In this paper, we discuss the adjacency graph of feedback shift registers (FSRs) whose characteristic polynomial can be written as $g=(x_0+x_1)*f$ for some linear function $f$. For $f$ contains an odd number of terms, we present a method to calculate the adjacency graph of FSR$_{(x_0+x_1)*f}$ from the adjacency graph of FSR$_f$. The parity of the weight of cycles in FSR$_{(x_0+x_1)*f}$ can also be determined easily. For $f$ contains an even number of terms, the theory is not so much complete. We need more information than the adjacency graph of FSR$_f$ to determine the adjacency graph of FSR$_{(x_0+x_1)*f}$.
Besides, some properties about the cycle structure of linear feedback shift registers (LFSR) are presented.
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