
In this section, we report progress on the `feasibility' and `uniqueness' of
the intersection arrays that were listed in the tables of `BCN'
\cite{bcn,BCNcoradd}, and some additional (larger) ones not in the tables. We
note that these tables are also available online in machine readable form
(although they are not exactly the same) \cite{tables}.

\subsection{Diameter \texorpdfstring{$3$}{3} and primitive}

\subsubsection{Uniqueness}

For the following intersection array, there is a unique distance-regular graph with that array:

\bigskip

\noindent $\{ 6,5,2; 1, 1,3\}$ ($v=57$): Perkel graph; Coolsaet and Degraer
\cite{CooDe05}.

\subsubsection{Existence}

For the following intersection arrays, there is a distance-regular graph with
that array:

\bigskip

\noindent $\{20,18,6; 1,1,15\}$ ($v=525$): unitary non-isotropics graph ($q=5$); \cite[Thm.~12.4.1]{bcn}.\\
$\{26,24,19; 1,3,8\}$ ($v=729$): Brouwer graph ($q=3$); Brouwer and Pasechnik \cite{BP2011}, see Section \ref{sec:Kasami}.\\
$\{31, 30, 17; 1, 2, 15\}$ ($v=1024$): Kasami graph ($q=2,j=2$); \cite[Thm.~11.2.1~(13)]{bcn}.\\
$\{110, 81, 12; 1, 18, 90\}$ ($v=672$): Moscow-Soicher graph (see Section \ref{sec: KoolenRiebeek}).

\subsubsection{Nonexistence}

The following intersection arrays are not feasible:

\bigskip

\noindent $\{5, 4, 3; 1, 1, 2\}$ ($v=56$): Fon-Der-Flaass \cite{FDF193}.\\
$\{13, 10, 7; 1, 2, 7\}$ ($v=144$): Coolsaet \cite{Co95}.\\
$\{19, 12, 5; 1, 4, 15\}$ ($v=96$): Coolsaet and Juri\v{s}i\'{c} \cite{CoJu08}, Neumaier \cite[p.~15]{BCNcoradd}.\\
$\{21, 16, 8; 1, 4, 14\}$ ($v=154$): Coolsaet \cite{Co05}.\\
$\{22, 16, 5; 1, 2, 20\}$ ($v=243$): Sumalroj and Worawannotai \cite{WS16}.\\
$\{35, 24, 8; 1, 6, 28\}$ ($v=216$): Juri\v{s}i\'{c} and Vidali \cite{JurVidpre}.\\
$\{36, 25, 8; 1, 4, 20\}$ ($v=352$): $\theta_1 = 14$ with multiplicity $32$ \cite[Thm.~4.4.4]{bcn}.\\
$\{40, 33, 8; 1, 8, 30\}$ ($v=250$): Juri\v{s}i\'{c} and Vidali \cite{JurVidpre}.\\
$\{44, 30, 5; 1, 3, 40\}$ ($v=540$): Koolen and Park \cite{KoPa10}, see Section \ref{sec:claws}.\\
$\{45, 30, 7; 1, 2, 27\}$ ($v=896$): Gavrilyuk and Makhnev \cite{GavMak2011T}.\\
$\{52, 35, 16; 1, 4, 28\}$ ($v=768$): Gavrilyuk and Makhnev \cite{GavMakpre}.\\
$\{55, 36, 11; 1, 4, 45\}$ ($v=672$): Bang \cite{Bapre} and Gavrilyuk \cite{Gav11}.\\
$\{56, 36, 9; 1, 3, 48\}$ ($v=855$): Bang \cite{Bapre} and Gavrilyuk \cite{Gav11}.\\
$\{65, 44,11; 1, 4, 55\}$ ($v=924$): Koolen and Park \cite{KoPa10}, see Section \ref{sec:claws}.\\
$\{69, 48,24; 1, 4, 46\}$ ($v=1330$): Gavrilyuk and Makhnev \cite{GavMakpre}.\\
$\{72, 45, 16; 1, 8, 54\}$ ($v=598$): $\theta_1 = 26$ with multiplicity $45$ \cite[Thm.~4.4.4]{bcn}.\\
$\{74, 54, 15; 1, 9,60\}$ ($v=630$): Coolsaet and Juri\v{s}i\'{c} \cite{CoJu08}.\\
$\{77, 60, 13; 1, 12, 65\}$ ($v=540$):  Coolsaet and Juri\v{s}i\'{c} \cite{CoJu08}.\\
$\{85, 54, 25; 1, 10, 45\}$ ($v=800$): $\theta_1 = 35$ with multiplicity $34$ \cite[Thm.~4.4.4]{bcn}.\\
$\{90,60, 12; 1, 12, 72\}$ ($v=616$): $\theta_1 = 27$ with multiplicity $48$ \cite[Thm.~4.4.4]{bcn}.\\
$\{104, 66, 8; 1, 12, 88\}$ ($v=729$): Urlep \cite{urlep12}.\\
$\{105, 102, 99; 1, 2, 35\}$ ($v=20608$): De Bruyn and Vanhove \cite{DV2012pre}.\\
$\{112,  77, 16; 1, 16, 88\}$ ($v=750$): $\theta_1 = 32$ with multiplicity $49$ \cite[Thm.~4.4.4]{bcn}.\\
$\{119, 96, 18; 1, 16, 102\}$ ($v=960$): Juri\v{s}i\'{c} and Vidali \cite{JurVidpre}.\\
$\{145, 84, 25; 1, 20, 105\}$ ($v=900$): $\theta_1 = 55$ with multiplicity $29$
\cite[Thm. 4.4.4]{bcn}.


