Combinatorics of a lattice

An $n \times n$ lattice has $n^{2}$ nodes.

For each row of the lattice, there are $n - 1$ horizontal links. For each column, there are $n - 1$ vertical links. Since there are $n$ rows there is a total of $n(n-1)$ horizontal links. Since there are $n$ columns there is a total of $n(n-1)$ vertical links. Therefore, for an $n \times n$ lattice there are $n(n-1) + n(n-1) = 2n(n-1)$ links.