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This question isn’t necessarily about Laplacian matrices!

Given the incidence matrix $B_S$ of an undirected graph $S$, we can calculate $B_SB_S^T=D_S-A_S$, where $D_S$ and $A_S$ are the degree and adjacency matrices, respectively, of $S$.

Furthermore, given the directed incidence matrix $B_T$ of a digraph $T$ with underlying undirected graph $S$, we can calculate $B_TB_T^T=D_S-A_S$, where $D_S$ and $A_S$ are the degree and adjacency matrices, respectively, of undirected $S$.

For a digraph $T$, let $A_T$ be $T$’s (directed) adjacency matrix and $D_T$ be defined such that ${D_T}_{i,j}=0$ if $i\neq j$ and ${D_T}_{i,j}={D_T}_{i,i}=\deg^-(i)+\deg^+(i)$ if $i = j$. From the literature, it is unclear whether $D_T-A_T$ can be expressed as a simple product of meaningful matrices (not in the $MM^T$ format) without using $D_T$ or $A_T$. I’m also interested in the same question for $D_T$ defined to instead represent the indegree or the outdegree of each vertex rather than the sum of the two. Any insight is appreciated.

(You can find the differences between the incidence matrices of undirected and directed graphs on the “Incidence matrix” Wikipedia page).

dheepthim
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1 Answers1

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If I read the question correctly, you are asking whether a 'Laplacian'-like matrix constructed for a directed graph as per your suggestion can be written as a product similar to what is possible for the Laplacian matrix using the incidence matrix.

I suggest you try it for a small graph.

I don't think there is a construction that works in general, as the adjacency matrix of a directed graph can be asymmetric, so will be your 'Laplacian' like matrix, while a $B^T B$ construction will always be symmetric as $(B^T B)^T = B^T B$.

References

Michael T
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  • I could have worded my question better - I am seeking an expression for the directed graph Laplacian involving the product of any number and type of meaningful matrices and not necessarily in the $MM^T$ format of the undirected graph Laplacian. – dheepthim Apr 05 '25 at 13:13
  • Hi dheepthim, welcome to stackexchange, you may want to have a look at the second comment you received from RdA... that may be what you are looking for. – Michael T Apr 05 '25 at 13:33
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    My apologies! I accidentally overlooked that comment - indeed, the linked post details a construction in the format I was looking for. – dheepthim Apr 18 '25 at 02:31