This problem occurred from my need to find all graphs with a certain topology inside a bigger one. I don't need the subgraphs but the graphs that have the exact topology I am searching.
We know for a fact that we can represent graph networks with a Laplacian matrix. If I wanted to check the entire graph it would take $n^2 \times n$ for the diagonal and $2^n$ for checking the row and column of each element.
Is there a better way to do this? Either by using matrices or graph theory?
For example below i want to find all the upper left matrices inside a big matrix and also check that the rest of their columns and rows are zero.
$$\begin{pmatrix} 2 & -1 & 0 & 0 & \cdots & 0 \\ -1 & 3 & -2 & 0 & \cdots & 0 \\ 0 & -2 & 3 & -1 & \cdots & 0 \\ 0 & 0 & 0 & 0 & \cdots & n \\ \vdots & \vdots & \vdots & \vdots & \ddots & n \\ 0 & 0 & 0 & 0 & \cdots & n \end{pmatrix}$$
