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Abstract. Let G be the circulant graph C n (S) with S ⊆ {1, 2, . . . , ⌊ n 2 ⌋}, and let I(G) denote its the edge ideal in the ring R = k[x 1 , . . . , x n ]. We consider the problem of determining when G is Cohen-Macaulay, i.e, R/I(G) is a Cohen-Macaulay ring. Because a Cohen-Macaulay graph G must be well-covered, we focus on known families of wellcovered circulant graphs of the form C n (1, 2, . . . , d). We also characterize which cubic circulant graphs are Cohen-Macaulay. We end with the observation that even though the well-covered property is preserved under lexicographical products of graphs, this is not true of the Cohen-Macaulay property.

A sign pattern Z (a matrix whose entries are elements of {+, −, 0}) is spectrally arbitrary if for any selfconjugate spectrum there is a real matrix with sign pattern Z having the given spectrum. Spectrally arbitrary sign patterns were introduced in [5], where it was (incorrectly) stated that if a sign pattern Z is reducible and each of its irreducible components is a spectrally arbitrary sign pattern, then Z is a spectrally arbitrary sign pattern, and it was conjectured that the converse is true as well; we present counterexamples to both of these statements. In [2] it was conjectured that any n × n spectrally arbitrary sign pattern must have at least 2n nonzero entries; we establish that this conjecture is true for 5 × 5 sign patterns. We also establish analogous results for nonzero patterns.

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