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Ternary cyclotomic polynomials are polynomials of the form Φpqr(z) = ρ (z − ρ), where p < q < r are odd primes and the product is taken over all primitive pqr-th roots of unity ρ. We show that for every p there exists an infinite family of polynomials Φpqr such that the set of coefficients of each of these polynomials coincides with the set of integers in(More)
Given an abelian group A, a graph G is said to be A-magic if there exists a labeling l : E(G) → A − {0} such that the induced vertex labeling l + : V (G) → A defined by l + (v) = u∈N (v) l(uv) is a constant map. A graph G is said to be non-magic if for any abelian group A, it is not A-magic. Also, a Z Z-magic graph G is said to be K-nonmagic if G is not Z Z(More)
A ternary inclusion-exclusion polynomial is a polynomial of the form Q {p,q,r} = (z pqr − 1)(z p − 1)(z q − 1)(z r − 1) (z pq − 1)(z qr − 1)(z rp − 1)(z − 1) , where p, q, and r are integers ≥ 3 and relatively prime in pairs. This class of polynomials contains, as its principle subclass, the ternary cyclotomic polynomials corresponding to restricting p, q,(More)