Let $G = (V, E)$ be an infinite undirected graph whose vertices are ordered pairs of integers, that is $V=\mathbb{Z}^2$. Given two natural numbers $a$ and $b$, two vertices $(x_1, y_1)$ and $(x_2, y_2)$ share an edge if and only if either one of these conditions hold: 1. $|x_1 - x_2| = a$ and $|y_1 - y_2| = b$, or 2. $|x_1 - x_2| = b$ and $|y_1 - y_2| = a$

Let's define the following functions -

$f(a, b)$ be $1$ if the graph is connected (there exists a path between any two vertices), and $0$ otherwise, and,

$g(i) = \sum_{j = 1}^{10^6}(f(i, j))$.

Evaluate $\sum_{i = 1}^{10^6}g(i)$