on the weak distance-regularity of moore-type digraphs
TRANSCRIPT
On the Weak Distance-regularity of Moore-type Digraphs∗
F. Comellas†, M. A. Fiol†, J. Gimbert‡, and M. Mitjana§
14th February 2005
Running Head:weak distance-regularity moore digraphs
Mailing Address:M.A. FiolDepartament de Matematica Aplicada IVUniversitat Politecnica de CatalunyaJordi Girona 1-3Modul C3, Campus Nord08034 Barcelona, SPAIN.
Email:[email protected]
∗Research supported by the Ministry of Science and Technology, Spain, and the European Regional Devel-opment Fund (ERDF) under project TIC-2001-2171.†Departament de Matematica Aplicada IV, Universitat Politecnica de Catalunya, Jordi Girona 1-3, Modul
C3, Campus Nord, 08034 Barcelona, Spain.‡Departament de Matematica, Universitat de Lleida, Jaume II 69, 25005 Lleida, Spain.§Departament de Matematica Aplicada I, Universitat Politecnica de Catalunya, Gregorio Maranon 44,
08028 Barcelona, Spain.
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Abstract
We prove that Moore digraphs, and some other classes of extremal digraphs, areweakly distance-regular in the sense that there is an invariance of the number of walksbetween vertices at a given distance. As weakly distance-regular digraphs, we then com-pute their complete spectrum from a ‘small’ intersection matrix. This is a very useful toolfor deriving some results about their existence and/or their structural properties. For in-stance, we present here an alternative and unified proof of the existence results on Mooredigraphs, Moore bipartite digraphs and, more generally, Moore generalized p-cycles. Inaddition, we show that the line digraph structure appears as a characteristic property ofany Moore generalized p-cycle of diameter D ≥ 2p.
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1 Introduction
Let us first introduce some basic notation (see [11]) and discuss the relevant background.Let Γ = (V,E) be a (strongly) connected digraph with order N := |V | and diameter D.We will denote by dist(u, v) the distance from vertex u to vertex v. Notice that dist(u, v)may not be equal to dist(v, u) since we are considering directed graphs. A graph will beidentified with the symmetric digraph obtained by replacing each undirected edge by a pairof opposite arcs, and viceversa. For any fixed integer 0 ≤ k ≤ D, we will denote by Γ+
k (v)(respectively, Γ−k (v)) the set of vertices at distance k from v (respectively, the set of verticesfrom which v is at distance k). Thus, the out-degree and in-degree of v are δ+(v) := |Γ+
1 (v)|and δ−(v) := |Γ−1 (v)|; and the digraph Γ is (∆-)regular if δ+(v) = δ−(v) = ∆ for every v ∈ V .Notice that if Γ is a symmetric digraph, then Γ+
k (v) = Γ−k (v) (:= Γk(v), for short). We recallthat the adjacency matrix A = (auv) of the digraph Γ is the N ×N matrix indexed by thevertices of Γ, with entries auv = 1 if u is adjacent to v, and auv = 0 otherwise. The distance-kmatrix Ak of a digraph Γ with diameter D, where 0 ≤ k ≤ D, is defined by
(Ak)uv :={
1 if dist(u, v) = k,0 otherwise.
(1)
In particular, A0 = I and A1 = A (we assume that Γ contains neither loops nor multipleedges).
Spectrum
The spectrum of a digraph Γ, denoted by sp Γ, consists of the eigenvalues of its adjacencymatrix A together with their (algebraic) multiplicities,
sp Γ := spA =(
λ0 λ1 . . . λdm(λ0) m(λ1) . . . m(λd)
).
The knowledge of the spectrum of a (di)graph is relevant for estimating some of its structuralproperties, which provide information on the topological and communication properties ofthe corresponding network. Among these properties we have, for instance, edge-expansionand node-expansion, bisection width, diameter, maximum cut, connectivity, and partitions(see [12, 36]). Besides, the estimation of connectivity and expansion parameters of graphsare, in general, very hard to obtain by other methods. Efficient graph partition algorithmshave been constructed based on the eigenvalues and eigenvectors (see [28, 29]). The spectrumof a graph has also been considered to provide load balancing algorithms (see [6]).
Distance-regularity
Distance-regular graphs were introduced by Biggs [8], by changing a symmetry-type require-ment, that of distance-transitivity, by a regularity-type condition concerning the cardinalityof some vertex subsets. To be more precise, recall that a graph Γ with diameter D is distance-regular if, for any pair of vertices u, v ∈ V such that dist(u, v) = k, 0 ≤ k ≤ D, the numbers
pki1(u, v) := |Γi(u) ∩ Γ1(v)| (2)
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do not depend on the chosen vertices u and v, but only on their distance k; in which case wejust write pki1(u, v) = pki1 for some constants pki1 called the intersection numbers (note thatpki1 = 0 when |i− k| > 1).
Damerell [17] adopted the strongest definition of distance regularity for digraphs. Thus,he defined a digraph Γ with diameter D to be (strongly) distance-regular if, for any pair ofvertices u, v ∈ V such that dist(u, v) = k, 0 ≤ k ≤ D, the numbers
ski1(u, v) := |Γ+i (u) ∩ Γ+
1 (v)| (3)
do not depend on the chosen vertices u and v, but only on their distance k; in which casethey are denoted by ski1 (note that ski1 = 0 when i > k + 1). Damerell [17] proved that everydistance-regular digraph Γ with girth g, say, is stable; that is,
dist(u, v) + dist(v, u) = g, if 0 < dist(u, v) < g. (4)
Thus, a stable digraph of girth two is a symmetric digraph and, consequently, can be identifiedwith a graph. Therefore, the case g = 2 corresponds to the distance-regular graphs. Damerellalso proved that the set of distance-matrices {Ak}Dk=0 of a distance-regular digraph Γ define acommutative association scheme on the vertices of Γ (see [17]). Then, using Higman’s resultson such configurations (see [30]), he showed that the standard theorems about distance-regular graphs (see e.g. [7, 26]) hold also for digraphs.
If we change Γ+1 (v) by Γ−1 (v) in the definition of distance-regularity (3), we get some new
parameters pki1. But now, as we showed in [14], the stability property (4) does not necessarilyhold, and a class of digraphs with less structure appears. In that paper, we considered suchdigraphs, referred to as ‘weakly distance-regular’, and studied some of their properties. Inparticular, we proved that every distance-regular digraph is also weakly distance-regular, sojustifying the name, and the defining parameters, ski1 and pki1, satisfy that Nkp
ki1 = Nis
ik1,
where Nk denotes the number of (ordered) vertex pairs u, v such that dist(u, v) = k. Further-more, we showed that distance-regular digraphs are precisely those weakly distance-regulardigraphs which are stable.
