Authors: Anđelić, Milica
Ghorbani, Ebrahim
Simić, Slobodan 
Title: Vertex types in threshold and chain graphs
Journal: Discrete Applied Mathematics
Volume: 269
First page: 159
Last page: 168
Issue Date: 30-Sep-2019
Rank: M22
ISSN: 0166-218X
DOI: 10.1016/j.dam.2019.02.040
Abstract: 
A graph is called a chain graph if it is bipartite and the neighbourhoods of the vertices in each colour 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 colour class. Given a graph G, let λ be an eigenvalue (of the adjacency matrix) of G with multiplicity k≥1. A vertex v of G is a downer, or neutral, or Parter depending whether the multiplicity of λ in G−v is k−1, or k, or k+1, respectively. We consider vertex types in the above sense in threshold and chain graphs. In particular, we show that chain graphs can have neutral vertices, disproving a conjecture by Alazemi et al.
Keywords: Adjacency spectrum | Chain graphs | Threshold graphs | Vertex types
Publisher: Elsevier
Project: IPM, Grant No. 96050211
Graph theory and mathematical programming with applications in chemistry and computer science 

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