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#W eigenv.gd Algebraic Graph Theory package Rhys J. Evans
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##
#Y Copyright (C) 2020
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## Declaration file for functions involving eigenvalues of graphs.
##

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#O LeastEigenvalueInterval( <gamma> , <eps> )
#O LeastEigenvalueInterval( <parms> , <eps> )
##
## <#GAPDoc Label="LeastEigenvalueInterval">
## <ManSection>
## <Oper Name="LeastEigenvalueInterval"
## Arg="gamma, eps"/>
## <Oper Name="LeastEigenvalueInterval"
## Arg="parms, eps" Label="for SRG parameters"/>
## <Returns>A list.</Returns>
##
## <Description>
## Given a graph <A>gamma</A> and rational number <A>eps</A>, this function
## returns an interval containing the least eigenvalue of <A>gamma</A>.
## 
## Given feasible strongly regular graph parameters <A>parms</A> and rational
## number <A>eps</A>, this function returns an interval containing the least
## eigenvalue of a strongly regular graph with parameters <A>parms</A>.
## 
## The interval returned is in the form of a list, <A>[y,z]</A> of rationals
## <M><A>y</A>\leq <A>z</A></M> with the property that
## <M><A>z</A>-<A>y</A>\leq <A>eps</A></M>. If the eigenvalue is rational this
## function will return a list <A>[y,z]</A>, where <M><A>y</A>=<A>z</A></M>.
## <Example>
## <![CDATA[
##gap> gamma:=EdgeOrbitsGraph(Group((1,2,3,4,5)),[[1,2],[2,1]]);;
##gap> LeastEigenvalueInterval(gamma,1/10);
##[ -13/8, -25/16 ]
##gap> parms:=SRGParameters(gamma);
##[ 5, 2, 0, 1 ]
##gap> LeastEigenvalueInterval(parms,1/10);
##[ -13/8, -25/16 ]
##gap> LeastEigenvalueInterval(JohnsonGraph(7,3),1/20);
##[ -3, -3 ]
## ]]>
## </Example>
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareOperation( "LeastEigenvalueInterval", [IsRecord, IsRat] );
DeclareOperation( "LeastEigenvalueInterval", [IsList, IsRat] );

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#O SecondEigenvalueInterval( <gamma> , <eps> )
#O SecondEigenvalueInterval( <parms> , <eps> )
##
## <#GAPDoc Label="SecondEigenvalueInterval">
## <ManSection>
## <Oper Name="SecondEigenvalueInterval"
## Arg="gamma, eps"/>
## <Oper Name="SecondEigenvalueInterval"
## Arg="parms, eps" Label="for SRG parameters"/>
## <Returns>A list.</Returns>
##
## <Description>
## Given a regular graph <A>gamma</A> and rational number <A>eps</A>, this
## function returns an interval containing the second largest eigenvalue
## of <A>gamma</A>.
## 
## Given feasible strongly regular graph parameters <A>parms</A> and
## rational number <A>eps</A>, this function returns an interval containing
## the second largest eigenvalue of a strongly regular graph with
## parameters <A>parms</A>.
## 
## The interval returned is in the form of a list, <A>[y,z]</A> of rationals
## <M><A>y</A>\leq <A>z</A></M> with the property that
## <M><A>z</A>-<A>y</A>\leq <A>eps</A></M>. If the eigenvalue is rational this
## function will return a list <A>[y,z]</A>, where
## <M><A>y</A>=<A>z</A></M>.
## <Example>
## <![CDATA[
##gap> gamma:=EdgeOrbitsGraph(Group((1,2,3,4,5)),[[1,2],[2,1]]);;
##gap> SecondEigenvalueInterval(gamma,1/10);
##[ 9/16, 5/8 ]
##gap> parms:=SRGParameters(gamma);
##[ 5, 2, 0, 1 ]
##gap> SecondEigenvalueInterval(parms,1/10);
##[ 9/16, 5/8 ]
##gap> SecondEigenvalueInterval(JohnsonGraph(7,3),1/20);
##[ 5, 5 ]
## ]]>
## </Example>
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareOperation( "SecondEigenvalueInterval" , [IsRecord, IsRat]);
DeclareOperation( "SecondEigenvalueInterval" , [IsList, IsRat] );