Weakly distance-regularity
The concept of a weakly distance-regular digraph was introduced by the authors [14], throughthe invariance of the number of walks between vertices at a given distance. (This approachhas already been used to characterize distance-regular graphs, see e.g. [38].) Formally, adigraph Γ of diameter D is weakly distance-regular if, for each non-negative integer l ≤ D,the number a(l)
uv of walks of length l from vertex u to vertex v only depends on their distancedist(u, v) = k, for any l = 0, 1, . . . , D.
From the definition of Γ as a weakly distance-regular digraph, and taking into accountthat a(l)
uv = (Al)uv, it follows that each matrix power Al, 0 ≤ l ≤ D, can be expressedas a linear combination of the distance matrices A0,A1, . . . ,AD. As a consequence, thesematrices belong to the adjacency algebra of Γ, which is defined by
A(Γ) := {p(A) : p ∈ C[x]}.
This leads to some characterizations of weakly distance-regular digraphs (see [14]) which areanalogous to those applying for distance-regular graphs (for a survey of the latter, see e.g.[19]).
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Theorem 1.1. Let Γ be a connected digraph of diameter D. Then, Γ is a weakly distance-regular digraph if and only if any of the following statements hold:
(a) The distance matrix Ak is a polynomial of degree k in the adjacency matrix A; that is,Ak = pk(A), for each k = 0, 1, . . . , D, where pk ∈ Q[x].
(b) The set of distance matrices {Ak}Dk=0 is a basis of the adjacency algebra A(Γ).
(c) For any two vertices u, v ∈ V (Γ) at distance dist(u, v) = k, the numbers
pkij(u, v) := |Γ+i (u) ∩ Γ−j (v)| (5)
do not depend on the vertices u and v, but only on their distance k; in which case theyare denoted by pkij (note that pkij = 0 when k > i+ j).
Some examples of weakly distance-regular digraphs, such as the butterfly and the cycleprefix digraph which are interesting for their applications, are analyzed in [14].
If Γ is a weakly distance-regular digraph, then∑D
k=0 pk(A) =∑D
k=0Ak = J , where Jdenotes the all-1 matrix. As a consequence, Γ is a regular digraph of degree ∆, say. Thepolynomials pk such that Ak = pk(A), 0 ≤ k ≤ D, are referred to as the distance polynomialsof Γ and the numbers pkij = |Γ+
i (u) ∩ Γ−j (v)|, where dist(u, v) = k, are also known as theintersection numbers of Γ. Since
AiAj =min{i+j,D}∑
k=0
pkijAk, (6)
the distance polynomials of Γ satisfy the recurrence relation (take j = 1)
pix =i+1∑k=0
pki1pk (0 ≤ i ≤ D − 1). (7)
As happens in the case of distance-regular graphs, the spectrum of a weakly distance-regular digraph can be retrieved from the information given by some different ‘small’ matricescharacterizing it, such as the intersection matrix or the multiplicity matrix (see [14]). Theintersection or recurrence matrix of a weakly distance-regular digraph Γ is the (D+1)×(D+1)matrix B whose entries are the intersection numbers pki1; that is (B)ik = pki1. In particular,all column sums of B equal to the degree ∆ of Γ since
∑Di=0 p
ki1(u, v) = |Γ−1 (v)|. We point out
that this definition of B coincides with the given in [14], (B)ik = pk1i, since A1Ai = AiA1
(Ai ∈ A(Γ), i = 0, 1, . . . , D, and A(Γ) is a commutative algebra). We also notice that, whilethe intersection matrix B of a weakly distance-regular digraph is a lower Hessenberg matrix(pki1 = 0 if k > i + 1), in the undirected case B is tridiagonal. The following result, provedin [14], shows how to compute the spectrum of a weakly distance-regular digraph given itsintersection matrix. We shall say that a vector u is standard if its first component is (u)0 = 1.
Proposition 1.2. Let Γ be a weakly distance-regular digraph of degree ∆, diameter D, or-der N , and distance polynomials {pk}Dk=0. Let A and B be, respectively, its adjacency andintersection matrices. Then, the following statements hold:
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(i) The minimum polynomials of A and B coincide with the characteristic polynomial ofB which is
det(xI −B) = 1αDD
(x−∆)∑D
k=0 pk,
where αDD is the leading coefficient of pD.
(ii) If λi is an eigenvalue of B, then vi = (p0(λi), p1(λi), . . . , pD(λi))> is a (right) standardeigenvector of B.
(iii) If each eigenvalue λi of B is simple, then λi has unique left and right standard eigen-vectors, u>i and vi, and the multiplicity of λi, as an eigenvalue of Γ, is
m(λi) =N
u>i vi(0 ≤ i ≤ D). (8)
We point out that the adjacency matrix A of a distance-regular digraph Γ is normal ,that is, AA∗ = A∗A, where A∗ denotes the transpose of its conjugate, since A> = Ag−1 =pg−1(A). This is equivalent to say that A diagonalizes by means of a unitary matrix (see e.g.[34]).
Moore graphs and digraphs
It is well known that the order N of a connected graph (respectively, digraph) Γ with maxi-mum degree (respectively, out-degree) ∆ and diameter D cannot be greater than the corre-sponding Moore bound ,
N ≤{MG(∆, D) := 1 + ∆ + ∆(∆− 1) + · · ·+ ∆(∆− 1)D−1 if Γ is a graph,MD(∆, D) := 1 + ∆ + ∆2 + · · ·+ ∆D if Γ is a digraph,
since the number of vertices of Γ at distance k ≤ D from a given vertex is at most ∆(∆−1)k−1
or ∆k, depending whether Γ is undirected or not, respectively. In the extremal case, N =MG(∆, D) or N = MD(∆, D), we have a Moore graph or a Moore digraph, respectively.Notice that in a Moore graph, if edges joining vertices at distance D from a distinguishedvertex are deleted, the residual graph is a ‘hierarchy’ and the same ‘structure’ results fromdistinguishing any vertex (see [33]). This makes a Moore graph to be distance-regular in theintuitive sense that, when we ‘hang’ the graph from a given vertex, we observe that vertices inthe same ‘layer’, or distance from the ‘root’, are ‘neighbourhood-indistinguishable’ from eachother, and the whole ‘configuration’ does not depend on the chosen vertex. More precisely, aMoore graph Γ is distance-regular with intersection matrix
B =
0 1∆ 0 1
∆− 1 0 .∆− 1 . .
