Partially metric association schemes with a multiplicity three

0.57 MB·

by  Edwin R. van Dam

PARTIALLY METRIC ASSOCIATION SCHEMES WITH A
MULTIPLICITY THREE
EDWINR.VANDAM,JACKH.KOOLEN,ANDJONGYOOKPARK
7
1
0 Abstract. An association scheme is called partially metric if it has a con-
2 nectedrelationwhosedistance-tworelationisalsoarelationofthescheme. In
this paper we determine the symmetric partially metric association schemes
n
with a multiplicity three. Besides the association schemes related to regular
a
J complete 4-partite graphs, we obtain the association schemes related to the
Platonic solids, the bipartite double scheme of the dodecahedron, and three
2
association schemes that are related to well-known 2-arc-transitive covers of
1
the cube: the Mo¨bius-Kantor graph, the Nauru graph, and the Foster graph

F048A.Inordertoobtainthisresult,wealsodeterminethesymmetricassoci-
]
O ationschemeswithamultiplicitythreeandaconnectedrelationwithvalency
three. Moreover, we construct an infinite family of cubic arc-transitive 2-
C
walk-regular graphs with an eigenvalue with multiplicity three that give rise
. tonon-commutativeassociationschemeswithasymmetricrelationofvalency
h
threeandaneigenvaluewithmultiplicitythree.
t
a
m
[

1 1. Introduction
v
BannaiandBannai[3]showedthattheassociationschemeofthecompletegraph
3
9 onfourverticesistheonlyprimitivesymmetricassociationschemewithamultiplic-
1 ityequaltothree. Theyalsoposedtheproblemofdeterminingallsuchimprimitive
3 symmetricassociationschemes. Asmanyproductconstructionscangiverisetosuch
0
schemes with a multiplicity three, we suggest and solve a more restricted problem.
.
1 Indeed, we will determine the symmetric partially metric association schemes with
0 a multiplicity three, where an association scheme is called partially metric if it has
7
a connected relation whose distance-two relation is also a relation of the scheme.
1
Besides the association schemes related to regular complete 4-partite graphs, we
:
v obtain the association schemes related to the Platonic solids, the bipartite dou-
i
X ble scheme of the dodecahedron, and three association schemes that are related
to well-known 2-arc-transitive covers of the cube: the M¨obius-Kantor graph, the
r
a Nauru graph, and the Foster graph F048A. In order to obtain this classification,
we also determine the symmetric association schemes with a multiplicity three and
a connected relation with valency three and build on work by C´amara and the
authors on 2-walk-regular graphs [6]. We furthermore construct an infinite family
of cubic arc-transitive 2-walk-regular graphs with an eigenvalue with multiplicity
threethatgiverisetonon-commutativeassociationschemeswithasymmetricrela-
tionofvalencythreeandaneigenvaluewithmultiplicitythree. Thelatterindicates
that the considered problem is completely different for non-symmetric association
2010 Mathematics Subject Classification. 05E30,05C50.
Keywordsandphrases. associationscheme,2-walk-regulargraph,smallmultiplicity,distance-
regulargraph,coverofthecube.
1
2 EDWINR.VANDAM,JACKH.KOOLEN,ANDJONGYOOKPARK
schemesandthatitmaybedifficulttoclassifythecubic2-walk-regulargraphswith
a multiplicity three.
Related work has been done by Yamazaki [23], who showed that if a symmetric
association scheme has a connected relation with valency three, then this relation
is bipartite or distance-regular. Hirasaka [17] classified the primitive commutative
association schemes with a non-symmetric relation of valency three. Distance-
regular graphs with a small multiplicity have also been classified; those with mul-
tiplicity three are the graphs of the five Platonic solids and the regular complete
4-partite graphs. For this and several other results on multiplicities of distance-
regulargraphs, wereferto[9, 14]. SeealsotheexpositorypaperbyBannai[2]on
§
among others the classification problem of association schemes.
