Deza graphs based on symplectic spaces

Let F q 2 ν be the 2 ν -dimensional symplectic space over a finite field F q . For 0 ≤ s ≤ ν − 1 and ν ≥ 2 , let W 0 be a fixed subspace of type ( ν + s , s ) in F q 2 ν , and let X be the set of the 1-dimensional subspaces in F q 2 ν not contained in W 0 . Denote by Γ the graph with the vertex set X , in which two vertices P and Q are adjacent if P + Q is non-isotropic. Note that Γ is a subgraph of the symplectic graph S p ( 2 ν , q ) . In this paper, we determine an equitable partition of X and compute its quotient matrix. Moreover, we show that Γ is a strictly Deza graph, compute its parameters and spectra. Finally, we prove that the (fractional) chromatic number of Γ is q ν .