. . 1. 0 1
∆− 1 ∆− 1
,
(see [7]). The question of the existence of (nontrivial) Moore graphs of degree ∆ > 2 anddiameter D = 2, 3 was ‘almost’ settled by Hoffman and Singleton [33], by showing that there
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is only one such graph for D = 2 and ∆ = 3, 7 and no other Moore graphs with the ‘possible’exception of a graph with degree ∆ = 57 and diameter D = 2. To prove the nonexistenceresult they used spectral techniques. More precisely, they worked with some graph invariantsgiven in terms of the eigenvalues of a Moore graph whose adjacency matrix A must satisfythe equation
FD(A) = J , where Fk = xFk−1 − (∆− 1)Fk−2, if k ≥ 2, and F0 = 1, F1 = x+ 1. (9)
The nonexistence of Moore graphs of diameter D > 3 (and degree ∆ > 2) was inde-pendently proved by Bannai and Ito [1] and Damerell [16], who derived the spectrum of aMoore graph Γ working with its intersection matrix B, in the light of the theory of distance-regularity.
Nevertheless, the classification of Moore digraphs, first given by Plesnık and Znam [37],was done by directly working with the corresponding matrix equation, which is
I +A+ · · ·+AD = J (10)
(see also [10, 15] for shorter proofs).So, a different approach has been used to solve the Moore graph existence problem for the
undirected and directed cases. One reason could be that Eq. (10) is much simpler than (9),since in the directed case we do not have to care about discounting walks that use a same edgetwice (forward and backward). Another reason may be that the theory of distance-regulardigraphs cannot be applied to this problem since nontrivial Moore digraphs are not (strongly)distance-regular.
We point out that, from (9) and (10), FD [respectively, 1 + x+ · · ·+ xD] is the Hoffmanpolynomial (see [31, 32]) of a Moore graph Γ of diameter D [respectively, Moore digraph],since it is the least degree polynomial satisfying H(A) = J . Moreover, FD coincides withthe sum of the distance polynomials of Γ, which can also be defined in the directed case aswe will see.
In this paper, we prove that Moore digraphs, and some other classes of extremal digraphs,are weakly distance-regular. This allow us to give an alternative and unified proof of theexistence of Moore digraphs, Moore bipartite digraphs and Moore generalized p-cycles (seeSections 2 and 3). Besides, we consider a certain subclass of ‘almost’ Moore digraphs whichare also weakly distance-regular (see Section 4). As another spectral application, we showthat the line digraph structure appears as a characteristic property of several Moore-typedigraphs (see Sections 3 and 4).
2 Moore digraphs revisited
In this section, we show that a Moore digraph of diameter D is the simplest example of aweakly distance-regular digraph, in the sense that its distance polynomials constitute thecanonical basis of CD[x].
Proposition 2.1. A Moore digraph Γ with maximum out-degree ∆ and diameter D is aweakly distance-regular digraph with distance polynomials pk = xk, k = 0, 1, . . . , D. Moreover,Γ is distance-regular if and only if ∆ = 1 or D = 1, which corresponds to the cycle digraphCD+1 and to the complete digraph K∆+1, respectively.
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Proof. A Moore digraph Γ of diameter D becomes a rooted tree at vertex u if the arcs incidentfrom vertices in Γ+
D(u) are removed. That is, for each vertex v ∈ V (Γ) there is exactly oneu → v walk of length k ≤ D (k = dist(u, v)). This means that the distance-k matrix Ak
of Γ is just Ak = Ak, where A is the adjacency matrix of Γ. Therefore, Γ is a weaklydistance-regular digraph with distance polynomials pk = xk, k = 0, 1, . . . , D. In particular,Γ is ∆-regular.
Now, let us assume that a Moore digraph Γ of degree ∆ and diameter D is distance-regular. Notice first that the girth of Γ is g = D + 1. Then, from the stability property (4),AD = A>1 and, consequently, AD = AD = A>. Therefore, ADJ = ∆DJ = A>J = ∆J ,since Γ is ∆-regular. Hence, ∆D = ∆, which implies that ∆ = 1 or D = 1. In such trivialcases, Γ is clearly distance-regular since it is either the cycle digraph of order D + 1 or thecomplete digraph of order ∆ + 1, respectively.
Now we compute the spectrum of a Moore digraph in the light of the theory of weakdistance-regularity.
Theorem 2.2. The spectrum of a Moore digraph Γ of degree ∆ and diameter D is
sp Γ =(
∆ ξ ξ2 . . . ξD
1 m(ξ) m(ξ2) . . . m(ξD)
),
where ξ is a primitive (D + 1)th root of unity, say ξ = e2πi/(D+1), and
m(ξk) = N∆
D + 1ξk − 1ξk −∆
, k = 1, 2, . . . , D,
where N = 1 + ∆ + · · ·+ ∆D.
Proof. Since a Moore digraph Γ of degree ∆ and diameter D is weakly distance-regular withdistance polynomials pk = xk, k = 0, 1, . . . , D, its intersection matrix B = (pki1) is
B =
0 10 0 1. . . .. . . . 10 0 0 . 0 1∆ ∆− 1 ∆− 1 . ∆− 1 ∆− 1
,
since pix =∑i+1
k=0 pki1pk = pi+1, which means that pi+1
i1 = 1 and pki1 = 0, otherwise, for eachi = 0, 1, . . . , D − 1. Taking into account that pkD1 = ∆−
∑D−1i=0 pki1, for each k = 0, 1, . . . , D,
the last row of B is derived. Then, from Proposition 1.2, the minimum polynomial of theadjacency matrix A of Γ coincides with the characteristic polynomial of B, which is
det(xI −B) = (x−∆)D∑k=0
pk = (x−∆)(1 + x+ · · ·+ xD) = (x−∆)xD+1 − 1x− 1
= (x−∆)D∏k=1
(x− ξk),
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where ξ is a primitive (D + 1)th root of unity, say ξ = e2πi/(D+1). Therefore, since all theeigenvalues of B are simple, we can apply (8) to obtain their multiplicities as eigenvalues ofΓ. Thus, the multiplicity m(ξ) of ξ can be computed as
m(ξ) =N
u>ξ vξ,
where N is the order of Γ and vξ = (1, ξ, . . . , ξD)>, u>ξ = (1, x1, . . . , xD) represent the onlyright and left standard eigenvectors of B corresponding to ξ, respectively. The conditionu>ξ B = ξu>ξ gives
xk =1ξxk−1 +
∆− 1∆
, k = 1, 2, . . . , D and x0 := 1,
whence
xk =1ξk
+∆− 1
∆ξk − 1ξ − 1
1ξk−1
, k = 0, 1, . . . , D.
Then, taking into account that ξD+1 = 1, we have
u>ξ vξ =D∑k=0
xkξk =
D∑k=0
(1 +
∆− 1∆
ξ
ξ − 1(ξk − 1)
)=D + 1
∆ξ −∆ξ − 1
. (11)
Since in the previous computation we have only used that ξ is a (D + 1)th root of unity,ξ 6= 1, we can extend (11) to any other root ξk, k = 1, 2, . . . , D. Hence, the multiplicity of ξk
is
m(ξk) =N
u>ξkvξk
= N∆
D + 1ξk − 1ξk −∆
.
