Authors: Anđelić, Milica
Du, Zhibin
da Fonseca, Carlos
Simić, Slobodan 
Affiliations: Mathematical Institute of the Serbian Academy of Sciences and Arts 
Title: Tridiagonal Matrices and Spectral Properties of Some Graph Classes
Journal: Czechoslovak Mathematical Journal
Issue Date: 23-Apr-2020
Rank: M23
ISSN: 0011-4642
DOI: 10.21136/CMJ.2020.0182-19
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 the characteristic polynomial of any chain graph and we show that it can be expressed using the determinant of a particular tridiagonal matrix. Then this fact is applied to show that in a certain interval a chain graph does not have any nonzero eigenvalue. A similar result is provided for threshold graphs.
Keywords: 05C50 | chain graph | eigenvalue-free interval | threshold graph | tridiagonal matrix
Publisher: Institute of Mathematics of the Czech Academy of Sciences; Springer Link

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