# The graphs with all but two eigenvalues equal to - 2 or 0

Sebastian M. Cioabă, Willem H. Haemers, Jason R. Vermette

Research output: Contribution to journalArticleScientificpeer-review

### Abstract

We determine all graphs whose adjacency matrix has at most two eigenvalues (multiplicities included) different from $\pm 1$ and decide which of these graphs are determined by their spectrum. This includes the so-called friendship graphs, which consist of a number of edge-disjoint triangles meeting in one vertex. It turns out that the friendship graph is determined by its spectrum, except when the number of triangles equals sixteen.
Original language English 153-163 Designs, Codes and Cryptography 84 1-2 https://doi.org/10.1007/s10623-016-0241-4 Published - 1 Jul 2017

### Fingerprint

Eigenvalue
Graph in graph theory
Triangle
p.m.
Multiplicity
Disjoint
Vertex of a graph

### Keywords

• Graph spectrum
• Spectral characterizations

### Cite this

Cioabă, Sebastian M. ; Haemers, Willem H. ; Vermette, Jason R. / The graphs with all but two eigenvalues equal to - 2 or 0. In: Designs, Codes and Cryptography. 2017 ; Vol. 84, No. 1-2. pp. 153-163.
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The graphs with all but two eigenvalues equal to - 2 or 0. / Cioabă, Sebastian M.; Haemers, Willem H.; Vermette, Jason R.

In: Designs, Codes and Cryptography, Vol. 84, No. 1-2, 01.07.2017, p. 153-163.

Research output: Contribution to journalArticleScientificpeer-review

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AU - Haemers, Willem H.

AU - Vermette, Jason R.

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AB - We determine all graphs whose adjacency matrix has at most two eigenvalues (multiplicities included) different from $\pm 1$ and decide which of these graphs are determined by their spectrum. This includes the so-called friendship graphs, which consist of a number of edge-disjoint triangles meeting in one vertex. It turns out that the friendship graph is determined by its spectrum, except when the number of triangles equals sixteen.

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KW - Spectral characterizations

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DO - 10.1007/s10623-016-0241-4

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JO - Designs, Codes and Cryptography

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