The multiplicity of the degree ∆ of Γ is 1 since JB = ∆J and, consequently, u>∆ =(1, 1, . . . , 1) and u>∆v∆ = N . Such a multiplicity can also be derived from Perron-FrobeniusTheorem (see [34]).
Thus, for instance, the spectrum of a Moore digraph Γ of degree ∆ = 1 and diameter Dis
sp Γ =(
1 ξ · · · ξD
1 1 · · · 1
),
which corresponds to the cycle digraph of order D + 1. In the case D = 1, the spectrum ofΓ is
sp Γ =(
∆ −11 ∆
),
since ξ = −1 and m(ξ) = ∆, which corresponds to the complete digraph of order ∆ + 1.
Corollary 2.3 ([37, 10, 15]). Moore digraphs of degree ∆ > 1 and diameter D > 1 do notexist.
Proof. Let Γ be a Moore digraph of degree ∆ and diameter D. From Theorem 2.2, we knowthat ξ = e2πi/(D+1) is an eigenvalue of Γ with multiplicity m(ξ) = N ∆
D+1ξ−1ξ−∆ . Therefore,
ξ−1ξ−∆ is rational, since m(ξ) is a positive integer, which implies that ξ = e2πi/(D+1) is rational,if ∆ > 1. But, this is so if and only if D = 1.
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3 Moore generalized p-cycles
A generalized p-cycle is a digraph Γ such that its set of vertices can be partitioned in p ≥ 1parts, V (Γ) =
⋃α∈Zp Vα, in such a way that the vertices in the partite set Vα are only adjacent
to vertices in Vα+1, where the sum is in Zp. Notice that bipartite digraphs are generalizedp-cycles with p = 2 and all digraphs are included in the case p = 1.
The diameter D of a strongly connected generalized p-cycle Γ is the minimum integer suchthat for any vertex v, all vertices of one of the partite sets of V (Γ) are at distance at mostD − (p − 1) from v. As a consequence, and taking also into account that the lengths of allwalks from one partite set to another are congruent modulo p, the order N of a generalizedp-cycle with maximum out-degree ∆ and diameter D is bounded above by the Moore-likebound
N ≤MGC(∆, D) := p∆r(1 + ∆p + · · ·+ ∆pq), (12)
where D − (p − 1) = pq + r and 0 ≤ r ≤ p − 1; see [27]. Notice that MGC(∆, D) coincideswith the (general) Moore bound MD(∆, D), if p = 1, and in the case p = 2 is equal to theknown Moore-like bound for bipartite digraphs (see [21])
MBD(∆, D) :={
2(1 + ∆2 + · · ·+ ∆2q) if D = 2q + 1,2∆(1 + ∆2 + · · ·+ ∆2q−2) if D = 2q.
A generalized p-cycle Γ = (V,E), V =⋃α∈Zp Vα, with maximum out-degree ∆, diameter
D and order N = MGC(∆, D) is called a Moore generalized p-cycle (in the case p = 2 it isreferred to as a Moore bipartite digraph). In fact, such a digraph must be out-regular without-degree ∆; that is, δ+(v) = ∆ for any v ∈ V (Γ). The problem of the Moore generalizedp-cycles existence was solved by Gomez et al. [27], as a natural extension of the techniquesused by Fiol and Yebra [21] to deal with the bipartite case. Thus, it is known that the boundMGC(∆, D) is only attained when p ≤ D ≤ 3p− 2, if ∆ > 1.
In this section, we show that a Moore generalized p-cycle is also weakly distance-regular.Then, using the theory of weak distance-regularity, we present another proof of the nonexis-tence of Moore generalized p-cycles of degree ∆ > 1 and diameter D > 3p − 2. Besides, wediscuss when a Moore generalized p-cycle is a line digraph.
Proposition 3.1. A Moore generalized p-cycle Γ with out-degree ∆ and diameter D, whereD−(p−1) = pq+r and 0 ≤ r ≤ p−1, is weakly distance-regular and its distance polynomialsare
pk = xk, k = 0, 1, . . . , D − (p− 1),
ppq+r+j =1
∆j (xpq+r+j − (∆j − 1)(xr+j + xp+r+j + · · ·+ xp(q−1)+r+j)) if r + j ≤ p− 1,
1∆j (xpq+r+j − (∆j − 1)(xr+j + xp+r+j + · · ·+ xp(q−1)+r+j))− xr+j−p if r + j ≥ p,
for each j = 1, 2, . . . , p − 1. Moreover, Γ is distance-regular if and only if ∆ = 1 or D = p,which corresponds to the cycle digraph Cp(q+1) and to the complete generalized p-cycle KGC∆,...,∆,respectively.
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Proof. Let Γ = (V0 ∪ · · · ∪ Vp−1, E) be a Moore generalized p-cycle with out-degree ∆ anddiameter D, where D − (p − 1) = pq + r and 0 ≤ r ≤ p − 1, and let A be its adjacencymatrix. From its definition as a Moore-type digraph, for any vertex u ∈ Vα, all vertices ofVα+r are reached from u in exactly one way in a number of steps (k) at most D − (p − 1)(k ≡ r mod p). This means that, for any vertex v ∈ V (Γ) at distance ≤ D− (p− 1) from u,the number of u→ v walks of lenght k, k = 0, 1, . . . , D− (p− 1), is a(k)
uv = 1, if dist(u, v) = k,and a
(k)uv = 0, otherwise. So, the distance k-matrix of Γ is
Ak = Ak, k = 0, 1, . . . , D − (p− 1). (13)
Then, taking into account that Γ has out-degree ∆,
a(pq+r+j)uv =
∆j if dist(u, v) = pq + r + j or dist(u, v) = r + j − p,∆j − 1 if dist(u, v) ≡ r + j mod p and dist(u, v) 6= pq + r + j, r + j − p,0 otherwise,
for each j = 1, 2, . . . , p− 1. In matrix terms,
Apq+r+j ={∆jApq+r+j + (∆j − 1)(Ar+j +Ap+r+j + · · ·+Ap(q−1)+r+j) if r + j ≤ p− 1,∆j(Apq+r+j +Ar+j−p) + (∆j − 1)(Ar+j +Ap+r+j + · · ·+Ap(q−1)+r+j) if r + j ≥ p.
(14)Hence, Γ is weakly distance-regular and, from (13) and (14), its distance polynomials areeasily derived. In particular, Γ is ∆-regular.
