The Spectrum of A Class of Graphs Derived From Grassmann Graphs

Let n; k be positive integers such that n ≥ 3, k < n2 . let q be a power of a prime p and fq be a finitefield of order q. let v (q; n) be a vector space of dimension n over fq. we define the graph s(q; n; k) asa graph with the vertex set v = vk [ vk+1, where vk and vk+1 are subspaces in v (q; n) of dimension kand k + 1 respectively, in which two vertices v and w are adjacent whenever v is a subspace of w or w isa subspace of v. it is clear that the graph s(q; n; k) is a bipartite graph. in this paper, we study someproperties of this graph. in particular, we determine the spectrum of the graph s(q; n; k).