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>.
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"/>.
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>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>.
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>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>
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