Nonlinear feedback shift registers (NFSRs) are widely used in communication and cryptography. How to construct more NFSRs with maximal periods, which can generate sequences with maximal periods, i.e., de Brujin sequences, is an attractive problem. Recently many results on constructing de Bruijn sequences from adjacency graphs of Linear feedback shift registers (LFSRs) by means of the cycle joining method have been obtained. In this paper we discuss a class of LFSRs with characteristic polynomial p ²(x), where p(x) is a primitive polynomial of degree n ≥ 2 over the finite field F ₂. As results, we determine their cycle structures and adjacency graphs, and further construct a class of new de Bruijn sequences from these LFSRs.