Let Gn be the number of graphs with n vertices let G n be th

Let Gn) be the number of graphs with n vertices, let G (n) be the number of connected graphs with n vertices, and let GD(n) be the number of cartesian prime graphs with n vertices. Prove that (G, is a commutative unital semiring

Solution

Let (S, +, ) be a commutative unital semiring and let : (S, +, ) (R0, +, ·) be a semiring homomorphism. We say that (S, +, , ) is an arithmetical semiring if (S, +, ) is an additive arithmetical semiring and (S +, , ) is a unique factorization semigroup such that S + (1) = {e}

Let (S, +, , ) be a monotonic arithmetical semiring for which G + 1 holds. The divisor function d : S + R0 that to each S S + assigns the total number d(S) of factorizations of the form S = S S for some S , S S +. Since d(S) = 2 for all S S d + max(n) 2 + X n/2 r=2 S +(r) 2 + S +(2) + S + jn 2 k + X n/3 r=3 S +(r)S + jn 3 k + 3 r = 2 + S +(2) + O S + jn 2 k . If we additionally assume that the sequence {S +(n)} is unbounded (which is the case if axiom the arithmetical semiring is strictly monotone), then d + max(n) = O S +

n 2 and thus d satisfy the assumptions of Corollary 26. We conclude that d has asymptotic meanvalue 2 and asymptotic variance 0 on S +.

. Let (S, +, , ) be a strictly monotonic arithmetical semiring for which axiom G + 1 holds. Let d : S + R0 be the unitary-divisor function that to each S S + assigns the number d(S) of factorizations S = S S for some S , S S + that are -coprime i.e. such that S and S have no common -prime factors. Since d(S) d(S) for each S S + and thus (d) + max d + max,. In particular d has asymptotic mean value 2 and asymptotic variance 0. Similarly, let : S + N be the prime-divisor function defined by the formula (S 1 1 S 2 2 · · · S m m ) = 12 · · ·m for all S1, . . . , Sm S and non-negative integers 1, . . . , m. Then (S) d(S) for each S S +, and thus has asymptotic mean value 1 and asymptotic variance 0.