Thispaperisorganizedasfollows: afterthisintroduction,wegivedefinitionsand
our main tools in Section 2. In particular, we will use a generalization of Godsil’s
multiplicity bound [13, Thm. 1.1] (Section 2.4), a generalization of the concept of
a light tail introduced by Juriˇsi´c, Terwilliger, and Zˇitnik [18] (Section 2.7), and
a lemma by Yamazaki [23] (Section 2.8). In Section 3, we describe the relevant
associationschemesandshowuniquenessornon-existenceoftheschemesthatoccur
in Section 4 in the proof of the classification result of association schemes with a
valencythreeandamultiplicitythree. InSection5,weobtainthefinalclassification
result of partially metric association schemes with a multiplicity three. Finally,
in Section 6, we construct an infinite family of cubic arc-transitive 2-walk-regular
graphswithaneigenvaluewithmultiplicitythreethatgiverisetonon-commutative
association schemes.
2. Definitions and tools
In this section we shall introduce notation, concepts, and useful tools that we
shall use in the remainder of the paper.
2.1. Graphs. Let Γ be a (simple and undirected) graph with vertex set V. The
distance dist(x,y) between two vertices x,y V is the length of a shortest path
∈
connecting x and y. The maximum distance between two vertices in Γ is the
diameter D. We use Γ (x) for the set of vertices at distance i from x and write,
i
for the sake of simplicity, Γ(x) := Γ (x). The degree of x is the number Γ(x) of
1
| |
vertices adjacent to it. A graph is regular with valency k if the degree of each of its
vertices is k.
For a graph Γ with diameter D, the distance-i graph Γ of Γ (1 i D) is
i
≤ ≤
the graph whose vertices are those of Γ and whose edges are the pairs of vertices
at mutual distance i in Γ. In particular, Γ = Γ. The distance-i matrix B of Γ
1 i
is the matrix whose rows and columns are indexed by the vertices of Γ and the
(x,y)-entry is 1 whenever dist(x,y)=i and 0 otherwise1. The adjacency matrix A
of Γ equals B and the eigenvalues of the graph Γ are those of A. The multiplicity
1
of an eigenvalue θ of Γ is denoted by m(θ). Let θ > θ > > θ be the
0 1 r
···
distinct eigenvalues of Γ. Then the minimal graph idempotent for θ is defined
j
by Fj := (cid:81)i(cid:54)=j Aθj−−θθiiI, i.e., this is the matrix representing the projection onto the
1NotethatwedonotusethenotationAi forthedistance-imatrixinordertoavoidconfusion
withtherelationmatrixofanassociationscheme;seeSection2.3
PARTIALLY METRIC ASSOCIATION SCHEMES WITH A MULTIPLICITY THREE 3
eigenspace for θ . The spectral decomposition theorem leads immediately to
j
r
(cid:88)
(1) A(cid:96) = θ(cid:96)F
j j
j=0
for every integer (cid:96) 0.
≥
2.2. Walk-regularity. Aconnectedgraphist-walk-regularifthenumberofwalks
of every given length (cid:96) between two vertices x,y V only depends on the distance
∈
between them, provided that dist(x,y) t (where it is implicitly assumed that the
≤
diameter of the graph is at least t). From (1), we obtain that a connected graph
is t-walk-regular if and only if for every minimal graph idempotent the (x,y)-entry
only depends on dist(x,y), provided that the latter is at most t (see Dalf´o, Fiol,
and Garriga [8]). In other words, for a fixed minimal graph idempotent F for θ,
there exist constants α := α (θ), for 0 i t, such that B F = α B , where
i i i i i
≤ ≤ ◦ ◦
is the entrywise product.
Given a vertex x in a graph Γ and vertex y at distance i from x, we consider
the numbers a (x,y) = Γ(y) Γ (x), b (x,y) = Γ(y) Γ (x), and c (x,y) =
i i i i+1 i
| ∩ | | ∩ |
Γ(y) Γ (x). A connected graph Γ with diameter D is distance-regular if these
i 1
| ∩ − |
parameters do not depend on x and y, but only on i, for 0 i D. If this is
≤ ≤
the case then these numbers are denoted simply by a , b , and c , for 0 i D,
i i i
≤ ≤
and they are called the intersection numbers of Γ. Also, if a connected graph Γ is
t-walk-regular, then the intersection numbers of Γ are well-defined for 0 i t
≤ ≤
(see Dalf´o et al. [7, Prop. 3.15]).