%{\it The following four intersection arrays occur in \cite{tables}, but not in \cite{bcn}:\\
%
%\noindent $\{ 14,10,3; 1,5,12\}$: Does not exist because local graph has second
%largest eigenvalue $+1$ (\cite[Theorem 4.4.3]{bcn} and hence is complement of
%line graph by \cite[Theorem 3.12.2]{bcn}, but that is not possible.  See also
%Koolen and Yu \cite{KoYu11}.\\
%$\{ 21, 12, 8; 1, 3, 8\}$: $\theta_1 = b_1 -1$ and not a Terwilliger graph, does not exist by \cite[Theorem 4.4.11]{bcn}\\
%$\{ 24, 18, 16; 1, 2, 9\}$: Does not exist by \cite[Theorem 5.2.1]{bcn} ($i=2$), as it can not be a Terwilliger graph.\\
%$\{42, 28, 5; 1,2 ,30\}$: Does not exist by \cite[page 6, Equation (6), and Proposition 4.3.3]{bcn}}

%Antipodal diameter 3: There are 2 distance-regular graphs with intersection array
%$\{13,8,1 ; 1,4,13\}$, 4 with intersection array $\{16,10,1 ;
%1,5,16\}$. This is mentioned in section 3.2.3.

\subsection{Diameter \texorpdfstring{$4$}{4} and primitive}

\subsubsection{Uniqueness}

For the following intersection array, there is a unique distance-regular graph with that array:

\bigskip

\noindent $\{ 280, 243, 144, 10; 1, 8, 90, 280\}$ ($v=22880$): Patterson graph
(see \cite[\S 13.7]{bcn}); Brouwer, Juri\v{s}i\'{c}, and Koolen
\cite{BrJuKo08}.


\subsubsection{Nonexistence}\label{sec:tablesD4prnon}
The following intersection arrays are not feasible:

\bigskip

\noindent $\{5,4,3,3;1,1,1,2\}$ ($v=176$): Fon-Der-Flaass \cite{FDF293}.\\
$\{39,32,20,2;1,4,16,30\}$ ($v=768$): Lambeck \cite{Lamb93}.\\
$\{ \mu(2\mu +1), (\mu-1)(2\mu +1), \mu^2, \mu; 1, \mu, \mu(\mu-1),
\mu(\mu+1)\}$ ($v=8\mu^2(\mu+1)$), $\mu \geq 2$: \linebreak Godsil and Koolen
\cite{GoKo95} (besides this family, in the tables also those with $\mu=4,5,6,7$
are explicitly mentioned: $\{ 36, 27, 16, 4; 1,4,12,36\}$,
$\{55,44,25,5; 1, 5,20, 55\}$, $\{ 78, 65, 36, 6;\linebreak 1, 6,
30, 78\}$, $\{105, 90, 49, 7; 1, 7, 42, 105\}$).\\ $\{50,48 ,48,
32; 1, 1, 9, 25\}$ ($v=31635$): De Bruyn \cite{DeB2010ElJC}.