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#F LeastEigenvalueFromSRGParameters( [ <v>, <k>, <a>, ] )
##
## <#GAPDoc Label="LeastEigenvalueFromSRGParameters">
## <ManSection>
## <Func Name="LeastEigenvalueFromSRGParameters"
## Arg='[ v, k, a, b ]'/>
## <Returns>An integer or an element of a cyclotomic field.</Returns>
##
## <Description>
## Given feasible strongly regular graph parameters <A>[v, k, a, b]</A>
## this function returns the least eigenvalue of a strongly regular graph
## with parameters <M>(<A>v,k,a,b</A>)</M>. If the eigenvalue is integer, the
## object returned is an integer. If the eigenvalue is irrational, the object
## returned lies in a cyclotomic field.
## <Example>
## <![CDATA[
##gap> LeastEigenvalueFromSRGParameters([5,2,0,1]);
##E(5)^2+E(5)^3
##gap> LeastEigenvalueFromSRGParameters([10,3,0,1]);
##-2
## ]]>
## </Example>
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareGlobalFunction( "LeastEigenvalueFromSRGParameters" );

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#F SecondEigenvalueFromSRGParameters( [ <v>, <k>, <a>, ] )
##
## <#GAPDoc Label="SecondEigenvalueFromSRGParameters">
## <ManSection>
## <Func Name="SecondEigenvalueFromSRGParameters"
## Arg='[ v, k, a, b ]'/>
## <Returns>An integer or an element of a cyclotomic field.</Returns>
##
## <Description>
## Given feasible strongly regular graph parameters <A>[v, k, a, b]</A>,
## this function returns the second largest eigenvalue of a strongly
## regular graph with parameters <M>(<A>v,k,a,b</A>)</M>. If the eigenvalue is
## integer, the object returned is an integer. If the eigenvalue is
## irrational, the object returned lies in a cyclotomic field.
## <Example>
## <![CDATA[
##gap> SecondEigenvalueFromSRGParameters([5,2,0,1]);
##E(5)+E(5)^4
##gap> SecondEigenvalueFromSRGParameters([10,3,0,1]);
##1
## ]]>
## </Example>
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareGlobalFunction( "SecondEigenvalueFromSRGParameters" );

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#O LeastEigenvalueMultiplicity( [ <v>, <k>, <a>, ] )
##
## <#GAPDoc Label="LeastEigenvalueMultiplicity">
## <ManSection>
## <Oper Name="LeastEigenvalueMultiplicity"
## Arg='[ v, k, a, b ]'/>
## <Returns>An integer.</Returns>
##
## <Description>
## Given feasible strongly regular graph parameters <A>[v, k, a, b]</A>
## this function returns the multiplicity of the least eigenvalue of a
## strongly regular graph with parameters <M>(<A>v,k,a,b</A>)</M>.
## <Example>
## <![CDATA[
##gap> LeastEigenvalueMultiplicity([16,9,4,6]);
##6
##gap> LeastEigenvalueMultiplicity([25,12,5,6]);
##12
## ]]>
## </Example>
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareOperation( "LeastEigenvalueMultiplicity" , [IsList] );

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#O SecondEigenvalueMultiplicity( [ <v>, <k>, <a>, ] )
##
## <#GAPDoc Label="SecondEigenvalueMultiplicity">
## <ManSection>
## <Oper Name="SecondEigenvalueMultiplicity"
## Arg='[ v, k, a, b ]'/>
## <Returns>An integer.</Returns>
##
## <Description>
## Given feasible strongly regular graph parameters <A>[v, k, a, b]</A>
## this function returns the multiplicity of the second eigenvalue of a
## strongly regular graph with parameters <M>(<A>v,k,a,b</A>)</M>.
## <Example>
## <![CDATA[
##gap> SecondEigenvalueMultiplicity([16,9,4,6]);
##9
##gap> SecondEigenvalueMultiplicity([25,12,5,6]);
##12
## ]]>
## </Example>
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareOperation( "SecondEigenvalueMultiplicity", [IsList] );

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#E

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