2.3. Association schemes. Let X be a finite set, say with n elements. An asso-
ciation scheme with rank d+1 on X is a pair (X, ) such that
R
(i) = R ,R , ,R is a partition of X X,
0 1 d
R { ··· } ×
(ii) R := (x,x) x X ,
0
{ | ∈ }
(iii) for each i (0 i d) R =R , i.e., if (x,y) R then (y,x) R ,
≤ ≤ i i(cid:62) ∈ i ∈ i
(iv) therearenumbersph —theintersectionnumbersof(X, )—for0 i,j,h
ij R ≤ ≤
d, such that for every pair (x,y) R the number of z X with (x,z) R
h i
∈ ∈ ∈
and (z,y) R equals ph.
∈ j ij
In the literature, more general definitions of association schemes are available.
We will use these also in Section 6. In particular, we will refer to them as non-
symmetric association schemes when not all relations are symmetric (in this case
(iii) is replaced by R =R for some i). A non-symmetric association scheme can
i i(cid:62)(cid:48) (cid:48)
even be non-commutative in the sense that ph = ph for some h,i,j. Association
ij (cid:54) ji
schemes in this broader sense are generalizations of so-called “Schurian schemes”
that arise naturally from the action of a finite transitive group on X; the orbitals
(the orbits on X X) of such a group action form the relations of a (possibly
×
non-commutative or non-symmetric) association scheme.
From now on, we will however assume that association schemes are symmetric
(as in the above definition), unless we specify explicitly that it is non-symmetric or
non-commutative.
The elements R (0 i d) of are called the relations of (X, ). For each
i
≤ ≤ R R
i > 0, the relation R can be interpreted as a graph Γ with vertex set X if we
i
call two vertices x and y adjacent whenever (x,y) R . We call Γ the scheme
i
∈
graph of R , that is regular with valency k := p0. The corresponding adjacency
i i ii
matrix A is called the relation matrix of R , for i > 0, and we let A = I be the
i i 0
4 EDWINR.VANDAM,JACKH.KOOLEN,ANDJONGYOOKPARK
relation matrix of R . It is easy to see that the conditions (i)-(iv) are equivalent to
0
conditions (i)’-(iv)’ on the relation matrices:
d
(cid:88)
(i)’ A =J, where J is the all-one matrix,
i
i=0
(ii)’ A =I, where I is the identity matrix,
0
(iii)’ (A ) =A for all i 0,1 ,d ,
i (cid:62) i
∈{ ··· }
d
(cid:88)
(iv)’ A A = phA .
i j ij h
h=0
The Bose-Mesner algebra of (X, ) is the matrix algebra generated by A
i
M R { |
i = 0,...,d . From (iv)’ we see that A i = 0,...,d is a basis of and
i
} { | } M
hence is (d + 1)-dimensional. Note that the Bose-Mesner algebra is closed
M
under both ordinary multiplication and entrywise multiplication . From (iii)’ and
◦
(iv)’, it follows that the relation matrices commute, and hence all the matrices in
are simultaneously diagonalizable. It follows that has a basis of minimal
M M
scheme idempotents E ,E , ,E , which we can order such that nE is the all-
0 1 d 0
···
ones matrix J. The rank of E is denoted by m and is called the multiplicity of
j j
E , for 0 j d.
j
≤ ≤
Now the Bose-Mesner algebra has two bases and we can express each basis
M
in terms of the other. Define constants P and Q (0 i,j d) by
ji ij
≤ ≤
d d
(cid:88) 1 (cid:88)
(2) A = P E and E = Q A .
i ji j j ij i
n
j=0 i=0
Note that m =rkE =trE =Q . From (2), we have
j j j 0j
(3) A E =P E ,
i j ji j
hence the numbers P are called the eigenvalues of (X, ). In this paper, we shall
ji
R
mainly focus on the eigenvalues of the scheme graph of R . In this case we call
1
P (0 j d) the corresponding eigenvalue on E and it is denoted by θ , i.e.,
j1 j j
≤ ≤
A E = θ E . Note that these eigenvalues θ need not be distinct, for example in
1 j j j j
the Johnson scheme J(7,3) defined on the triples of a 7-set, the relation defined
by “intersecting in 1 point” has this property (because it is strongly regular in an
association scheme with rank 4).
