<Chapter Label="regsub">
<Heading>Regular induced subgraphs</Heading>

In this chapter we give methods to investigate regular induced subgraphs
of regular graphs.
<P/>
Let <M>\Gamma</M> be a graph, and consider a subset <M>U</M> of its vertices.
The <E>induced subgraph</E> of <M>\Gamma</M> on <M>U</M>, 
<M>\Gamma[U]</M>, is the graph with vertex set <M>U</M>, and vertices in 
<M>\Gamma[U]</M> are adjacent if and only if they are adjacent in <M>\Gamma</M>.


<Section Label="Spectral bounds">
<Heading>Spectral bounds</Heading>

In this section, we introduce some bounds on regular induced subgraphs of 
regular graphs, which depend on the spectrum of the graph.
<P/>
Let <M>\Gamma</M> be a graph. A <E>coclique</E>, or <E>independent set</E>, 
of <M>\Gamma</M> is a subset of vertices for which each pair of distinct vertices
are non-adjacent. A <E>clique</E> of <M>\Gamma</M> is a subset of vertices for 
which each pair of distinct vertices are adjacent. 

<#Include Label="HoffmanCocliqueBound">

<#Include Label="HoffmanCliqueBound">

<#Include Label="HaemersRegularUpperBound">

<#Include Label="HaemersRegularLowerBound">

</Section>

<Section Label="Block intersection polynomials and bounds">
<Heading>Block intersection polynomials and bounds</Heading>

In this section, we introduce functions related to the block intersection 
polynomials, defined in <Cite Key="S_2010"/>. If you would like to know more about the
properties of these polynomials, see <Cite Key="S_2010"/>, <Cite Key="S_2015"/>
and <Cite Key="GS_2016"/>.

<#Include Label="CliqueAdjacencyPolynomial">

<#Include Label="CliqueAdjacencyBound">

<#Include Label="RegularAdjacencyPolynomial">

<#Include Label="RegularAdjacencyUpperBound">

<#Include Label="RegularAdjacencyLowerBound">

</Section>

<Section Label="Regular sets">
<Heading>Regular sets</Heading>

In this section we give functions to investigate regular sets, with a focus on
regular sets in strongly regular graphs. 
<P/>
Let <M>\Gamma</M> be a graph and <M>U</M> be a subset of its vertices. Then 
<M>U</M> is <M>m</M><E>-regular</E> if every vertex in 
<M>V(\Gamma)\backslash U</M> is adjacent to the same number <M>m&gt;0</M> of 
vertices in <M>U</M>. In this case we say that 
<M>U</M> has <E>nexus</E> <M>m</M>.
<P/>
The set <M>U</M> is a <M>(d,m)</M><E>-regular set</E> 
if <M>U</M> is an <M>m</M>-regular set and <M>\Gamma[U]</M> is a <M>d</M>-regular graph.
Then we call <M>(d,m)</M> the <E>regular set parameters</E> of <M>U</M>.

<#Include Label="Nexus">

<#Include Label="RegularSetParameters">

<#Include Label="IsRegularSet">

<#Include Label="RegularSetSRGParameters">

</Section>

<Section Label="Neumaier graphs">
<Heading>Neumaier graphs</Heading>

In this section, we give functions to investigate regular cliques in edge-regular
graphs.
<P/>
A clique <M>S</M> in <M>\Gamma</M> is <M>m</M><E>-regular</E>, for some 
<M>m&gt;0</M>, if <M>S</M> is an <M>m</M>-regular set. A graph <M>\Gamma</M> is 
a <E>Neumaier graph</E> with <E>parameters</E> <M>(v,k,a;s,m)</M> if it is 
edge-regular with parameters <M>(v,k,a)</M>, and <M>\Gamma</M> contains an 
<M>m</M>-regular clique of size <M>s</M>. For more information on Neumaier 
graphs, see <Cite Key="E_2020"/>.

<#Include Label="NGParameters">

<#Include Label="IsNG">

<#Include Label="IsFeasibleNGParameters">

<#Include Label="RegularCliqueERGParameters">

</Section>

</Chapter>
