The Spectrum of a Class of Graphs Derived From Grassmann Graph

Let n , k be positive integers such that n ≥ 3, k < n/2. Let q be a power of a prime p and F _ q be a finite field of order q. Let V(q,n) be a vector space of dimension n over F_q. We define the graph S( q , n , k )as a graph with the vertex set V = V _ k ⋃ V _ (k+1), where V _ k and V _ (k+1) are subspaces in V( q , n )of dimension k and k+1 respectively, in which two vertices v and ware adjacent whenever v is a subspace of w or w is a subspace of v. It is clear that the graph S(q , n , k )is a bipartite graph. In this paper, we study some properties of this graph. In particular, we determine the spectrum of the graph S(q,n,k).