Since the minimal scheme idempotents form a basis of , we have
M
d
1 (cid:88)
(4) E E = qhE
i◦ j n ij h
h=0
for certain real numbers qh (0 h,i,j d) that are called Krein parameters. The
ij ≤ ≤
Krein parameters are nonnegative and q0 = δ m , where δ is 1 whenever i = j
ij ij j ij
and 0 otherwise. From (2), we also have
Q
ij
(5) E A = A .
j i i
◦ n
It follows that (E ) = Q0j = mj for all x X. For (x,y) R , let ω =
j xx n n ∈ ∈ i xy
ω (j) = (Ej)xy = Qij/n = Qij. We call these numbers ω = ω (j) = Qij the
xy (Ej)xx mj/n mj i i mj
PARTIALLY METRIC ASSOCIATION SCHEMES WITH A MULTIPLICITY THREE 5
cosines corresponding to E , and note that ω = 1. From (3) and (5), it follows
j 0
that if (x,y) R , then
h
∈
d
(cid:88) (cid:88)
(6) P ω =P ω = ω = phω ,
ji h ji xy zy i(cid:96) (cid:96)
z∈Ri(x) (cid:96)=1
where (here and elsewhere) R (x)= z (x,z) R .
i i
{ | ∈ }
FromastandardpropertyoftheentriesofQ,see[5,Lemma2.2.1.(iv)],weobtain
that
d
(cid:88)
(7) m k ω2 =n.
j i i
i=0
For more background on association schemes, see [4], [5, Ch. 2], and [19].
2.4. Partially metric association schemes and Godsil’s bound. An associ-
ation scheme (X, ) with rank d+1 is called t-partially metric (with respect to
R
the connected relation R) if — possibly after reordering of the relations — A is
i
a polynomial of degree i in A for i = 1,2,...,t, where A is the relation matrix of
R=R (where implicitly it is assumed that t d). This is equivalent to R being
1 i
≤
the distance-i graph of the scheme graph of R for i t. Note that the distance-i
≤
graphΓ ofaschemegraphΓisalwaysaunionofrelationsR . Thescheme(X, )
i j
R
is called metric if it is d-partially metric; in this case R is a distance-regular graph.
For the sake of readability, we will assume in the remainder of the paper that for
a partially t-metric scheme, the relations are ordered according to distance, up to
distancet(asintheabovedefinition),unlessspecifieddifferently. Wenotethatat-
partiallymetricschemeisclearlyalsos-partiallymetricfors t. Everyassociation
≤
scheme with at least one connected relation is 1-partially metric; we therefore call
an association scheme partially metric if it is at least 2-partially metric. To ensure
that every metric association scheme is also partially metric, we also say that an
association scheme with rank 2 (where there is no distance-2 relation) is partially
metric. We finally note that the concept of t-partially metric can be extended to
non-symmetricschemeswithrespecttoasymmetricrelation. Sucht-partiallymet-
ric (possibly non-symmetric) schemes would arise naturally from t-arc-transitive
graphs, for example; see also Section 6.
If the association scheme (X, ) is t-partially metric, then the corresponding
R
scheme graph Γ of R is called a t-partially metric scheme graph. This scheme
1
graph is t-partially distance-regular in the sense of [7], and even stronger, it is t-
walk-regular. Thus, the intersection numbers of Γ are well-defined for 0 i t.
≤ ≤
In this case we have a = pi ,b = pi and c = pi for 0 i t 1, and
a = pt , c = pt aind b1i=ib 1a,i+1 c , whiere b1,i−=1 k = p≤0 is≤the−valency
oft Γ. A1tn itllustra1t,ti−n1g examptle of0a−3-tpa−rtiatlly metric0 scheme g1r1aph is given by
the so-called flag graph of the 11-point biplane; see Figure 1 in [6] or [7] for the
corresponding “relation-distribution diagram”. Such a diagram is similar as the
distance-distributiondiagramofadistance-regulargraph. Therelation-distribution
diagramofanassociationschemewithrespecttoaschemegraphR hasa“bubble”
1
for each relation R , inside of which we depict k , and we connect the bubble of R
i i i
by an edge to the bubble of R if pi > 0, and depict this intersection number on
j 1j
top of the edge; see for example Figure 3.
