<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. 


<#Include Label="LeastEigenvalueInterval">

<#Include Label="SecondEigenvalueInterval">

<#Include Label="LeastEigenvalueFromSRGParameters">

<#Include Label="SecondEigenvalueFromSRGParameters">

<#Include Label="LeastEigenvalueMultiplicity">

<#Include Label="SecondEigenvalueMultiplicity">

</Section>


</Chapter>
