Let $G$ be a connected graph with $V$ vertices and let say I have an adjacency matrix of order $N$, how can I make sum of each column even?
Like I have a graph with $4$ vertices and $4$ edges as
$\begin{bmatrix}0 & 1 & 0 &0 \\
0 & 0 & 1 & 0\\
0 & 0 & 0 & 1 \\
1 & 0 & 0 & 0 \end{bmatrix}$
And I want to convert it like this,
$\begin{bmatrix}0 & 0 & 0 &0 \\
1 & 0 & 1 & 0\\
0 & 0 & 0 & 0\\
1 & 0 & 1 & 0 \end{bmatrix}$
For reference, if $i^{th}$ element of the $j^{th}$ row is $1$ then the edge is directed from $j$ to $i$.
You can reverse any edge between two vertices such that the graph remains connected. And the adjacency matrix's indexing starts from 1 rather than from 0.
Best Answer
As mentioned in the comments, this depends on what operations you allow, because you are clearly allowing your underlying graph to change --- the graph after your transformation isn't isomorphic to the first graph.
For your first matrix, you have the following graph:
For your second matrix, you have the following:
Taken as directed graphs, these are not isomorphic (however, if you relax them to merely undirected, they are). As such, it's difficult to determine what operations you're allowing as legal transformations.