%{\it In the tables \cite{tables} there are 3 intersection arrays which do not occur in \cite{bcn}:
%$\{ 21, 14, 9, 3; 1, 3, 6, 21\}$, $\{22, 15, 9, 4; 1, 3, 6, 10\}$, and
%$\{63, 48, 45, 3; 1, 3, 16, 45\}$. They are all ruled out by \cite[Thm. 5.2.3, 5.2.1, Cor. 1.16.6]{bcn}.}

\subsection{Diameter \texorpdfstring{$4$}{4} and bipartite}

\subsubsection{Existence}

For the following intersection array, there is a distance-regular graph with that array:

\bigskip

\noindent $\{45, 44, 36, 5; 1, 9, 40, 45\}$ ($v=486$): Koolen-Riebeek graph (see Section \ref{sec:
KoolenRiebeek}).

\subsubsection{Nonexistence}

The following intersection arrays are not feasible:

\bigskip

\noindent $\{36,35,27,6; 1,9,30,36\}$ ($v=324$): Galazidis (see \cite{tables})\footnote{$\G_1\cup \G_4$ would be a strongly regular graph with parameters $(324,57,0,12)$, which does not exist \cite{GavMak2005}.}.\\
 $\{36,35,33,3; 1,3,33,36\}$ ($v=912$): Huang (see \cite{tables})\footnote{The halved graphs would be the complement of a strongly regular graph with parameters $(456,35,10,2)$, which does not exist \cite{BN88} by Proposition \ref{shillaterw}.}.\\
  $\{88,87,77,4; 1,11,84,88\}$ ($v=1452$): Huang (see \cite{tables})\footnote{The halved graphs would be the complement of a strongly regular graph with parameters $(726,29,4,1)$, which does not exist by Proposition \ref{shillaterw}.}.

%
%\subsubsection{Uniqueness and nonexistence}
%
%$\{k,k-1,k-2,k-5; 1,2,5,k\}$: Brouwer \cite{Br94} addressed the question of
%uniqueness and existence of distance-regular graphs with these intersection
%arrays. Such graphs may exist only for $k \in \{6,10,12,20\}$; for $k=6$ and
%$k=12$, examples of such graphs are known, and these have automorphism groups
%$3.$Sym$(6).2$ (see also \cite[Thm. 13.2.2]{bcn}) and $M_{12}.2$, respectively.
%Brouwer proved (by a nice purely combinatorial and computer-free argument) that
%for $k=6$ and $k=12$, distance-regular graphs with the above intersection
%arrays are unique; and that there is no such graph for $k=10$. The proof
%actually provides constructions for these graphs, and extensive information on
%their structure. Nothing is known for $k=20$.

%%the above is already in BCN (k=6) or the corrections and additions (k=12).

\subsection{Diameter \texorpdfstring{$4$}{4} and antipodal}

\subsubsection{Uniqueness}

For the following intersection arrays, there is a unique distance-regular graph with that array:

\bigskip

\noindent $\{32, 27, 8, 1; 1, 4, 27, 32\}$ ($v=315$): Soicher graph (see Section \ref{sec: Soicher and Meixner}); Soicher \cite{Soicher15}.\\
$\{ 45, 32, 12, 1,; 1, 6, 32, 45\}$ ($v=378$): $3.O_6^-(3)$-graph
(see \cite[\S
13.2C]{bcn}); Juri\v{s}i\'{c} and Koolen \cite{JuKo11}.\\
$\{56, 45, 16, 1; 1, 8, 45, 56\}$ ($v=486$): Soicher graph (see Section \ref{sec: Soicher and Meixner});
Brouwer \cite{AEBSoicher} (see also \cite[Thm.~11.4.6]{BCNcoradd}).\\
$\{117, 80, 24, 1; 1, 12, 80, 117\}$ ($v=1134$): $3.O_7(3)$-graph (see \cite[\S
13.2D]{bcn}); Juri\v{s}i\'{c} and Koolen \cite{JuKopre}.\\
$\{176, 135, 36, 1; 1, 12, 135, 176\}$ ($v=2688$): Meixner 4-cover (see Section
\ref{sec: Soicher and Meixner} and \cite[\S 12.4A]{BCNcoradd}); Juri\v{s}i\'{c}
and Koolen \cite{JuKopre}.