Now, let us assume that Γ is a distance-regular digraph with degree ∆ > 1. Let us take avertex u ∈ V0 and let v and w be two distinct out-neighbours of u; that is, v, w ∈ Γ+
1 (u) ⊆ V1.Taking into account that the adjacency matrix A of Γ is normal,
|Γ+1 (v) ∩ Γ+
1 (w)| = (AA>)vw = (A>A)vw = |Γ−1 (v) ∩ Γ−1 (w)| ≥ 1,
which means that there is a vertex z ∈ V2 adjacent from v and w. As a consequence, Γ hastwo u → z walks of length two, which is impossible unless D − (p − 1) < 2; that is, D = p.Hence, Γ cannot be distance-regular unless ∆ = 1 (D = pq+ p− 1) or D = p (∆ > 1), whichcorrespond to the cycle digraph Cp(q+1) and to the complete generalized p-cycle KGC∆,...,∆,respectively.
As we will see, the intersection matrix of a Moore generalized p-cycle of diameter D hasthe eigenvalue 0 that is not simple, if D − (p − 1) 6≡ 0, 1 mod p. To cope with this case weneed to extend the applicability of (8).
Proposition 3.2. Let Γ be a weakly distance-regular digraph of diameter D, order N and dis-tance polynomials {pk}Dk=0. Let B be the intersection matrix of Γ and let λi be an eigenvalueof B. If λi has a left standard eigenvector u>i then the multiplicity of λi, as an eigenvalueof Γ, is
m(λi) =N
u>i vi,
where vi = (p0(λi), p1(λi), . . . , pD(λi))>.
11
Proof. Let {λj}dj=0 be the distinct eigenvalues of B, where 0 ≤ d ≤ D, with algebraicmultiplicities {m(λj)}dj=0 as eigenvalues of Γ. Let {Ak}Dk=0 be the distance matrices of Γ andlet A be the adjacency matrix of Γ. Then,
trAk = tr pk(A) =d∑j=0
m(λj)pk(λj) = Nδk0, (0 ≤ k ≤ D).
Such a linear system can be written in matrix terms as follows
Pm = Ne0,
where e0 = (1, 0, . . . , 0)>, m = (m(λ0),m(λ1), . . . ,m(λd))> and P denotes the D× d matrixwhose (k, j)-element is pk(λj), 0 ≤ k ≤ D and 0 ≤ j ≤ d. We know that the jth column of P ,vj = (p0(λj), p1(λj), . . . , pD(λj))>, is a right standard eigenvector of B corresponding to theeigenvalue λj . Let U be a matrix whose jth row, u>j , is a left eigenvector of B correspondingto the eigenvalue λj , 0 ≤ j ≤ d. We assume that the first component of uj is 1, if j = i.Since UPm = NUe0, we have
d∑j=0
(u>i vj)m(λj) = N.
Notice that u>i vj = 0, if i 6= j, since λiu>i vj = u>i Bvj = λju>i vj . Therefore,
(u>i vi)m(λi) = N,
which implies (8).
In [13] we determined the spectrum of a wrapped butterfly digraph by using the submatrixP ev = (pk(λj))0≤k,j≤d.
Theorem 3.3. The spectrum of a Moore generalized p-cycle Γ of degree ∆ and diameter D,where D − (p− 1) = pq + r and 0 ≤ r ≤ p− 1, is
sp Γ =(
∆ζj ξζj ξ2ζj . . . ξqζj 01 m(ξ) m(ξ2) . . . m(ξq) N
∆r (∆r − 1)
), j = 0, 1, . . . , p− 1, (15)
where ζ = e2πi/p and ξ is a primitive (D + 1− r)th root of unity, say ξ = e2πi/(p(q+1)), and
m(ξk) = N∆p−r
D + 1− rξpk − 1ξpk −∆p
, k = 1, 2, . . . , q,
where N = p∆r(1 + ∆p + · · ·+ ∆pq).Moreover, the minimum polynomial of Γ is
mΓ = xr(xp −∆p)xD+1−r − 1xp − 1
. (16)
In particular, λ = 0 is an eigenvalue of Γ unless r = 0.
12
Proof. From Proposition 1.2 and using the distance polynomials of Γ, {pk}Dk=0, given inProposition 3.1, the minimum polynomial of Γ is
mΓ = ∆p−1(x−∆)∑D
k=0 pk = xr(xp −∆p)xD+1−r−1xp−1
= xrp−1∏j=0
(x− ζj∆)q∏
k=1
p−1∏j=0
(x− ζjξk),
where ζ is a primitive pth root of unity and ξ is a primitive (D + 1 − r)th root of unity.Notice that, since Γ is a connected generalized p-cycle, the whole spectrum of Γ, regarded asa system of points in the complex plane, goes over into itself under a rotation of the planeby the angle 2π/p (Perron-Frobenius Theorem; see [34]).
Besides, it can be checked that the non-null coefficients of the intersection matrix B =(pki1) of Γ are
pki1 =
1 if k = i+ 1 and i ≤ D − p,∆− 1 if k ≡ i+ 1 mod p and 0 < k < i+ 1 and D − (p− 1) ≤ i ≤ D,∆ if k ≡ i+ 1 mod p and k = 0, i+ 1 and D − (p− 1) ≤ i ≤ D.
The standard right eigenvector of B corresponding to ξ is vξ = (p0(ξ), p1(ξ), . . . , pD(ξ))>,where
pk(ξ) ={ξk if k ≤ D − r,0 if D − r + 1 ≤ k ≤ D (r ≥ 1).
It turns out that B has also a (unique) standard left eigenvector corresponding to ξ, u>ξ =(1, x1, . . . , xD), where
xk = 1ξ (xk−1 + (∆− 1)xD−j)
= 1ξ
(xk−1 + (∆− 1)
(ξ∆
)j+1−r), where j ≡ r − k mod p and 0 ≤ j ≤ p− 1,
whence,xk = 1
ξk+ (∆r − 1) ξk−1
ξk(ξp−1)+ ∆p−r−1
∆p−rξk−1
ξk−p(ξp−1)
= ∆r
ξk+ ∆p−1
∆p−rξk−p−1
ξk−p(ξp−1)+ ∆p−r−1
∆p−r , if k ≡ 0 mod p,
and
xk =
1ξjxk−j + 1
ξj(∆j − 1) if k ≡ j mod p and 1 ≤ j ≤ r,
1ξjxk−j + 1
ξj(∆j − 1) + 1
ξj(∆j−r − 1) ξ
p−∆p
∆p−r if k ≡ j mod p and r + 1 ≤ j ≤ p− 1.
Taking into account that ξp(q+1) = 1,
u>ξ vξ =D−r∑k=0
xkξk =
p−1∑j=0
q∑l=0
xpl+jξpl+j
= p
q∑l=0
xplξpl =
D + 1− r∆p−r
ξp −∆p
ξp − 1.
