Let $ F_q $ be a finite field of characteristic 2. Let $ x^2 + Sx +P \in F_q[x] $ be an irreducible polynomial over $ F_q $, and let $ g $ be one of its roots in $ F_{q^2} $. Define a map $ M: F_{q^2}^* \rightarrow F_{q^2}^* $ by $$ M(c_1 + c_2 g) = c_1 + c_2 + c_2 g, $$ where $ c_1 $ and $ c_2 $ are in $ F_q $. Consider a directed graph over the vertex set $ F_{q^2}^* $, where there is an edge from $ a $ to $ b $ iff there is $ \alpha\in F_q $ such that $ b = M( (\alpha +g) a) $. What are the eigenvalues and eigenvectors of the adjacency matrix of this graph? Note that without M, the graph is just a Cayley graph over $ F_{q^2}^* $, thus its eigenvalues and eigenvectors can be determined easily.
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$\begingroup$ What is your motivation? $\endgroup$– Nick GillCommented May 30, 2014 at 17:01
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$\begingroup$ It comes up in one of my research problems $\endgroup$– user3208Commented May 30, 2014 at 17:05
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1$\begingroup$ Well, yes, I kind of guessed that! The question as presented though seems a bit random (to me, others may well disagree) - it might benefit from some context. Plus the way it arose in your research may help to shed light on how to solve it. $\endgroup$– Nick GillCommented May 30, 2014 at 17:10
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$\begingroup$ You are right that it looks random. I have simplified it. $\endgroup$– user3208Commented May 30, 2014 at 19:43
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