Let q(G) be the minimum number of distinct eigenvalues taken over all real symmetric matrices whose underlying graph is G . In this note, we investigate the parameter q(G) when G is a sunlet graph. A ( n, ℓ )-sunlet graph, denoted by S n ℓ , is a graph obtained from a cycle C n connecting one pendant path P ℓ to each vertex of the cycle. Our main result is to determine the exact value of q(S n ℓ ) when n is even.

Let f ∈ K(t) be a univariate rational function. It is well known that any non-trivial decomposition g o h, with g,h ∈ K(t), corresponds to a non-trivial subfield K(f(t)) ⊊ L ⊊ K(t) and vice-versa. In this paper we use the idea of principal subfields and fast subfield-intersection techniques to compute the subfield lattice of K(t)/K(f(t)). This yields a Las Vegas type algorithm with improved complexity and better run times for finding all non-equivalent complete decompositions of f.