Which graphs have integral spectra?

Harary, Frank; Schwenk, Allen J.

doi:10.1007/bfb0066434

Cited by 185 publications

(141 citation statements)

References 3 publications

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“…According to HARARY and SCHWENK [14], a graph G is defined to be integral if all of its eigenvalues are integers. Integral graphs have been a focus of research for some time; see [4] for a survey.…”

Section: Eigenspace Bases Of Gcd-graphsmentioning

confidence: 99%

GCD-graphs and NEPS of complete graphs

Klotz

Sander

2012

Ars Math. Contemp.

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A gcd-graph is a Cayley graph over a finite abelian group defined by greatest common divisors. Such graphs are known to have integral spectrum. A non-complete extended psum, or NEPS in short, is well-known general graph product. We show that the class of gcd-graphs and the class of NEPS of complete graphs coincide. Thus, a relation between the algebraically defined Cayley graphs and the combinatorially defined NEPS of complete graphs is established. We use this link to show that gcd-graphs have a particularly simple eigenspace structure, to be precise, that every eigenspace of the adjacency matrix of a gcdgraph has a basis with entries −1, 0, 1 only.

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Section: Eigenspace Bases Of Gcd-graphsmentioning

confidence: 99%

GCD-graphs and NEPS of complete graphs

Klotz

Sander

2012

Ars Math. Contemp.

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“…Some graph operations such as the Cartesian product and the strong product may be used to generate new integral graphs from given ones [8]. In this section, we give a new general method for constructing integral graphs using the Kronecker product and commuting sets of matrices with integral eigenvalues.…”

Section: A Construction Using Commuting Matricesmentioning

confidence: 99%

“…The zeros of the characteristic polynomial of A(G) are called the eigenvalues of G. The graph G is said to be integral if all the eigenvalues of G are integers. The notion of integral graphs was first introduced in [8]. Recently, integral graphs have found applications in quantum networks allowing perfect state transfer [12].…”

Section: Introductionmentioning

confidence: 99%

Some constructions of integral graphs

Mohammadian¹,

Tayfeh-Rezaie²

2011

Linear and Multilinear Algebra

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“…Eigenvalues of an undirected graph Γ are the eigenvalues of an arbitrary adjacency matrix of Γ. Harary and Schwenk [10] defined Γ to be integral, if all of its eigenvalues are integers. For a survey of integral graphs see [3].…”

Section: Introductionmentioning

confidence: 99%