gamma\) is the bilinear forms graph \(H_q(n,d)\). Corollary 1.3. Suppose that the distance regular graph \(\Gamma\) has classical parameters \((d,q,\alpha,\beta)\) with \(d\geq 3\) and \(\Gamma\) is not a Hamming graph, a Johnson graph, a Grassmann graph, or a bilinear forms graph. Then \(\beta\) is bounded in terms of \(d,\;q\) and \(\alpha\).
scientific article; zbMATH DE number 1310848