6 EDWINR.VANDAM,JACKH.KOOLEN,ANDJONGYOOKPARK
From (6), we now obtain that
θ =b ω
0 1
θω =c ω +a ω +b ω (1 h t 1),
h h h 1 h h h h+1
− ≤ ≤ −
whereω (0 h d)arethecosinescorrespondingtoaminimalschemeidempotent
h
≤ ≤
E for corresponding eigenvalue θ. It follows in particular that if t 2, then
≥
θ2 a θ k
1
(8) ω =1, ω =θ/k, ω = − − .
0 1 2
kb
1
As an immediate consequence of [6, Thm. 4.3], we find the following generaliza-
tion of Godsil’s bound [13, Thm. 1.1].
Theorem2.1. Let(X, )beapartiallymetricassociationschemeandassumethat
R
the corresponding scheme graph Γ has valency k 3. Let E be a minimal scheme
≥
idempotent of (X, ) with multiplicity m for corresponding eigenvalue θ = k. If
R (cid:54) ±
Γ is not complete multipartite, then k (m+2)(m−1).
≤ 2
This result implies that if k 3 then m 3. For k = 2, we only have the
≥ ≥
polygons and they have multiplicity 2 for all minimal scheme idempotents except
those for corresponding eigenvalue 2. If Γ is complete multipartite, then d = 2.
±
Inthiscase,itfollowsthatifk 3,thenmultiplicity2onlyoccursforthecomplete
≥
tripartite graphs, and multiplicity 1 only occurs for eigenvalue k of the complete
±
bipartite graphs. Multiplicity 3 occurs only for the complete 4-partite graphs and
the complete tripartite cocktail party graph, also known as the octahedron.
2.5. Product schemes and the bipartite double. Let (X, ) be an associa-
R
tion scheme with rank d+1 and relation matrices A for i = 0,1,...,d, and let
i
(X , ) be an association scheme with rank d +1 with relation matrices A for
(cid:48) R(cid:48) (cid:48) (cid:48)j
j =0,1,...,d. Thedirect productof(X, )and(X , )istheassociationscheme
(cid:48) (cid:48) (cid:48)
R R
with relation matrices A A for i = 0,1,...,d and j = 0,1,...,d. It is easy to
i⊗ (cid:48)j (cid:48)
seethattheminimalidempotentsofthisdirectproductschemearealsoallpossible
Kroneckerproductsoftheminimalidempotentsof(X, )and(X , ); seealso[1,
(cid:48) (cid:48)
R R
Chapter 3]. Starting from an association scheme (X, ) with a multiplicity three,
R
one can construct other association schemes with a multiplicity three by taking
the direct product of (X, ) with any other scheme. Also other kinds of product
R
constructions for association schemes are possible, giving rise to many association
schemes with a multiplicity three, and suggesting that classifying all association
schemes with a multiplicity three may be impossible. Likewise, multiplicity two
may be too hard, although in this case our result in [6, Prop. 6.5] should be useful.
Thebipartite double schemeBD(X, )of(X, )isthedirectproductof(X, )
R R R
and the rank two association scheme on two vertices. In this way, every minimal
idempotentof(X, )withmultiplicitymcorrespondstotwominimalidempotents
R
of BD(X, ) with multiplicity m. For a connected graph Γ with vertex set V, the
R
bipartite double of Γ is the graph whose vertices are the symbols x+,x (x V)
−
∈
and where x+ is adjacent to y if and only of x is adjacent to y in Γ. If Γ is the
−
scheme graph of a relation R in (X, ), then the bipartite double of Γ is a scheme
R
graph in the bipartite double scheme BD(X, ).
R
If (X, ) is t-partially metric with corresponding scheme graph Γ having odd-
R
girth at least 2t+1, then the bipartite double of (X, ) is also t-partially metric.
R
PARTIALLY METRIC ASSOCIATION SCHEMES WITH A MULTIPLICITY THREE 7
This result follows from the arguments given in the proof of the analogous result
for t-walk-regular graphs in [6, Prop. 3.1].
