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##
#W regprop.gd Algebraic Graph Theory package Rhys J. Evans
##
##
#Y Copyright (C) 2020
##
## Declaration file for functions involving regularity properties of graphs.
##
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##
#F RGParameters( <gamma> )
##
## <#GAPDoc Label="RGParameters">
## <ManSection>
## <Func Name="RGParameters"
## Arg='gamma'/>
## <Returns>A list or <K>fail</K>.</Returns>
##
## <Description>
## Given a graph <A>gamma</A>, this function returns the regular graph
## parameters of <A>gamma</A>. If <A>gamma</A> is not a regular graph,
## the function returns <K>fail</K>.
## <Example>
## <![CDATA[
##gap> gamma:=EdgeOrbitsGraph(Group((2,3,4,5)),[[1,2],[2,1]]);;
##gap> RGParameters(gamma);
##fail
##gap> gamma:=HammingGraph(3,4);;
##gap> RGParameters(gamma);
##[ 64, 9 ]
## ]]>
## </Example>
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareGlobalFunction( "RGParameters" );
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##
#F IsRG( <gamma> )
##
## <#GAPDoc Label="IsRG">
## <ManSection>
## <Func Name="IsRG"
## Arg='gamma'/>
## <Returns><K>true</K> or <K>false</K>.</Returns>
##
## <Description>
## Given a graph <A>gamma</A>, this function returns <K>true</K> if
## <A>gamma</A> is a regular graph, and <K>false</K> otherwise.
## <Example>
## <![CDATA[
##gap> gamma:=NullGraph(Group(()),5);;
##gap> IsRG(gamma);
##true
##gap> gamma:=EdgeOrbitsGraph(Group((2,3,4,5)),[[1,2],[2,1]]);;
##gap> IsRG(gamma);
##false
##gap> gamma:=TriangularGraph(6);;
##gap> IsRG(gamma);
##true
## ]]>
## </Example>
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareSynonym( "IsRG", IsRegularGraph );
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##
#F IsFeasibleRGParameters( [ <v>, <k> ] )
##
## <#GAPDoc Label="IsFeasibleRGParameters">
## <ManSection>
## <Func Name="IsFeasibleRGParameters"
## Arg='[ v, k ]'/>
## <Returns><K>true</K> or <K>false</K>.</Returns>
##
## <Description>
## Given a list of integers of length 2, <A>[v,k]</A>, this
## function returns <K>true</K> if <M>( <A>v</A>, <A>k</A> )</M> is a feasible
## parameter tuple for a regular graph. Otherwise, the function
## returns <K>false</K>.
## <P/>
## The tuple <M>(v, k)</M> is a <E>feasible</E>
## parameter tuple for a regular graph if it satisfies the following
## well-known conditions:
## <List>
## <Item><M>v>k\geq 0</M>;</Item>
## <Item><M>2</M> divides <M>vk</M>.</Item>
## </List>
## Any regular graph must have parameters that satisfy these
## conditions (see <Cite Key="BCN_1989"/>).
## <Example>
## <![CDATA[
##gap> IsFeasibleRGParameters([15,9]);
##false
##gap> IsFeasibleRGParameters([16,9]);
##true
## ]]>
## </Example>
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareGlobalFunction( "IsFeasibleRGParameters" );
#############################################################################
##
#F ERGParameters( <gamma> )
##
## <#GAPDoc Label="ERGParameters">
## <ManSection>
## <Func Name="ERGParameters"
## Arg='gamma'/>
## <Returns>A list or <K>fail</K>.</Returns>
##
## <Description>
## Given a graph <A>gamma</A>, this function returns the edge-regular graph
## parameters of <A>gamma</A>. If <A>gamma</A> is not an edge-regular graph,
## the function returns <K>fail</K>.
## <Example>
## <![CDATA[
##gap> gamma:=NullGraph(Group(()),5);;
##gap> ERGParameters(gamma);
##fail
##gap> gamma:=JohnsonGraph(7,3);;
##gap> ERGParameters(gamma);
##[ 35, 12, 5 ]
## ]]>
## </Example>
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareGlobalFunction( "ERGParameters" );
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##
#F IsERG( <gamma> )
