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?...] 



Are these three different notions of a graph Laplacian?
These are usually known as the Laplacian, the normalized Laplacian and the unsigned Laplaian. All three are positive semidefinite. If the graph is regular, they all provide the same information.


If the graph is not regular they are, in general, independent. The normalized Laplacian is the right tool for the analysis of random walks. The spectral information provided by the unsigned Laplacian is equivalent to what you get from the spectrum of the line graph of the original graph.


To expand on the last comment: if $B$ is the vertex-edge incidence matrix, then $BB^T$ is the unsigned Laplacian and $B^TB=2I+A(L(G))$. This appears for example on page 16 of the first edition of Cvetkovic et al "Spectra of Graphs", but it is older. (I know I did not learn it from there.) Note that it follows that $BB^T$ and $B^TB$ have the same non-zero eigenvalues with the same multiplicities.


