Find the no. of edges of a graph G.

Question

Consider an undirected graph G where self-loops are not allowed. The vertex set of G is {(i,j):1≤ i ≤12 , 1≤ j ≤12}.There is an edge between (ܽa,b) and (ܿc,d) if |ܽa−ܿc| ≤1 and |ܾb−݀d| ≤1. What is the number of edges in this graph G ?

Answer 1

If you lay this out on a $12\times 12$ grid (think the squares of a $12\times 12$ chess board, for instance), then every vertex has an edge to its $8$ (or $5$ or $3$) neighbours.

There are $10\cdot 10$ vertices with $8$ neighbours (any vertex not along an edge), $4\cdot 10$ vertices with $5$ neighbours (any vertex along an edge but not in a corner) and $4$ vertices with $3$ neighbours (the corner vertices). Adding up, that's $100\cdot 8 + 40 \cdot 5 + 4\cdot 3 = 800 + 200 + 12 = 1012$ edges. Now, naturally, we've counted each edge twice (once for each of the two vertices it connects), so the number of edges is half of this, or $506$.