2.6. Quotient schemes and covers. Anassociationschemeiscalledimprimitive
ifanon-trivialunionofsomeoftherelationsisanequivalencerelation. Inthiscase,
there is a subscheme on each of the equivalence classes, and a quotient scheme on
the set of equivalence classes. The original scheme is called a cover of the quotient
scheme. TheintersectionnumbersandKreinparametersofthesubschemesandthe
quotient scheme follow from those of the original scheme. Like all direct product
schemes, the bipartite double scheme BD(X, ) is an example of an imprimitive
R
association scheme; it is a double cover of (X, ). For details, we refer the reader
R
to [4, 2.9], [5, 2.4], or [10].
§ §
A particular way to construct covers of graphs is by using voltage graphs. Let
Γ=(V,E) be a graph and let (G,+) be a group. Let E(cid:126) be the set of arcs of Γ (for
everyedge x,y ,therearetwooppositearcs: (x,y)and(y,x)). Amapα:E(cid:126) G
{ } →
such that α(x,y) = α(y,x) for every edge x,y is called a voltage assignment,
− { }
and (V,E,α) is called a voltage graph. The derived graph Γ of this voltage graph
(cid:48)
is a cover of Γ; it has vertex set V G, and if x,y is an edge in Γ, then Γ has
(cid:48)
× { }
edges (x,g),(y,g+α(x,y)) for every g G. Every double cover is the derived
graph{of a voltage graph wit}h group Z . In∈this case, the situation is simpler, and
2
we can put voltages on the edges instead of the arcs. For example, the bipartite
double can be obtained by putting voltage 1 on every edge.
2.7. A light tail. Let(X, )beapartiallymetricassociationschemeandletAbe
R
therelationmatrixofR . AminimalschemeidempotentE :=E forcorresponding
1 j
(cid:88)
eigenvalue θ is called a light tail if the matrix F := qhE is nonzero and
jj h
h=0
AF = ηF for some real number η. Thus, if qh = 0,(cid:54) then the corresponding
jj (cid:54)
eigenvalue on E is equal to η, for all h = 0. Because R is connected, this also
h 1
(cid:54)
implies that η =k. We call F the associated matrix for E and η the corresponding
(cid:54)
eigenvalue on F. We call the light tail degenerate if θ = η and non-degenerate
otherwise. Thisgeneralizestheconceptoflighttailsindistance-regulargraphsthat
wasintroducedbyJuriˇsi´c,Terwilliger,andZˇitnik[18]. NotethatF =nE E mE
0
◦ −
by (4), where m = m is the rank of E, which implies that F = 1m(m 1).
i xx n −
Because F is positive semidefinite, it follows that F = 0 if and only if m = 1. By
Theorem2.1andtheremarksthereafterthisisequivalenttoθ = k,wherek isthe
±
valency of R1. Let us now define F(cid:101) := m(mn 1)F, so that F(cid:101)xx =1. Because F(cid:101) is in
−
the Bose-Mesner algebra of (X, ), there are ρ0,ρ1,...,ρd such that F(cid:101) Ai =ρiAi
R ◦
for all i = 0,1,...,d. Similar as for minimal scheme idempotents, we call these
numbers the cosines corresponding to F, and we let ρ = ρ for (x,y) R .
xy i i
∈
Similar as (6), the following now holds for (x,y) R :
h
∈
d
(cid:88) (cid:88)
ηρ = ρ = ph ρ .
h zy 1(cid:96) (cid:96)
z∈R1(x) (cid:96)=1
In particular, this implies that ρ0 = 1,ρ1 = η/k, and ρ2 = η2−kab11η−k. It moreover
follows from the equation F =nE E mE that
0
◦ −
(9) (m 1)ρ =mω2 1,
− i i −
8 EDWINR.VANDAM,JACKH.KOOLEN,ANDJONGYOOKPARK
where ω are the cosines corresponding to E, for i = 0,1,...,d. Working out this
i
equation for i=1 gives that
m
(10) (m 1)η = θ2 k.
− k −
Ourgeneralizationoflighttailsismotivatedbythecharacterizationofthecaseof
equalityinthefollowingresultonthemultiplicitiesofminimalschemeidempotents.
For distance-regular graphs, this bound was derived by Juriˇsi´c, Terwilliger, and
Zˇitnik [18], and their proof can be followed almost completely.
