Discrete Mathematics 310 (2010) 1812–1819 Contents lists available at ScienceDirect Discrete Mathematics journal homepage: www.elsevier.com/locate/disc On graphs with complete multipartite µ-graphs Aleksandar Jurišić a , Akihiro Munemasa b,∗ , Yuki Tagami b a Faculty of Computer and Informatic Sciences, University of Ljubljana, and Institute of Mathematics, Physics and Mechanics, Ljubljana, Slovenia b Graduate School of Information Sciences, Tohoku University 6-3-09 Aramaki-Aza-Aoba, Aoba-ku, Sendai 980-8579, Japan article info Article history: Received 1 October 2007 Accepted 14 December 2009 Available online 18 January 2010 Keywords: Distance-regular graph µ-graph Complete multipartite graph Local graph Generalized quadrangle Regular point abstract Jurišić and Koolen proposed to study 1-homogeneous distance-regular graphs, whose µ-graphs (that is, the graphs induced on the common neighbours of two vertices at distance 2) are complete multipartite. Examples include the Johnson graph J (8, 4), the halved 8-cube, the known generalized quadrangle of order (4, 2), an antipodal distance-regular graph constructed by T. Meixner and the Patterson graph. We investigate a more general situation, namely, requiring the graphs to have complete multipartite µ-graphs, and that the intersection number α exists, which means that for a triple (x, y, z ) of vertices in Γ , such that x and y are adjacent and z is at distance 2 from x and y, the number α(x, y, z ) of common neighbours of x, y and z does not depend on the choice of a triple. The latter condition is satisfied by any 1-homogeneous graph. Let Kt ×n denote the complete multipartite graph with t parts, each of which consists of an n-coclique. We show that if Γ is a graph whose µ-graphs are all isomorphic to Kt ×n and whose intersection number α exists, then α = t, as conjectured by Jurišić and Koolen, provided α ≥ 2. We also prove t ≤ 4, and that equality holds only when Γ is the unique distance-regular graph 3.O7 (3). © 2009 Elsevier B.V. All rights reserved. 1. Introduction Let Γ be a connected graph with a pair of nonadjacent vertices. Then the subgraph induced on the set of common neighbours of two vertices at distance 2 is called a µ-graph of Γ . Let Kt ×n denote the complete multipartite graph consisting of t parts of size n, i.e., the complement of t copies of Kn ; see Fig. 1.1(a). In this paper, we investigate graphs whose µ-graphs are isomorphic to Kt ×n , where t and n are positive integers independent of the choice of a µ-graph. When n = 1, such graphs are known as Terwilliger graphs and we refer the reader to [1, Sect. 1.16] for the current status of research on these graphs. The graph Kt ×2 is called a cocktail party graph, and distance-regular graphs whose µ-graphs are Kt ×2 have been investigated by Jurišić and Koolen [5]. They extended their work to the more general case, where µ-graphs are Kt ×n in [6]. The main motivation came from antipodal tight distance-regular graphs of diameter 4. While not all such distanceregular graphs have complete multipartite µ-graphs, these graphs enjoy an extra property that is a consequence of the 1-homogeneous property (see Fig. 1.1): we say that the intersection number α(Γ ) exists if, for a triple of vertices (x, y, z ) of Γ such that x and y are adjacent and z is at distance 2 from both x and y, the number of common neighbours of x, y, z is a constant independent of (x, y, z ), and we denote this constant by α = α(Γ ). To avoid the degenerate case, we assume that there exists at least one such triple (x, y, z ) in Γ when we say α(Γ ) exists. In the case of distance-regular graphs this assumption is equivalent to a2 6= 0. Jurišić and Koolen [6] proposed the following problem. Problem 1 ([6]). Let Γ be a distance-regular graph with diameter at least 2, whose µ-graphs are the complete multipartite graph Kt ×n with n ≥ 2, and the intersection number α(Γ ) exists with α(Γ ) ≥ 2. Then show α(Γ ) = t. ∗ Corresponding author. E-mail addresses: ajurisic@valjhun.fmf.uni-lj.si (A. Jurišić), munemasa@math.is.tohoku.ac.jp (A. Munemasa). 