<Chapter Label="eigenv">
<Heading>Spectra of graphs</Heading>
In this chapter we give methods for investigating the eigenvalues of a graph.
<P/>
Let <M>\Gamma</M> be a graph of order <M>v</M>. The <E>adjacency matrix</E> of
<M>\Gamma</M>, <M>A(\Gamma)</M>, is the <M>v\times v</M> matrix indexed by
<M>V(\Gamma)</M> such that <M>A(\Gamma)_{xy}=1</M> if <M>xy\in E(\Gamma)</M>,
and <M>A(\Gamma)_{xy}=0</M> otherwise.
<P/>
The <E>spectrum</E> of <M>\Gamma</M>,
<M>Spec(\Gamma)</M>, is the multiset of eigenvalues of its adjacency matrix,
and an <E>eigenvalue of </E><M>\Gamma</M> is a member of <M>Spec(\Gamma)</M>.
The <E>multiplicity</E> of an eigenvalue <M>\alpha</M> of <M>\Gamma</M> is the
number of times <M>\alpha</M> appears in <M>Spec(\Gamma)</M>.
For information on most of the objects and results discussed in this chapter,
see <Cite Key="BH_2011"/>.
<Section Label="Eigenvalues of regular graphs">
<Heading>Eigenvalues of regular graphs</Heading>
In this section, we introduce methods for investigating eigenvalues of regular graphs.
The input for these methods will be a specific graph or the parameters
of a graph.
<P/>
Let <M>\Gamma</M> be a regular graph with parameters <M>(v,k)</M>. Then
<M>\Gamma</M> has largest eigenvalue <M>k</M> (see <Cite Key="BH_2011"/>). Therefore we do not implement a
<Q>LargestEigenvalue</Q> function for regular graphs.
<P/>
Let <M>\Gamma</M> be a strongly regular graph with parameters <M>(v,k,a,b)</M>.
The eigenvalues of <M>\Gamma</M> and their corresponding multiplicities
are uniquely determined by the parameters <M>(v,k,a,b)</M> (see <Cite Key="BH_2011"/>).
Using this knowledge, we provide methods which take as input feasible strongly
regular graph parameters <M>(v,k,a,b)</M>. We also give methods
which return an exact representation of the eigenvalues of a strongly regular
graph with parameters <M>(v,k,a,b)</M>, and their multiplicities.
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