13
So, applying (8),
m(ξ) =N
u>ξ vξ= N
∆p−r
D + 1− rξp − 1ξp −∆p
.
Such a formula can be extended to any other eigenvalue ζjξk, k = 1, 2, . . . , q. It only remainsto compute the multiplicity of 0 as an eigenvalue of Γ, when 1 ≤ r ≤ p − 1. It can beseen that the subspace of left eigenvectors of B corresponding to 0 is generated by u>0 =(x0, x1, . . . , xD), whose non-null components are xj = 1, j ≡ r − 1 mod p and j < D, andxD = 1/(1−∆). So, only when r = 1 we get a standard vector u0, in which case
m(0) =N
u>0 v0=
N
∆/(∆− 1)= N − N
∆,
since v0 = (1, 0, . . . , 0,−1)>. If 2 ≤ r ≤ p− 1, then
m(0) = N − p− pq∑
k=1
m(ξk) = N − p− (N
∆r− p) = N − N
∆r.
Corollary 3.4 ([27]). Moore generalized p-cycles of degree ∆ > 1 and diameter D do notexist unless p ≤ D ≤ 3p− 2.
Proof. Let D−(p−1) = pq+r, where 0 ≤ r ≤ p−1. From the multiplicity of ξ = e2πi/(D+1−r)
as an eigenvalue of a Moore generalized p-cycle of degree ∆ and diameter D, given by Theorem3.3, (ξp − 1)/(ξp −∆p) must be rational. So, if ∆ > 1, then ξp = e2πi/(q+1) is rational and,consequently, q = 0, 1; that is, p−1 ≤ D ≤ 3p−2. Notice that the case D = p−1 correspondsto the multidigraph C∆
p obtained by replacing each arc of the cycle digraph Cp by ∆ > 1multiple arcs.
Corollary 3.5 ([21]). Moore bipartite digraphs of degree ∆ > 1 and diameter D do not existunless 2 ≤ D ≤ 4.
We remark that the enumeration of Moore digraphs can also be derived from Corollary3.4, taking p = 1. Nevertheless, we have decided to keep their study apart since it providesa simpler understanding of the techniques used.
Next, we study when Moore generalized p-cycles are line digraphs. First, notice that,from Theorem 3.3, the spectrum of a Moore generalized p-cycle Γ of degree ∆ and diameterD, where D − (p − 1) = pq + r and 1 ≤ r ≤ p − 1, is the same as the spectrum of a Mooregeneralized p-cycle Γ′ with equal degree and diameter D − r, apart from the multiplicity ofthe eigenvalue 0. Thus, sp Γ = spLrΓ′, where LrΓ′ is the rth iterated line digraph of Γ′. Inthe bipartite case (p = 2), Fiol et al. [20] proved that the line digraph structure is inherentto any Moore bipartite digraph Γ of diameter D = 4. Here, we extend such a structuralproperty, when p ≥ 2 and 2p ≤ D ≤ 3p−2, using the following result that somehow describesthe “hereditary character” of the regular line digraph structure (see [25]):
Proposition 3.6. Let Γ be a regular digraph of degree ∆ and order N and let A be itsadjacency matrix. If there exists an integer r ≥ 1 satisfying the following conditions:
(i) Ar+1 is a (0, 1)-matrix;
14
(ii) rankAr = N/∆r;
then Γ is an rth iterated line digraph.
We point out that, in the regular case, the condition rankAr = N/∆r is equivalent tosay that the associated digraph to the (0, 1)-matrix Ar is a line digraph (see [24]).
Theorem 3.7. Every Moore generalized p-cycle of diameter D = 2p− 1 + r, where 1 ≤ r ≤p− 1, is the rth iterated line digraph of a Moore generalized p-cycle of diameter 2p− 1.
Proof. Let A be the adjacency matrix of a Moore (regular) generalized p-cycle Γ of diameterD = 2p−1 + r, 1 ≤ r ≤ p−1, and degree ∆ (∆ > 1). Since the multiplicity of the factor x inthe minimum polynomial of A is r (see (16)), dim kerAr is equal to the algebraic multiplicityof 0 as an eigenvalue of Γ. So, from (15),
rankAr = N −m(0) = N/∆r.
Besides, Ar+1 is a (0, 1)-matrix, since Ar+1 = Ar+1 (r + 1 ≤ D − (p − 1)). Hence, usingProposition 3.6, we conclude that Γ = Lr Γ′. Since the line digraph operator preserves allcycle lengths, Γ′ must be a p-generalized cycle (every cycle of Γ′ must have length multipleof p). Moreover, we can assume that Γ′ has no isolated vertices, whence Γ′ must be a regulardigraph with degree ∆. Taking into account the relationship between the diameters andorders of a regular digraph and those of its line digraph (see [22]), we derive that Γ′ hasdiameter D − r = 2p− 1 and order
MGC(∆, D)/∆r = p(1 + ∆ + · · ·+ ∆pq) = MGC(∆, D − r).
Hence Γ′ is also a Moore generalized p-cycle.
Corollary 3.8 ([20]). Every Moore bipartite digraph of diameter four is the line digraph ofa Moore bipartite digraph of diameter three.
Notice that the unique Moore generalized p-cycle with diameter D = p is also a linedigraph, KGC∆,...,∆ = LC∆
p . Moreover, it turns out that the enumeration of generalized p-cycles Γ = (V0 ∪ · · · ∪ Vp−1, E) with degree ∆ > 1 and diameter D = p − 1 + r, where2 ≤ r ≤ p− 1, is equivalent to the search of all binary submatrices {Aα,α+1}p−1
α=0 of order ∆r
such that
Aα,α+1J = ∆J andr−1∏j=0
Aα+j,α+j+1 = J , α = 0, 1, . . . , p− 1,
where the coefficients of Aα,α+1 represent the adjacencies from Vα to Vα+1. Thus, if r dividesp, we can construct Aα,α+1 as a circulant matrix with Hall polynomial1
θAα,α+1(x) = 1 + x∆j
+ · · ·+ x(∆−1)∆j,
where α ≡ j mod r and 0 ≤ j ≤ r − 1, providing a Moore generalized p-cycle of diameterD = p− 1 + r, which is not a line digraph.
1In a circulant matrix A each row (excluding the first one) is the previous row moved to the right 1 place,with wraparound. In such a case, A is uniquely determined by its first row (a0, a1, . . . , an−1) that can berepresented by the polynomial
θA(x) :=
n−1∑i=0
aixi,
which is called the Hall polynomial of A (see [18]).
