Exercise 4.6.1

The Laplacian matrix for a triangle graph is: $A^{T}A=\left[\begin{array}{ccc}\hfill 2\hfill & \hfill -1\hfill & \hfill -1\hfill \\ \hfill -1\hfill & \hfill 2\hfill & \hfill -1\hfill \\ \hfill -1\hfill & \hfill -1\hfill & \hfill 2\hfill \end{array}\right]$

The Laplacian matrix for a square graph is: $A^{T}A=\left[\begin{array}{cccc}\hfill 2\hfill & \hfill -1\hfill & \hfill 0\hfill & \hfill -1\hfill \\ \hfill -1\hfill & \hfill 2\hfill & \hfill -1\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill -1\hfill & \hfill 2\hfill & \hfill -1\hfill \\ \hfill -1\hfill & \hfill 0\hfill & \hfill -1\hfill & \hfill 2\hfill \end{array}\right]$