The Euler characteristic states that V - E + F = 2
where V = vertices, E = Edges and F = Faces.
An octahedron has 8 faces, so the Euler characteristic leads to
V - E + 8 = 2
or E = V + 6
An octahedron can be any one of
a heptagonal pyramid (V = 8, E = 14)
a hexagonal prism (V = 12, E = 18)
a quadrilateral dipyramid (V = 6, E = 12)
There are other octahedral shapes as well.