15
4 Almost Moore digraphs
The nonexistence of digraphs with order equal to the Moore bound MD(∆, D), when ∆, D >1, motivated the problem of the existence of almost Moore digraphs, namely digraphs without-degree ∆ > 1, diameter D > 1 and whose order N misses such unattainable bound byjust one vertex; that is N = ∆ + ∆2 + · · ·+ ∆D (see e.g. [5, 3]). Every almost Moore digraphΓ has the property that for each vertex v ∈ V (Γ) there exists only one vertex, denoted byr(v) and called the repeat of v, such that there are exactly two v → r(v) walks of length lessthan or equal to D (one of them must be of length D). If r(v) = v, which means that v iscontained in (exactly) one cycle of length D, v is called a selfrepeat of G. Since Miller et al.[35] proved that such extremal digraphs must be regular, it turns out that the map r, whichassigns to each vertex v ∈ V (Γ) its repeat r(v), is an automorphism of Γ (see [2]).
Now we characterize the subclass of almost Moore digraphs that are weakly distance-regular.
Proposition 4.1. An almost Moore digraph Γ of degree ∆ > 1 and diameter D > 1 isweakly distance-regular if and only if each vertex of Γ is contained in exactly one cycle oflength D. In such a case, the distance polynomials of Γ are pk = xk, k = 0, 1, . . . , D− 1, andpD = xD − 1. Moreover, an almost Moore digraph can never be distance-regular.
Proof. Let A be the adjacency matrix of an almost Moore digraph Γ and P be the per-mutation matrix associated with the map r of repeats of Γ such that its (i, j) entry is 1iff r(i) = j. Let {Ak}Dk=0 be the set of distance matrices of Γ. Then, Ak = Ak, for eachk = 0, 1, . . . , D − 1, and AD = AD − P , since there is a unique u→ v walk of length l ≤ D(l = dist(u, v)) unless v = r(u), in which case there is an extra walk of length D. Therefore,if Γ is weakly distance-regular, then P = AD −AD belongs to the adjacency algebra A(Γ),which has the set of distance matrices of Γ as one of its basis. As a consequence, P is a linearcombination of {Ak}Dk=0, which implies that P = A0 = I, since P is a permutation matrix.Hence, each vertex v ∈ V (Γ) is a selfrepeat (r(v) = v), which means that v is contained inexactly one cycle of length D. In such a case, AD = AD−I; that is, the distance polynomialof Γ of degree D is pD = xD−1. Clearly, the remaining distance polynomials of Γ are pk = xk,for each k = 0, 1, . . . , D − 1.
Since the girth g of an almost Moore digraph Γ with all selfrepeat vertices is g = D, inorder to satisfy the stability property (4) it is necessary that AD−1 = A>1 , which implies that∆D−1 = ∆. As a consequence, Γ cannot be distance-regular when D > 2. In the case D = 2,Γ would be a symmetric digraph (A = A>) and, consequently, each vertex v ∈ V (Γ) wouldbe included in ∆ > 1 cycles of length D = 2, which is impossible.
Theorem 4.2. Let Γ be an almost Moore digraph with degree ∆ > 1, diameter D > 1 andsuch that all its vertices are selfrepeats. Then, the spectrum of Γ is
sp Γ =(
∆ ξ ξ2 . . . ξD−1 01 m(ξ) m(ξ2) . . . m(ξD−1) ∆D − 1
), where ξ = e2πi/D and
m(ξk) =N
D
ξk − 1ξk −∆
, where N = ∆ + ∆2 + · · ·+ ∆D.
Moreover, the minimum polynomial of Γ is
mΓ = x(x−∆)(1 + x+ · · ·+ xD−1).
16
Proof. From the distance polynomials of Γ, pk = xk, if k < D, and pD = xD − 1, theconstruction of the intersection matrix B of Γ is derived,
B =
0 10 0 1. . . .. . . . 11 0 0 . 0 1
∆− 1 ∆− 1 ∆− 1 . ∆− 1 ∆− 1
.
Besides, the minimum polynomial of Γ is
det(xI −B) = (x−∆)D∑k=0
pk = (x−∆)x(1 + x+ · · ·+ xD−1) = (x−∆)xD−1∏k=1
(x− ξk),
where ξ = e2πi/D.In order to obtain the multiplicity m(ξ) of the Dth root of unity ξ, as an eigenvalue of Γ,
we need to compute the left standard eigenvector u>ξ = (1, x1, . . . , xD) of B correspondingto ξ. Since the condition u>ξ B = ξB is equivalent to the recurrence relation
xk =1ξ
(xk−1 + ∆− 1), k = 1, 2, . . . , D and x0 := 1,
it can be checked that
xk =1ξk
+ (∆− 1)ξk − 1ξ − 1
1ξk, k = 0, 1, . . . , D.
Therefore, since the right eigenvector vξ of B corresponding to ξ is vξ = (1, ξ, . . . , ξD−1, 0)>,since pD(ξ) = ξD − 1 = 0,
u>ξ vξ =D−1∑k=0
xkξk =
D−1∑k=0
(1 + (∆− 1)
ξk − 1ξ − 1
)= D
ξ −∆ξ − 1
,
and, consequently,
m(ξ) =N
u>ξ vξ=N
D
ξ − 1ξ −∆
,
where N is the order of Γ. As before, this formula also holds for any other Dth root of unityξk, if 1 ≤ k ≤ D − 1.
The multiplicity of the eigenvalue λ = 0 is derived taking into account that u>0 =(1, 1, . . . , 1, 1/(1−∆)) and v0 = (1, 0, . . . , 0,−1)>, which gives
m(0) =N
u>0 v0=
∆ + ∆2 + · · ·+ ∆D
∆/(∆− 1)= ∆D − 1.
From the knowledge of the spectrum of almost Moore digraphs with all selfrepeat verticeswe derive their classification. We point out that their nonexistence for diameter D > 2 wasfirstly proved by Bosak [9] and, subsequently, Baskoro et al. [4] provided a shorter proof.
17
Corollary 4.3. Almost Moore digraphs with all selfrepeats do not exist unless the diameteris two, in which case there is a unique solution, namely the Kautz digraph.
Proof. Let Γ be an almost Moore digraph of degree ∆ > 1 and diameter D > 1, such that allits vertices are selfrepeats. From Theorem 4.2, the multiplicity of the eigenvalue ξ = e2πi/D
of Γ is m(ξ) = ND
ξ−1ξ−∆ , where N is the order of G. Therefore, ξ = e2πi/D is rational, which
implies that D = 2.Now, we consider the case of diameter D = 2, where the minimum polynomial of Γ is
mΓ = x(x−∆)(x+ 1) and its spectrum is
sp Γ =(
∆ −1 01 ∆ ∆2 − 1
),
since ξ = eπi = −1 and m(ξ) = ∆+∆2
22
1+∆ = ∆. This is the spectrum of the line digraph ofthe complete digraph of order ∆ + 1, LK∆+1, which is isomorphic to the Kautz digraph ofdiameter two and degree ∆ (see [22]). Since the multiplicity of the factor x in mΓ is 1,
rankA = N −m(0) = ∆ + 1 =N
∆.
