Hello everyone, i am stuck with the problem 1279 — Graph Coloring which can be solved computing the value $$$K ^ {(n - r)}$$$, where $$$r$$$ is the rank of matrix $$$a$$$, according with this blog, but i cant understand how it works, specifically what's the sense of the matrix $$$a$$$ builded in this code bellow:

//G is the adjancent list
for(int i =0 ;i < n ; i ++ ) {
	a[i][i] = 1;
	for(int j = 0; j < G[i].size() ;j  ++) {
		a[i][G[i][j]] = k-1;
	}
	a[i][n] = 0;
}