Exercise 4.6.2

According to the property of Laplacian matrix $G=A^{T}A$, the diagonal entry counts the edges meeting at node $i$, for complete graph, each node has connection with all other nodes, so the count of edges is $n-1$.

Also, the off-diagonal entry is $-1$ when an edge connects nodes $i$ and $j$, for complete graph, every node is connected with another node, so for each pair of $i,j$, where $i\ne j$, there’s an edge connecting them, and corresponding entry should be -1.

So for complete graph with $n$ nodes, the diagonal entries are all $n-1$, all other entries are -1.