##
## <#GAPDoc Label="IsERG">
## <ManSection>
## <Func Name="IsERG"
## Arg='gamma'/>
## <Returns><K>true</K> or <K>false</K>.</Returns>
##
## <Description>
## Given a graph <A>gamma</A>, this function returns <K>true</K> if
## <A>gamma</A> is an edge-regular graph, and <K>false</K> otherwise.
## <Example>
## <![CDATA[
##gap> gamma:=NullGraph(Group(()),5);;
##gap> IsERG(gamma);
##false
##gap> gamma:=JohnsonGraph(7,3);;
##gap> IsERG(gamma);
##true
## ]]>
## </Example>
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareGlobalFunction( "IsERG" );
#############################################################################
##
#F IsFeasibleERGParameters( [ <v>, <k>, <a> ] )
##
## <#GAPDoc Label="IsFeasibleERGParameters">
## <ManSection>
## <Func Name="IsFeasibleERGParameters"
## Arg='[ v, k, a ]'/>
## <Returns><K>true</K> or <K>false</K>.</Returns>
##
## <Description>
## Given a list of integers of length 3, <A>[v,k,a]</A>, this
## function returns <K>true</K> if <M>( <A>v, k, a</A> )</M> is a feasible
## parameter tuple for an edge-regular graph. Otherwise, the function
## returns <K>false</K>.
## <P/>
## The tuple <M>( v, k, a )</M> is a <E>feasible</E>
## parameter tuple for an edge-regular graph if it satisfies the following
## well-known conditions:
## <List>
## <Item><M>(v,k)</M> is a feasible regular graph parameter tuple;</Item>
## <Item><M>k>a\geq 0</M>;</Item>
## <Item><M>2</M> divides <M>ka</M> and <M>6</M>
## divides <M>vka</M>;</Item>
## <Item><M>v-2k+a \geq 0</M>.</Item>
## </List>
## Any edge-regular graph must have parameters which satisfy these
## conditions (see <Cite Key="BCN_1989"/>).
## <Example>
## <![CDATA[
##gap> IsFeasibleERGParameters([15,9,6]);
##false
##gap> IsFeasibleERGParameters([16,9,4]);
##true
## ]]>
## </Example>
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareGlobalFunction( "IsFeasibleERGParameters" );
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##
#F SRGParameters( <gamma> )
##
## <#GAPDoc Label="SRGParameters">
## <ManSection>
## <Func Name="SRGParameters"
## Arg='gamma'/>
## <Returns>A list or <K>fail</K>.</Returns>
##
## <Description>
## Given a graph <A>gamma</A>, this function returns the strongly regular
## graph parameters of <A>gamma</A>. If <A>gamma</A> is not a strongly
## regular graph, the function returns <K>fail</K>.
## <Example>
## <![CDATA[
##gap> gamma:=CompleteGraph(Group(()),5);;
##gap> SRGParameters(gamma);
##fail
##gap> gamma:=JohnsonGraph(5,3);;
##gap> SRGParameters(gamma);
##[ 10, 6, 3, 4 ]
## ]]>
## </Example>
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareGlobalFunction( "SRGParameters" );
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##
#F IsSRG( <gamma> )
##
## <#GAPDoc Label="IsSRG">
## <ManSection>
## <Func Name="IsSRG"
## Arg='gamma'/>
## <Returns><K>true</K> or <K>false</K>.</Returns>
##
## <Description>
## Given a graph <A>gamma</A>, this function returns <K>true</K> if
## <A>gamma</A> is a strongly regular graph, and <K>false</K> otherwise.
## <Example>
## <![CDATA[
##gap> gamma:=CompleteGraph(Group(()),5);;
##gap> IsSRG(gamma);
##false
##gap> gamma:=JohnsonGraph(5,3);;
##gap> IsSRG(gamma);
##true
## ]]>
## </Example>
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareGlobalFunction( "IsSRG" );
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##
#F IsFeasibleSRGParameters( [ <v>, <k>, <a>, <b> ] )
##
## <#GAPDoc Label="IsFeasibleSRGParameters">
## <ManSection>
## <Func Name="IsFeasibleSRGParameters"
## Arg='[ v, k, a, b ]'/>
## <Returns><K>true</K> or <K>false</K>.</Returns>
##
## <Description>
## Given a list of integers of length 4, <A>[v,k,a,b]</A>, this
## function returns <K>true</K> if <M>( <A>v, k, a, b</A> )</M> is a feasible
## parameter tuple for a strongly regular graph. Otherwise, this function
## returns <K>false</K>.
## <P/>
## The tuple <M>(v,k,a,b)</M> is a <E>feasible</E>
## parameter tuple for a strongly regular graph if it satisfies the following
## well-known conditions:
## <List>
## <Item><M>(v,k,a)</M> is a feasible edge-regular graph parameter tuple;</Item>
## <Item><M>k\geq b</M>;</Item>
## <Item><M>(v-k-1)b = k(k-a-1)</M>;</Item>
## <Item><M>v-2-2k+b \geq 0</M>;</Item>
## <Item>the formulae for the multiplicities of the eigenvalues of a strongly
## regular graph with these parameters evaluate to positive integers
## (see <Cite Key="BH_2011"/>).</Item>
## </List>
## Any strongly regular graph must have parameters which satisfy these
## conditions (see <Cite Key="BCN_1989"/>).
## <Example>
## <![CDATA[
##gap> IsFeasibleSRGParameters([15,9,4,7]);
##false
##gap> IsFeasibleSRGParameters([10,3,0,1]);
##true
## ]]>
## </Example>
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareGlobalFunction( "IsFeasibleSRGParameters" );
#############################################################################
##
#E
