Characterization of projective graphs

Abstract

We denote the distance between vertices x and y of a graph by d( x, y), and p ij ( x, y) = ∥ { z : d( x, z) = i, d( y, z) = j} ∥. The ( s, q, d)-projective graph is the graph having the s-dimensional subspaces of a d-dimensional vector space over GF( q) as vertex set, and two vertices x, y adjacent iff dim(x ⌢ y) = s − 1 . These graphs are regular graphs. Also, there exist integers λ and μ > 4 so that μ is a perfect square, p 11( x, y) = λ whenever d( x, y) = 1, and p 11( x, y) = μ whenever d( x, y) = 2. The ( s, q, d)-projective graphs where 2d 3 ≤ s < d − 2 and (s, q, d) ≠ ( 2d 3 , 2, d) , are characterized by the above conditions together with the property that there exists an integer r satisfying certain inequalities.