0012-365X/$ – see front matter © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.disc.2009.12.009 A. Jurišić et al. / Discrete Mathematics 310 (2010) 1812–1819 a 1813 b Fig. 1.1. The distance partition corresponding to a pair of adjacent vertices in Γ . When this partition is equitable, we say that the graph is 1-homogeneous. This means that we can write numbers beside the edges connecting any two cells that indicate how many neighbours a vertex from the closer cell has in the other cell and also put beside each cell the valency of the graph induced by its vertices. Finally, we put inside the cell the number of its vertices. (a) the complete multipartite graph K(t +1)×n , with the intersection array {tn, n − 1; 1, tn}, (b) the collinearity graph of GQ(s, t ) with the intersection array {s(t + 1), st ; 1, t + 1} and a1 = s − 1, a2 = (s − 1)(t + 1). In [5, Lemma 2.1] it was shown that α(Γ ) ∈ {t − 1, t }. See also Lemma 7 and [5, Remark 3.6, Conjecture 3.7] for some partial information in the case when α(Γ ) = 1. One of the main purposes of this paper is to provide an affirmative answer to the above problem. In fact, we do not need to assume that Γ is distance-regular. The key step in the proof is the reduction lemma, which states that by taking a local graph of Γ one obtains a graph whose µ-graphs are isomorphic to K(t −1)×n and α exists, although the value of the intersection number α is smaller by 1 than that of Γ . This reduction lemma allows us to restrict the possible values of t and n, since the case α = t = 1 corresponds to generalized quadrangles whose extensions have been investigated in detail. In particular, we show t ≤ 4, and that equality holds only when Γ is the unique distance-regular graph 3.O7 (3). 2. Convexity and generalized quadrangles We use the standard terminology in graph theory, as in [1]. Let Γ = (V , E ) be a graph. If S is a subset of V , then by abuse of notation, we also denote by S the induced subgraph of Γ on S. A subset C of the vertices V is said to be convex if it contains all the shortest paths between any two of its vertices. The convex closure of a subset of vertices is the smallest convex subset containing it. For vertices v1 , . . . , vn of Γ we denote by Γ (v1 , . . . , vn ) the set of their common neighbours. If S = {v1 , . . . , vn }, then we also denote this set by Γ (S ). For a vertex v ∈ V we denote by Γi (v) the set of vertices at distance i from v . Let Γ be a graph. For a vertex x of Γ we call the graph induced by Γ (x) the local graph of Γ with respect to x. If X is a graph, or a family of graphs, or a property of graphs, then we say Γ is locally X if every local graph of Γ is isomorphic to X , is isomorphic to a member of X , has the property X , respectively. Recall some regularity properties of graphs. (R1) Any two adjacent vertices have precisely λ common neighbours, i.e., every local graph is λ-regular. (R2) Any two vertices at distance 2 have precisely µ common neighbours. (R3) Any two nonadjacent vertices have precisely µ common neighbours. A regular graph with v vertices and valency k is called edge-regular with parameters (v, k, λ) if (R1) holds, amply regular with parameters (v, k, λ, µ) if (R1) and (R2) hold, co-edge-regular