TITLE: Are these three different notions of a graph Laplacian?
QUESTION [11 upvotes]: I seem to see three different things that are being called the Laplacian of a graph, 

One is the matrix $L_1 = D - A$ where $D$ is a diagonal matrix consisting of degrees of all the vertices and $A$ is the (possibly signed) adjacency matrix. 
The other is to say that the matrix $L_2$ is a $\vert V \vert \times \vert V \vert$ matrix such that $(L_2)_{ii} = deg(v_i)$ and $(L_2)_{ij} = -\frac{1}{\sqrt{deg(v_i)deg(v_j) } }$
The third is to say that $L_3 = BB^T$ (where $B$ is the incidence matrix) 

Can someone kindly clarify what is the relation between these three pictures (and or may be all these are the same somehow!?). [..for example for which of them would the positive-semi-definiteness and the the heat-equation intuition hold?...]

REPLY [2 votes]: Let me comment a bit on Chris Godsil's answer. The fact that $L_1\ne L_3$ follows from the easy-to-check fact that $L_1=\mathcal I\mathcal I^T$, where $\mathcal I$ is the incidence matrix of an arbitrary orientation of the graph. The nullity of $\mathcal I^T$ is the number of connected components of the graph, whereas the nullity of the transpose of the signless incidence matrix (which is what you presumably denote by $B^T$) is the number of bipartite connected components. So these matrices are clearly different. However, as pointed out by Chris Godsil, they yield the same information in the regular case: the difference is that the upper end of the spectrum of $L_3$ corresponds to the lower end of the spectrum of $L_1$, and vice versa: This was in fact, as far as I can judge, the main reasons for the introduction of $L_3$ in a 1994 paper by Desai and Rao.
The fact that the nullities of $L_1,L_2$ coincide can be seen by an argument that might be interesting for your purposes: $L_2$ (or at least the matrix $L_2:=D^{-\frac12}L_1 D^{-\frac12} $ suggested in the comment by Aaron Meyerowitz) is similar to $\tilde{L}_2:=D^{-1}L_1$, which is often called normalized Laplacian as well. 
It turns out that both $L_1,\tilde{L}_2$ can be seen as Fréchet derivatives of the same energy functional $\mathcal E:f\mapsto\|\mathcal I^T f\|_{\ell^2(E)}^2$, but with respect to two different Hilbert spaces -- more precisely, to the vector space $\mathbb R^V$ endowed with the inner products
$$
(f|g):=\sum_{v\in V}f(v)g(v)
$$
and
$$
(f|g):=\sum_{v\in V}f(v)g(v)\deg(v)
$$
respectively (obvious modifications are needed in the case of infinite graphs).