\subsubsection{Existence}

For the following intersection arrays, there is a distance-regular graph with
that array:

\bigskip

\noindent
$\{176, 135, 24, 1; 1, 24, 135, 176\}$ ($v=1344$): Meixner 2-cover (see Section \ref{sec: Soicher and
Meixner} and \cite[\S 12.4A]{BCNcoradd}).\\
$\{416,315,64,1;1,32,315,416\}$ ($v=5346$): Soicher graph (see Section \ref{sec: Soicher and Meixner}).

\subsubsection{Nonexistence}

The following intersection arrays are not feasible:

\bigskip

\noindent $\{32, 27, 6, 1; 1, 6, 27, 32\}$ ($v=210$): Soicher \cite{Soicher15}.\\
$\{32, 27, 9, 1; 1, 3, 27, 32\}$ ($v=420$): Soicher \cite{Soicher15}.\\
$\{45, 32, 9, 1; 1, 9, 32, 45\}$ ($v=252$): Juri\v{s}i\'{c}
and Koolen \cite{JuKo00EuJC}.\\
$\{45, 32, 15, 1; 1, 3, 32, 45\}$ ($v=756$):  Juri\v{s}i\'{c}
and Koolen \cite{JuKo00EuJC}.\\
$\{45, 40, 11, 1; 1, 1, 40, 45\}$ ($v=2352$): $\overline{\G}$ is of order $(5,8)$ with $c_2=12$, \cite[Thm.~4.2.7]{bcn}.\\
$\{ 56, 45, 12, 1; 1, 12, 45, 56\}$ ($v=324$): Brouwer \cite[Thm.~11.4.6]{BCNcoradd}.\\
$\{ 56, 45, 18, 1; 1, 6, 45, 56\}$ ($v=648$): Brouwer \cite[Thm.~11.4.6]{BCNcoradd}.\\
$\{ 56, 45, 20, 1; 1, 4, 45, 56\}$ ($v=972$): Brouwer \cite[Thm.~11.4.6]{BCNcoradd}.\\
$\{ 56, 45, 21, 1; 1, 3, 45, 56\}$ ($v=1296$): Brouwer \cite[Thm.~11.4.6]{BCNcoradd}.\\
$\{81, 56, 18, 1; 1, 9, 56, 81\}$ ($v=750$): Juri\v{s}i\'{c}
and Koolen \cite{JuKo11}.\\
$\{81, 56, 24, 1; 1, 3, 56, 81\}$ ($v=2250$): Juri\v{s}i\'{c}
and Koolen \cite{JuKo00EuJC}.\\
$\{ 96, 75, 24, 1; 1, 8, 75, 96\}$ ($v=1288$): Juri\v{s}i\'{c}
and Koolen \cite{JuKo11}.\\
$\{ 96, 75, 28, 1; 1, 4, 75, 96\}$ ($v=2576$): Juri\v{s}i\'{c}
and Koolen \cite{JuKo00EuJC}.\\
$\{115, 96, 32, 1; 1, 8, 96, 115\}$ ($v=1960$): Juri\v{s}i\'{c}
and Koolen \cite{JuKo00EuJC}.\\
$\{115, 96, 35, 1; 1, 5, 96, 115\}$ ($v=3136$): Juri\v{s}i\'{c}
and Koolen \cite{JuKo00EuJC}.\\
$\{115, 96, 36, 1; 1, 4, 96, 115\}$ ($v=3920$): Juri\v{s}i\'{c}
and Koolen \cite{JuKo00EuJC}.\\
$\{117, 80, 27, 1; 1, 9, 80, 117\}$ ($v=1512$): Juri\v{s}i\'{c}
and Koolen \cite{JuKo00EuJC}.\\
$\{117, 80, 30, 1; 1, 6, 80, 117\}$ ($v=2268$): Juri\v{s}i\'{c}
and Koolen \cite{JuKo00EuJC}.\\
$\{117, 80, 32, 1; 1, 4, 80, 117\}$ ($v=3402$): Juri\v{s}i\'{c}
and Koolen \cite{JuKo00EuJC}.\\
$\{175, 144, 25, 1; 1, 25, 144, 175\}$ ($v=1360$): Juri\v{s}i\'{c}
and Koolen \cite{JuKo00EuJC}.\\
$\{175, 144, 40, 1; 1, 10, 144, 175\}$ ($v=3400$): Juri\v{s}i\'{c}
and Koolen \cite{JuKo11}.\\
$\{176, 135, 40, 1; 1,8,135, 176\}$ ($v=4032$): Juri\v{s}i\'{c}
and Koolen \cite{JuKo00EuJC}.\\
$\{189, 128, 27, 1; 1, 27, 128, 189\}$ ($v=1276$): Juri\v{s}i\'{c}
and Koolen \cite{JuKo00EuJC}.\\
$\{189, 128, 36, 1; 1, 18, 128, 189\}$ ($v=1914$): Juri\v{s}i\'{c}
and Koolen \cite{JuKo11}.\\
$\{189, 128, 45, 1; 1, 9, 128, 189\}$ ($v=3828$): Juri\v{s}i\'{c}
and Koolen \cite{JuKo00EuJC}.\\
$\{204, 175, 40, 1; 1, 20, 175, 204\}$ ($v=2400$): Juri\v{s}i\'{c}
and Koolen \cite{JuKo00EuJC}.\\
$\{204, 175, 45, 1; 1, 15, 175, 204\}$ ($v=3200$): Juri\v{s}i\'{c}
and Koolen \cite{JuKo00EuJC}.\\
$\{261, 176, 54, 1; 1, 18, 176, 261\}$ ($v=3600$): Juri\v{s}i\'{c}
and Koolen \cite{JuKo00EuJC}.\\
$\{414, 350, 45, 1; 1, 45, 350, 414\}$ ($v=4050$): Juri\v{s}i\'{c} and Koolen
\cite{JuKo00EuJC}.



