Given a weighted undirected graph $G=(V,E)$, which adjacency matrix is $A$. The weight of edges can be negative.
The graph laplacian is defined as
$l_{ij}=-a_{ij},\ \text{if } i\neq j$
$l_{ij}=\sum_{j=1,j\neq i}^na_{ij},\ \text{if } i=j$
Note that since this matrix may not enjoy the nice properties like positive semi-definiteness, and so on, the graph laplacian is usually modified as follows,
$l_{ij}=-a_{ij},\ \text{if } i\neq j$
$l_{ij}=\sum_{j=1,j\neq i}^n|a_{ij}|,\ \text{if } i=j$
see Spectral Analysis of Signed Graphs for Clustering, Prediction and Visualization
However, I am interested in the first definition of laplacian matrix. I am wondering is there any approach or work studying its spectrum, especially the smallest eigenvalue?
Thanks a lot for suggestions!