Theorem 2.2. Let (X, ) be a partially metric association scheme with rank d+
R
1 3, and assume that the corresponding scheme graph Γ has valency k 3. Let E
≥ ≥
be a minimal scheme idempotent with multiplicity m for corresponding eigenvalue
θ = k. Then
(cid:54) ±
k(θ+1)2a (a +1)
1 1
(11) m k ,
≥ − ((a +1)θ+k)2+ka b
1 1 1
with equality if and only if E is a light tail.
Proof. We give a sketch of the proof of the first part, as most details are the same
as in the case of distance-regular graphs; see [18, Thm 3.2 and 4.1]. Let j be such
that E =E . Then the bound (11) follows from applying Cauchy-Schwarz to
j
(cid:104)(cid:113) (cid:113) (cid:105) (cid:104) (cid:113) (cid:113) (cid:105)
v = q1 m ,..., qd m and v = θ q1 m ,...,θ qd m .
0 jj 1 jj d 1 1 jj 1 d jj d
The bound is tight if and only if v and v are linearly dependent, which is the
0 1
case if and only if θ is the same for all h=0 such that qh =0, in other words, if
and only if E is a lhight tail. (cid:54) jj (cid:54) (cid:3)
j
2.8. Yamazaki’s lemma. The following result was shown by Yamazaki [23] and
is analogous to the result that a cubic 1-walk-regular graph is 2-walk-regular [6].
For convenience and because the terminology in [23] is different, we give a proof of
this result.
Lemma 2.3. (cf. [23, Lemma2.4]) Let (X, ) be an association scheme with rank
R
d+1 3. IfthereexistsaconnectedrelationR withvalencythree, then(X, )
≥ ∈R R
is partially metric with respect to R.
Proof. Let Γ be the scheme graph of R =: R . Because the rank of the scheme is
1
at least 3, Γ is not the complete graph on 4 vertices, and so a = p1 = 0. If Γ
1 11 2
is not a relation of the scheme, then it must be the union of two relations, R and
2
R say, and then p1 = p1 = 1. Now let x be a vertex of Γ and let y ,y ,y be
3 12 13 1 2 3
the three neighbors of x. Clearly these three are mutually at distance 2. Without
loss of generality, we may assume that (y ,y ) R and (y ,y ) R because
1 2 2 1 3 3
∈ ∈
p1 =p1 =1. But then (y ,y ) should be contained in both R and R , which is
12 13 2 3 2 3
a contradiction. Thus, (X, ) is partially metric with respect to R. (cid:3)
R
Thefinallemma,whichweshallcallYamazaki’slemma,isalsofrom[23]. Again,
we give a proof for convenience and because of the different terminology in [23].
The result is depicted in Figure 1.
Lemma 2.4. (cf. [23, Lemma 2.8]) Let (X, ) be an association scheme with
R
rank d+1 4 and a connected scheme graph Γ of R . Let x,z be vertices
1
≥ ∈ R
such that (x,z) R and dist(x,z) = i 2. Assume that there exist two distinct
2
∈ ≥
PARTIALLY METRIC ASSOCIATION SCHEMES WITH A MULTIPLICITY THREE 9
ci+1(x;z3)=1
z3
R3 R3
i(cid:21)2 i(cid:21)2
x x
R2 R2 R5
R4 R4
Figure 1. A graphical interpretation of Yamazaki’s lemma 2.4
R3 R3 z3
i 2
≥
v3 x z
R4 z4
R5
Figure 2. The configuration of vertices in the proof of Yamazaki’s lemma 2.4
neighbors z ,z of z and two distinct relations R ,R such that (x,z ) R ,
3 4 3 4 3 3
∈ R ∈
(x,z ) R , dist(x,z )=dist(x,z )=i+1, and c (x,z )=1. Then there exists
4 4 3 4 i+1 3
∈
a relation R such that p1 =0, p1 =0, and R Γ = .
5 ∈R 35 (cid:54) 45 (cid:54) 5∩ i ∅
Proof. Because the association scheme is symmetric, there exist a neighbor v of
3
x such that (z,v ) R , dist(z,v ) = i+1, and c (z,v ) = 1. Let R be the
3 3 3 i+1 3 5
∈
relation containing (v ,z ). See Figure 2 for a picture of this configuration. Then
3 4
p1 = 0 as (z,z ) R , (z,v ) R , and (v ,z ) R , and similarly p1 = 0 as
35 (cid:54) 4 ∈ 1 3 ∈ 3 3 4 ∈ 5 45 (cid:54)
(x,v ) R , (x,z ) R , and (v ,z ) R .
