# 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}
}
• Published 1 March 2018
• Mathematics, Computer Science
• Discret. Appl. Math.
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… 3 Citations ## Figures, Tables, and Topics from this paper Wiener Index of Chain Graphs A bipartite graph is called a chain graph if the neighborhoods of the vertices in each partite set form a chain with respect to set inclusion. Chain graphs are discovered and re-discovered by various Tridiagonal Matrices and Spectral Properties of Some Graph Classes • Mathematics • 2020 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. In this paper we give an explicit formula for On main eigenvalues of chain graphs • Computer Science Comput. Appl. Math. • 2021 ## References SHOWING 1-10 OF 28 REFERENCES Vertex types in some lexicographic products of graphs • Mathematics Linear and Multilinear Algebra • 2018 ABSTRACT Let be a symmetric matrix, or equivalently, a weighted graph whose edge ij has the weight . The eigenvalues of are the eigenvalues of M. We denote by the principal submatrix of M obtained by On the multiplicities of eigenvalues of graphs and their vertex deleted subgraphs: old and new results • Mathematics • 2015 Given a simple graph G, let A(G) be its adjacency matrix. A principal submatrix of A(G) of order one less than the order of G is the adjacency matrix of its vertex deleted subgraph. It is well-known Eigenvalue location for chain graphs • Mathematics • 2016 Abstract Chain graphs (also called double nested graphs) play an important role in the spectral graph theory since every connected bipartite graph of fixed order and size with maximal largest Graphs for which the least eigenvalue is minimal, II • Mathematics • 2008 Let G be a connected graph whose least eigenvalue λ(G) is minimal among the connected graphs of prescribed order and size. We show first that either G is complete or λ(G) is a simple eigenvalue. In On the Spectrum of Threshold Graphs • Mathematics • 2011 The antiregular connected graph on 𝑟 vertices is defined as the connected graph whose vertex degrees take the values of 𝑟−1 distinct positive integers. We explore the spectrum of its adjacency Some new considerations about double nested graphs • Mathematics • 2015 Abstract In the set of all connected graphs with fixed order and size, the graphs with maximal index are nested split graphs, also called threshold graphs. It was recently (and independently) The structure of threshold graphs Manca has derived an efficient matrix method for testing a given graph to see whether or not it is a threshold graph.Chvátal and Hammer introduced these graphs which can be defined by the condition Eigenvalues and energy in threshold graphs • Mathematics • 2015 Abstract Assuming a uniform random model of selecting creation sequences, we show that almost every connected threshold graph has more negative than positive eigenvalues. We show that no threshold On a simple characterisation of threshold graphs AbstractIn their very interesting paper “Set-packing problems and threshold graphs” [1] V. Chvatal and P. L. Hammer have shown that the constraints$\$\begin{gathered} \sum\limits_{j = 1}^n {a_{ij}
Spectra of graphs obtained by a generalization of the join graph operation
• Mathematics, Computer Science
Discret. Math.
• 2013
A more general result is deduced and applied to the determination of adjacency and Laplacian spectra of graphs obtained by a generalized join graph operation on families of graphs.