Hence, Γ is a line digraph; that is, Γ = LK∆+1 (this was independently proved in [24] and[39]).
Furthermore, LK∆+1 is the only almost Moore digraph of diameter D = 2 and degree∆ > 2 (see [23]).
References
[1] E. Bannai and T. Ito, On finite Moore graphs, J. Fac. Sci. Univ. Tokyo 20 (1973),191–208.
[2] E.T. Baskoro, M. Miller and J. Plesnık, On the structure of digraphs with order closeto the Moore bound, Graphs Combin. 14 (1998), 109–119.
[3] E.T. Baskoro, M. Miller, J. Siran and M. Sutton, Complete characterisation of almostMoore digraphs of degree three, J. Graph Theory 48 (2005), no. 2, 112–126.
[4] E.T. Baskoro, M. Miller, J. Plesnık and S. Znam, Digraphs of degree 3 and order closeto the Moore bound, J. Graph Theory 20 (1995), 339–349.
[5] E.T. Baskoro, M. Miller, J. Plesnık and S. Znam, Regular digraphs of diameter 2 andmaximum order, Australas. J. Combin. 9 (1994), 291–306.
[6] S. Bezrukov, R. Elsasser, B. Monien, R. Preis and J.-P. Tillich, New spectral lowerbounds on the bisection width of graphs, Lecture Notes in Comput. Sci. 1928 (2000),23–34.
[7] N. Biggs, Algebraic Graph Theory, Cambridge University Press, Cambridge, 1974; secondedition: 1993.
18
[8] N. Biggs, Intersection matrices for linear graphs, in: Combinatorial Mathematics andits Applications (ed. D.J.A. Welsh), 15–23, Academic Press, London, 1971.
[9] J. Bosak, Directed graphs and matrix equations, Math. Slovaca 28 (1978), 189–202.
[10] W.G. Bridges and S. Toueg, On the impossibility of directed Moore graphs, J. Combin.Theory Ser. B 29 (1980), 339–341.
[11] G. Chartrand and L. Lesniak, Graphs & Digraphs, 3rd ed. Chapman & Hall, London,1996.
[12] F.R.K. Chung, Diameters and eigenvalues, J. Amer. Math. Soc. 2 (1989), 187–196.
[13] F. Comellas, M.A. Fiol, J. Gimbert and M. Mitjana, The spectra of wrapped butterflydigraphs, Networks 42 (2003), no. 1, 15–19.
[14] F. Comellas, M.A. Fiol, J. Gimbert and M. Mitjana, Weakly distance-regular digraphs,J. Combin. Theory Ser. B 90 (2004), no. 2, 233–255.
[15] J.H. Conway and M.J.T. Guy, Message graphs, Ann. Discrete Math. 13 (1982), 61–64.
[16] R.M. Damerell, On Moore graphs, Proc. Camb. Phil. Soc. 74 (1973), 227–236.
[17] R.M. Damerell, Distance-transitive and distance-regular digraphs, J. Combin. TheorySer. B 31 (1981), no. 1, 46–53.
[18] P.J. Davis, Circulant Matrices, Wiley-Interscience, New York, 1979.
[19] M. A. Fiol, Algebraic characterizations of distance-regular graphs, Discrete Math. 246(2002), no. 1, 111–129.
[20] M.A. Fiol, J. Gimbert, J. Gomez, and Y. Wu, On Moore bipartite digraphs, J. GraphTheory 43 (2003), no. 3, 171–187.
[21] M.A. Fiol and J.L.A. Yebra, Dense bipartite digraphs, J. Graph Theory 14 (1990),687–700.
[22] M.A. Fiol, J.L.A. Yebra, and I. Alegre, Line digraphs iterations and the (d, k) digraphproblem, IEEE Trans. Comput. C-32 (1984), 400–403.
[23] J. Gimbert, Enumeration of almost Moore digraphs of diameter two, Discrete Math. 231(2001), no. 1-3, 177–190.
[24] J. Gimbert, On digraphs with unique walks of closed lengths between vertices, Australas.J. Combin. 20 (1999), 77–90.
[25] J. Gimbert and Y. Wu, The underlying line digraph structure of some (0, 1)-matrixequations, Discrete Appl. Math. 116 (2002), 289–296.
[26] C. D. Godsil, Algebraic Combinatorics, Chapman and Hall, New York, 1993.
[27] J. Gomez, C. Padro, and S. Perennes, Large generalized cycles, Discrete Appl. Math. 89(1998), no. 1-3, 107–123.
19
[28] L. Hagen and A. Kahng, Fast spectral methods for ratio cut partitioning and clustering,International Conference on Computer-Aided Design, 1991 (ICCAD-91). IEEE Digestof Technical Papers (1991), 10–13.
[29] L. Hagen and A. Kahng, New spectral methods for ratio cut partitioning and clustering,IEEE Trans. CAD/ICS 11 (1992), no. 9, 1074–1085.
[30] D. G. Higman, “Coherent configurations, I”, Geom. Dedicata 4 (1975), 1–32.
[31] A.J. Hoffman, On the polynomial of a graph, Amer. Math. Monthly 70 (1963), 30–36.
[32] A.J. Hoffman and M.H. McAndrew, The polynomial of a directed graph, Proc. Amer.Math. Soc. 16 (1965), 303–309.
[33] A.J. Hoffman and R.R. Singleton, On Moore graphs with diameter 2 and 3, IBM Res.Develop. 4 (1960), 497–504.
[34] P. Lancaster and M. Tismenetsky, The Theory of Matrices, Academic Press, Orlando,FL; second edition: 1985.
[35] M. Miller, J. Gimbert, J. Siran and Slamin, Almost Moore digraphs are diregular, Dis-crete Math. 218 (2000), 265–270.
[36] B. Mohar, Isoperimetric number of graphs, J. Combin. Theory Ser. B 47 (1989), no. 3,274–291.
[37] J. Plesnık and S. Znam, Strongly geodetic directed graphs, Acta Fac. Rerum. Natur.Univ. Comenian. Math. Publ. 29 (1974), 29–34.
[38] P. Rowlinson, Linear algebra, in: Graph Connections (ed. L.W. Beineke and R.J. Wil-son), Oxford Lecture Ser. Math. Appl., Vol. 5, 86–99, Oxford Univ. Press, New York,1997.
[39] Y. Wu and Q. Li, On the matrix equation Al + Al+k = Jn, Linear Algebra Appl. 277(1998), 41–48.
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