I'm trying to determine the number of vertex in a graph $G=(V,H)$ where:
$\displaystyle V = \left\{ v = (x_1, x_2, x_3) \in \mathbb{Z}_p^3:\sum_{i=1}^3x_i\equiv0\right\}$
with $p \ge 3$. Equivalent is modulo $p$.
e.g. some of the vertex for $ p = 3 $ are:
$v_1 = (0,0,0)\\ v_2 = (1,-1,0)\\ v_3 = (2,1,0)\\$
Assuming the worst case when I want all the possible permutations, I know the answer is $3^{2p+1}$, but I really was expecting to find a polinomial solution on $p$.
Thanks for the help.