with parameters (v, k, µ) if (R3) holds, and strongly regular with parameters (v, k, λ, µ) if (R1) and (R3) hold. For a regular graph Γ , (R1) and (R3) are complementary, in the sense that Γ satisfies (R1) if and only if the complement of Γ satisfies (R3). We have already mentioned in the introduction that graphs with all µ-graphs isomorphic to Kt ×n turn out to be related to generalized quadrangles. A generalized quadrangle GQ(s, t ) is an incidence structure of points and lines such that • on each line there are exactly s + 1 points (sitting), • through each point there are exactly t + 1 lines, and • for every nonincident pair (p, ℓ) of a point and a line there is exactly one line through the point p that intersects the line ℓ (or equivalently, there is exactly one point on the line ℓ that is collinear with the point p). For a detailed treatment of generalized quadrangles see Payne and Thas [8], Hirschfeld and Thas [4]. There a distinguished property of a point being regular is studied. For a point p of an incidence structure of points and lines we denote by p⊥ the set of all points that are collinear with p, and for a subset A of points we denote the intersection of the sets p⊥ for p ∈ A by A⊥ . To define this property in a generalized quadrangle GQ(s, t ) we consider two noncollinear points p, q and note that |{p, q}⊥ | = t + 1 and |{p, q}⊥⊥ | ≤ t + 1. As usual, we pay a special attention to the case when an inequality is satisfied with equality. So we say that a point p is regular when equality holds in this inequality for every point q that is noncollinear with p. Let Γ be the collinearity graph (i.e., the point graph) of GQ(s, t ). Then the graph Γ is strongly regular with parameters v = (s + 1)(st + 1), k = s(t + 1), λ=s−1 and µ = t + 1, (1) A. Jurišić et al. / Discrete Mathematics 310 (2010) 1812–1819 1814 and α(Γ ) = 1; see Fig. 1.1(b). The convex closure of the vertices in Γ that correspond to the points p and q with |{p, q}⊥⊥ | = t + 1, is the complete bipartite graph Kt +1,t +1 . A generalized quadrangle with all points regular is called regular. In regular generalized quadrangles GQ(s, t ) with s = t, the sets |{p, q}⊥⊥ |, where (p, q) is a pair of noncollinear points, have the same size as lines and can be used to construct new generalized quadrangles. The following result provides some explanation why regular generalized quadrangles motivate our study. Lemma 1 (Convexity). For integers t , n ≥ 2 let Γ be a connected graph of diameter at least 2, in which every µ-graph is isomorphic to Kt ×n . For vertices x, y at distance 2 in Γ , set K (x, y) = Γ (x, y) ∪ Γ (Γ (x, y)). Then the following statements hold: (i) Γ (Γ (x, y)) ∼ = Kn ; (ii) K (x, y) ∼ = K(t +1)×n ; (iii) K (x, y) is the convex closure of the set {x, y}. Proof. Set ∆ = Γ (x, y) ∼ = Kt ×n . Since n ≥ 2, there exist nonadjacent vertices x′ , y′ ∈ ∆. Setting ∆′ = Γ (x′ , y′ ), we find ∆ ∩ ∆′ ∼ = K(t −1)×n 6= ∅ by the assumption. Set C = ∆ \ (∆ ∩ ∆′ ) = Γ (x, y) ∩ Γ (∆ ∩ ∆′ ) ∼ = Kn ⊇ {x′ , y′ }, C ′ = ∆′ \ (∆ ∩ ∆′ ) = Γ (x′ , y′ ) ∩ Γ (∆ ∩ ∆′ ) ∼ = Kn ⊇ {x, y}, with other words, C (resp. C ′ ) is the maximal independent set of vertices in ∆ (resp. ∆′ ) that contains x′ , y′ (resp. x, y). Since t ≥ 2 there exists a pair of nonadjacent vertices x′′ , y′′ ∈ ∆ ∩ ∆′ . Then C ∪ C ′ ⊂ Γ (x′′ , y′′ ) ∼ = Kt ×n . Since C ∩ C ′ = ∅, we ′ have C ⊂ Γ (C ). Thus C ′ = C ′ ∩ Γ (C ) = Γ (x′ , y′ ) ∩ Γ (∆ ∩ ∆′ ) ∩ Γ (C ) = Γ ({x′ , y′ } ∪ (∆ ∩ ∆′ ) ∪ C ) = Γ (∆) = Γ (Γ (x, y)). This proves the statement (i). Furthermore, K (x, y) = ∆ ∪ C ′ = ∆ ∪ Γ (∆) by the above equality, which means K (x, y) ∼ = K(t +1)×n and so we have also proved (ii). The convex closure of {x, y} contains ∆ and thus also Γ (∆), so altogether it contains K (x, y). We have K (x, y) ∼ = K(t +1)×n by (ii), the graph K(t +1)×n has diameter 2 and µ(K(t +1)×n ) = nt = µ(Γ ), so we conclude that the set K (x, y) is convex. Therefore, we have shown (iii) as well.  