%\subsection{Diameter \texorpdfstring{$4$}{4} and antipodal (but not bipartite)}

%%In the tables \cite{tables} there are 32 intersection arrays which do not occur in \cite{bcn}. These are \\
%%$\{9,8,2,1; 1,2,8,9\}$
%%$\{9,8,3,1; 1,1,8,9\}$
%%$\{21,16,6,1; 1,6,16,21\}$
%%$\{21,16,8,1; 1,4,16,21\}$
%%$\{21,16,9,1; 1,3,16,21\}$\\
%%$\{21,16,10,1; 1,2,16,21\}$
%%$\{21,20,8,1; 1,2,20,21\}$
%%$\{21,20,9,1; 1,1,20,21\}$
%%$\{48,45,14,1; 1,2,45,48\}$\\
%%$\{56,54,9,1; 1,9,54,56\}$
%%$\{56,54,12,1; 1,6,54,56\}$
%%$\{56,54,15,1; 1,3,54,56\}$
%%$\{56,54,16,1; 1,2,54,56\}$\\
%%$\{56,54,17,1; 1,1,54,56\}$
%%$\{108,104,18,1; 1,9,104,108\}$
%%$\{135,128,18,1; 1,18,128,135\}$\\
%%$\{135,128,24,1; 1,12,128,135\}$
%%$\{135,128,27,1; 1,9,128,135\}$
%%$\{135,128,30,1; 1,6,128,135\}$\\
%%$\{154,150,15,1; 1,15,150,154\}$
%%$\{154,150,20,1; 1,10,150,154\}$
%%$\{182,180,40,1; 1,20,180,182\}$\\
%%$\{182,180,45,1; 1,15,180,182\}$
%%$\{182,180,48,1; 1,12,180,182\}$
%%$\{200,189,48,1; 1,24,189,200\}$\\
%%$\{200,189,54,1; 1,18,189,200\}$
%%$\{243,224,36,1; 1,36,224,243\}$
%%$\{243,224,48,1; 1,24,224,243\}$\\
%%$\{243,224,54,1; 1,18,224,243\}$
%%$\{261,176,48,1; 1,24,176,261\}$
%%$\{264,250,30,1; 1,30,250,264\}$\\
%%$\{264,250,40,1; 1,20,250,264\}$.
%%\\
%%\\
%%Except for $\{261,176,48,1; 1,24,176,261\}$, the arrays do not satisfy the Krein condition.
%%The intersection array $\{261,176,48,1; 1,24,176,261\}$ does not satisfy the absolute bound.
%%\\
%%\\
%%There are 4 intersection arrays that occur in \cite{bcn} but not in \cite{tables}:
%%\\
%%$\{ 56, 45, 12, 1; 1, 12, 45, 56\}$
%%$\{ 56, 45, 18, 1; 1, 6, 45, 56\}$
%%$\{ 56, 45, 20, 1; 1, 4, 45, 56\}$
%%$\{ 56, 45, 21, 1; 1, 3, 45, 56\}$. \\
%%(But see below)
%%



