$Q$ is $(0,1,-1)$ vertex edge incidence matrix of a simple directed graph. $M$ is $(0,1)$ vertex edge incidence matrix of a simple non directed graph. $A$ is vertex vertex incidence matrix of a graph. Is there any relation between these matrix?
I know that there is relation between Laplacian matrix and $Q$. i.e $$L=QQ^T$$
If $D$ is the diagonal matrix with the out-degrees of the vertices as its diagonal entries, then $$ MM^T = A + D.$$ The matrix $A+D$ is often called the unsigned Laplacian. On the other hand, $$ M^TM = L +2I$$ where $L$ denotes the adjacency matrix of the line graph. In general there is no useful relation between the spectrum of $A$ and the spectrum of $A+D$, unless the graph is regular. If your graph is bipartite, then $D+A$ and $D-A$ are similar via a diagonal matrix with $\pm1$-entries on the diagonal.]
[Edit: for the similarity, argue as follows. Suppose the graph is bipartite on $n$ vertices, and let $S$ be the $n\times n$ diagonal matrix with $S_{i,i}=1$ if $i$ is in the first colour class and $S_{i,i}=-1$ otherwise. Then $Q=SM$ and so $QQ^T = SMM^TS$ and therefore $D_A=S(A+D)S$. Since $S$ is orthogonal, $D+A$ and $D-A$ are similar.