  In 1983, Conway-Gordon showed that for every spatial complete graph on 6
vertices, the sum of the linking numbers over all of the constituent
2-component links is congruent to 1 modulo 2, and for every spatial complete
graph on 7 vertices, the sum of the Arf invariants over all of the Hamiltonian
knots is also congruent to 1 modulo 2. In this article, we give integral lifts
of the Conway-Gordon theorems above in terms of the square of the linking
number and the second coefficient of the Conway polynomial. As applications, we
give alternative topological proofs of theorems of Brown-Ramirez Alfonsin and
Huh-Jeon for rectilinear spatial complete graphs which were proved by
computational and combinatorial methods.
