Vertex types in threshold and chain graphs

@article{Andelic2019VertexTI,
  title={Vertex types in threshold and chain graphs},
  author={Milica Andelic and Ebrahim Ghorbani and Slobodan K. Simic},
  journal={Discret. Appl. Math.},
  year={2019},
  volume={269},
  pages={159-168}
}
A graph is called a chain graph if it is bipartite and the neighborhoods of the vertices in each color class form a chain with respect to inclusion. A threshold graph can be obtained from a chain graph by making adjacent all pairs of vertices in one color class. Given a graph $G$, let $\lambda$ be an eigenvalue (of the adjacency matrix) of $G$ with multiplicity $k \geq 1$. A vertex $v$ of $G$ is a downer, or neutral, or Parter depending whether the multiplicity of $\lambda$ in $G-v$ is $k-1… 
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TLDR
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