%The tables of \cite{bcn} and \cite{tables} agree for diameter at least 5. \\

\subsection{Diameter \texorpdfstring{$5$}{5} and antipodal} %(but not bipartite)}

\subsubsection{Uniqueness}

For the following intersection array, there is a unique distance-regular graph with that array:

\bigskip

\noindent $\{ 22, 20, 18, 2,1 ; 1,2, 9, 20, 22\}$ ($v=729$): coset graph of the
dual of the ternary Golay code; Blokhuis, Brouwer, and Haemers
\cite{BlokhuisBH07}.

\subsubsection{Nonexistence}

The following intersection arrays are not feasible:

\bigskip

\noindent $\{ 2 \mu^2 + \mu, 2\mu^2 + \mu -1, \mu^2, \mu, 1; 1, \mu, \mu^2,
2\mu^2 + \mu -1, 2\mu^2 + \mu\}$ ($v=4\mu^2(2\mu+3)$), $\mu \geq
2$: Coolsaet, Juri\v{s}i\'{c}, and Koolen \cite{CoJuKo08EuJC}
(besides this family, in the tables also those with $\mu=3,4,5,6,7$
are explicitly mentioned: $\{21, 20, 9,3,1; 1,3,9,20,21\}$,
$\{36,35,16,4,1;1,4,16,35,36\}$,\linebreak$\{55, 54, 25, 5, 1;
1,5,25,54,55\}$, $\{78, 77, 36, 6, 1; 1, 6, 36, 77, 78\}$, $\{105,
104, 49, 7, 1; 1, 7, 49, \linebreak104, 105\}$).\\ $\{105, 90, 49,
7, 1; 1, 7, 49, 90, 105\}$ ($v=2912$): $\theta_1 = 35$ with
multiplicity $78$ \cite[Thm.~4.4.4]{bcn}.

\subsection{Diameter \texorpdfstring{$5$}{5} and bipartite} %(but not antipodal)}

\subsubsection{Uniqueness}

For the following intersection arrays, there is a unique distance-regular graph with that array:

\bigskip

\noindent $\{ 7, 6,6,4,4; 1,1, 3, 3, 7\}$ ($v=310$): Doubled Grassmann ($q=2$); Cuypers \cite{Cuypers92}, see Section \ref{sec:otherinfinite}.\\
$\{13, 12, 12, 9, 9; 1, 1, 4, 4, 13\}$ ($v=2420$): Doubled Grassmann ($q=3$);
Cuypers \cite{Cuypers92}, see Section \ref{sec:otherinfinite}.

\subsubsection{Nonexistence}

The following intersection array is not feasible:

\bigskip

\noindent $\{55, 54, 50, 35, 10; 1, 5, 20, 45, 55\}$ ($v=3500$): Vidali \cite{Vidali}.

\subsection{Diameter \texorpdfstring{$6$}{6} and imprimitive}

\subsubsection{Nonexistence}

The following intersection arrays are not feasible:

\bigskip

\noindent $\{15, 14, 12, 6, 1, 1; 1, 1, 3, 12, 14, 15\}$ ($v=1518$, antipodal):
Ivanov and Shpectorov \cite{IvaShp90}\\
$\{7, 6, 6, 5, 4, 3; 1, 1, 2, 3, 4, 7\}$ ($v=686$, bipartite): Koolen \cite{Ko292}.
