I read the definition of Laplace matrix recently. The definition are a little difference depended on different resources.
Generally, Laplacian matrix $L=\Delta-A$, where $\Delta$ is degree matrix of graph $G$ and $A$ is adjacency matrix. Another definition is $L=BB^T$, where $B$ is incidence matrix of $G$ with rows are nodes and columns are edges.
Graphs usually is considered as no-selfloop, unweighted, no-multi-edges.
$N$ nodes and $M$ edges.
The problem I got:
directed network:
$\Delta=\Delta_{in}+\Delta_{out}$, the degree is combined by indegree and outdegree. Well, what about $A$? Does that mean $A=A_{in}+A_{out}$? Caz, $\Delta-A$ is different with $\Delta - A_{in}$ or $\Delta - A_{out}$. If $A=A_{in}+A_{out}$, then the adjacency matrix is same with undirected network -- symmetric.
The oriented incidence matrix $B_{oriented}$, $N\times M$. $b_{im}=1$ if edge $m$ start from $i$. $b_{im}=-1$ if edge $m$ ended to $i$. $b_{im}=0$ otherwise.
I tested that $B_{oriented}B_{oriented}^T$ is a symmetric matrix with degrees on diagonal elements. And $(\Delta_{in}+\Delta_{out})-(A_{in}+A_{out})= B_{oriented}B_{oriented}^T$
But what is actually $\Delta-A$ in directed graph for Laplacian matrix definition?
undirected network:
That is nothing to talk about $\Delta-A$ for undirected network.
Well, there are two kinds of incidence matrix, oriented and unoriented.
oriented: The graph will be treated as a directed graph. So $B_{oriented}$ have dimension $N\times 2M$. In my test the result shows that $B_{oriented}B_{oriented}^T=2(\Delta-A)$.
unoriented: $b_{im}=1$ if link $m$ incident -- start from $i$ or end to $i$. $b_{im}=0$ otherwise. $B_{oriented}B_{oriented}^T=\Delta +A$.
Many definitions just give relations like $L=BB^T=\Delta-A$. But I am confusing. Does $BB^T$ always satisfy this equation? What are the actually full assumptions to support this equation?
If the question is not clearly enough please comment below. I will edit it.
Thank you.