3 1 4 4 3 4 5
∈ ∈ ∈
In order to show that R Γ = , it suffices to show that dist(v ,z ) = i.
5 i 3 4
∩ ∅ (cid:54)
From dist(z,v ) = i+1 and c (z,v ) = 1, it is clear that Γ(v ) Γ (z) = x .
3 i+1 3 3 i
∩ { }
By symmetry and because Γ is a union of relations of , there exists a unique
i
R
vertex y such that Γ(z) Γ (v ) = y , and it follows that dist(x,y) = i 1. But
i 3
dist(x,z )=i+1, hence∩dist(v ,z ){=}i. − (cid:3)
4 3 4
(cid:54)
3. Uniqueness and non-existence of the relevant association schemes
In this section, we will discuss some of the relevant association schemes having
a multiplicity three that occur in the proof of the classification result in Section 4.
3.1. The dodecahedron. The dodecahedron graph is a distance-regular graph
with spectrum 31,√53,15,04, 24, √53 . Thus, both the corresponding metric
{ − − }
association scheme and its bipartite double scheme have minimal scheme idempo-
tents with a multiplicity three. Note however that the bipartite double graph does
not have an eigenvalue with multiplicity three; its spectrum is 31,√56,24,15,08,
{
15, 24, √56, 31 . The relation-distribution diagram of the bipartite double
− − − − }
scheme is given in Figure 3, where we also included the cosines for eigenvalue √5
that we obtained in the proof of Theorem 4.3. We note that the bipartite double
10 EDWINR.VANDAM,JACKH.KOOLEN,ANDJONGYOOKPARK
graphisalsotheschemegraphofa3-partiallymetricfusionschemeofthebipartite
double scheme. This scheme can be obtained by fusing three times a pair of rela-
tions(i.e.,R R ,R R ,andR R ;seeFigure3). However,alsothreepairs
3 4 5 8 11 12
∪ ∪ ∪
ofidempotentsare“fused”,inparticulartwopairsofidempotentswithmultiplicity
three, leaving no multiplicity three in this fusion scheme.
1 3
3 1
2
1 w5=√35 w11=1
6 1
1
1 3 1 3 2 1 6 1 w3=13 1 6 1 2 3 1 3 1
1 1
w0=1 w1=√35 w2=13 1 1 w6=w7=−13 w9=−√35 w10=−1
6
1
w4=−13 2
1 3
3 1
w8=−√35 w12=−1
Figure 3. Relation-distribution diagram of the bipartite double
of the dodecahedron
Proposition 3.1. The bipartite double of the association scheme of the dodeca-
hedron graph is the unique association scheme with scheme graph having relation-
distribution diagram as in Figure 3.
Proof. Because R has valency 1, the relation R R is clearly an equivalence
11 0 11
∪
relation. If we take the quotient scheme with respect to this equivalence relation,
weobtainanassociationschemeforwhichtheschemegraphobtainedfromR R
1 5
∪
is distance-regular with valency three and distance-distribution diagram as that
of the dodecahedron; this follows from Figure 3. Because the dodecahedron and
the corresponding association scheme is determined by its intersection numbers,
this quotient scheme is indeed the metric association scheme of the dodecahedron.
But then (the scheme graph) R is a bipartite double cover of the dodecahedron,
1
and hence it must be the bipartite double graph of the dodecahedron. Moreover,
the association scheme is therefore the bipartite double scheme of the association
scheme of the dodecahedron. (cid:3)
3.2. TheM¨obius-Kantorgraph. TheM¨obius-Kantorgraphistheuniquedouble
cover of the cube without 4-cycles [5, p. 267]. It is isomorphic to the generalized
Petersen graph GP(8,3) and has spectrum 31,√34,13, 13, √34, 31 . It is 2-
{ − − − }
arc-transitive and also known as the Foster graph F016A [22]. It generates an
association scheme with scheme graph having relation-distribution diagram as in