The statements (ii), (iii) in the above result and the following lemma were first shown in [7, Lemma 4.1] and [5, Lemma 2.1] for distance-regular graphs. We included short proofs in order to make our presentation self-contained. Lemma 2 (Bounds). For integers t , n ≥ 2 let Γ be a connected graph in which every µ-graph is isomorphic to Kt ×n . If the intersection number α(Γ ) exists, then for any triple of vertices (x, y, z ) of Γ such that d(x, y) = 1, d(x, z ) = d(y, z ) = 2, we have Γ (x, y, z ) ∼ = Kα(Γ ) . Moreover, α(Γ ) ∈ {t − 1, t }. Proof. By the definition of the intersection number α(Γ ), we have |Γ (x, y, z )| = α(Γ ). If Γ (x, y, z ) = Γ (x, z ) ∩ Γ (y) contains a pair of nonadjacent vertices u and v , then the vertices x, y and z all belong to Γ (u, v) ∼ = Kt ×n . Since Kt ×n cannot contain three vertices x, y, z with d(x, y) = 1, d(x, z ) = d(y, z ) = 2, we obtain a contradiction. Thus Γ (x, y, z ) is a clique. Since the largest clique of Γ (x, z ) ∼ = Kt ×n has size t, we conclude α(Γ ) ≤ t. If α(Γ ) ≤ t − 2, then there are at least two cocliques of Γ (x, z ) ∼ = Kt ×n which are adjacent to all the vertices of Γ (x, y, z ). Thus there are adjacent vertices u, w ∈ Γ (x, z ) ∩ Γ (Γ (x, y, z )) ⊂ Γ (x, z ) \ Γ (x, y, z ) ⊂ Γ2 (y), so α(Γ ) = |Γ (u, w, y)| ≥ |{x} ∪ Γ (x, y, z )| = 1 + α(Γ ). This is a contradiction. Hence α(Γ ) ≥ t − 1.  3. Reduction to local graphs The complete multipartite graph K(t +1)×n has the property that its µ-graphs are the complete multipartite graphs Kt ×n , however, the intersection number α is not defined. It is maybe even more important that its local graphs are of the same form as the original graph, namely they are complete multipartite graphs Kt ×n . Before we use this recursive property in our situation, we need the following result that turned out to be very important. Lemma 3 (Regularity). Let µ, µ′ be nonnegative integers satisfying µ > µ′ + 1. Let Γ be a connected graph in which for every pair of vertices x1 , x2 at distance 2, the graph Γ (x1 , x2 ) is regular of valency µ′ on µ vertices. Then Γ is regular. Proof. Since Γ is connected, it suffices to show |Γ (x)| = |Γ (y)| for every pair of adjacent vertices x, y of Γ . We do it by a two-way counting of the edges between the sets Γ (x) ∩ Γ2 (y) and Γ (y) ∩ Γ2 (x) (this also shows that one of these two sets is empty if and only if the other one is). A vertex z ∈ Γ (x) ∩ Γ2 (y) (if it exists) has one neighbour in Γ (y) ∩ Γ0 (x) (namely the vertex x), µ′ neighbours in Γ (y) ∩ Γ1 (x) (by the definition of µ′ ) and all other neighbours in Γ (y) are (there are µ − µ′ − 1 of them) in Γ (y) ∩ Γ2 (x) (by the triangle inequality Γ (y) ∩ Γi (x) = ∅ for i > 2). Therefore, we have (µ − µ′ − 1) |Γ (x) ∩ Γ2 (y)| = X z ∈Γ (x)∩Γ2 (y) |Γ (y, z ) ∩ Γ2 (x)| A. Jurišić et al. / Discrete Mathematics 310 (2010) 1812–1819 1815 and by symmetry also (µ − µ′ − 1) |Γ2 (x) ∩ Γ (y)| = X w∈Γ2 (x)∩Γ (y) |Γ (x, w) ∩ Γ2 (y)|. Since these two numbers both count the edges between the sets Γ (x) ∩ Γ2 (y), Γ2 (x) ∩ Γ (y), and µ 6= µ′ + 1 by our assumption, we conclude |Γ (x) ∩ Γ2 (y)| = |Γ2 (x) ∩ Γ (y)|, which implies |Γ (x)| = 1 + |Γ (x) ∩ Γ (y)| + |Γ (x) ∩ Γ2 (y)| = 1 + |Γ (x) ∩ Γ (y)| + |Γ2 (x) ∩ Γ (y)| = |Γ (y)|.  ′ A special case (µ = 0) of the above result is due to Enomoto [2]; see also [1, p.4]. Lemma 4 (Reduction). For integers t , n ≥ 2 let Γ be a connected graph of diameter at least 2, in which every µ-graph is isomorphic to Kt ×n . Then Γ is regular. Moreover, for an arbitrary vertex x of Γ , the subgraph ∆ = Γ (x) satisfies the following properties: (i) (ii) (iii) (iv) ∆ is regular; ∆ has diameter 2 and every µ-graph of ∆ is isomorphic to K(t −1)×n ; ∆ is strongly regular if t ≥ 3; if the intersection number α(Γ ) exists, then α(Γ ) > 0 and the intersection number α(∆) exists with α(∆) = α(Γ ) − 1. Proof. We have µ = tn, the valency of the µ-graph is µ′ = (t − 1)n and µ − µ′ − 1 = n − 1 > 0, so the graph Γ is regular by Lemma 3. Since Γ is connected, noncomplete and regular, there exists a vertex u ∈ Γ2 (x), so ∆ contains Γ (x, u) ∼ = K t ×n . In particular, as n ≥ 2, the subgraph ∆ is noncomplete. Let y, z ∈ ∆ be nonadjacent vertices. Then ∆(y, z ) = Γ (x, y, z ) is a local graph of Γ (y, z ) ∼ = K(t −1)×n . Thus (ii) holds by t ≥ 2. We have µ(∆) = (t − 1)n, µ′ (∆) = (t − 2)n = Kt ×n , so ∆(y, z ) ∼ ′ and µ(∆) − µ (∆) − 1 = n − 1 > 0, so we obtain (i) by Lemma 3. If t ≥ 3, then applying (i) to the graph ∆, we see that it is edge-regular. This, together with (i) and (ii), establishes (iii). Finally, suppose that the intersection number α(Γ ) exists. A graph which is locally a complete multipartite graph is, by [1, Proposition 1.1.5], either triangle-free or complete multipartite. Neither is possible in our situation, since we assumed t , n ≥ 2 and that α(Γ ) exists. So the valency of ∆ is at most |∆| − 3 and being at distance 0 or 2 is not an equivalence relation in ∆. (i.e., ∆ is not antipodal). Therefore, there exists a pair of adjacent vertices at distance 2 from a vertex in ∆ (note that these distances are the same in Γ as in ∆, since the diameter of ∆ is equal to 2). This means that we can pick vertices y, z , w ∈ Γ (x) with d(z , w) = 1, d(y, z ) = d(y, w) = 2 and that α(Γ ) is at least 1, since x is their common neighbour. Then x ∈ Γ (y, z , w) ∼ = Kα(Γ ) by Lemma 2, so |∆(y, z , w)| = α(Γ ) − 1. Thus (iv) holds.  By Lemma 4, we are led to consider the case where the value of t is the smallest, namely t = 1. In this case, µ-graphs of Γ are cocliques, i.e., Γ contains no K1,1,2 as an induced subgraph, hence the following lemma applies, provided Γ contains a triangle. Lemma 5. Let Γ be a connected regular graph. Assume that Γ contains a triangle but no K1,1,2 as an induced subgraph. If the intersection number α(Γ ) exists, then α(Γ ) = 1. Proof. Suppose the intersection number α(Γ ) exists. If α(Γ ) ≥ 2, then for a triple (x, y, z ) of vertices with d(x, y) = 1, d(x, z ) = d(y, z ) = 2, we have |Γ (x, y, z )| ≥ 2. Let u, v be distinct vertices in Γ (x, y, z ). If the vertices u, v are adjacent, then {x, z , u, v} ∼ = K1,1,2 , which is = K1,1,2 , which is a contradiction. If the vertices u, v are nonadjacent, then {x, y, u, v} ∼ also a contradiction. Therefore α(Γ ) ≤ 1. Next suppose that α(Γ ) = 0. Since the graph Γ is not triangle-free, we can choose a triangle (w, x, y) of Γ . Then Γ (w)∩Γ2 (x)∩Γ2 (y) = ∅. Since Γ contains no induced subgraphs K1,1,2 , we have Γ (w, x)∩Γ2 (y) = ∅ and Γ (w, y)∩Γ2 (x) = ∅. Therefore, Γ (w) ∪ {w} ⊆ {x, y} ∪ Γ (x, y) ⊆ {x} ∪ Γ (x). Since Γ is regular, equality holds above, hence Γ (w) ∪ {w} = Γ (x) ∪ {x}. Now if we take z ∈ Γ (x) \ {w}, then (z , w, x) is a triangle, so we can argue just as above to conclude Γ (z ) ∪ {z } = Γ (x) ∪ {x}. We have shown that Γ (x) ∪ {x} is a clique. Since Γ is connected, this forces the graph Γ to be complete, contradicting the hypothesis that the intersection number α(Γ ) exists.  4. A solution of the problem The following result is essentially [5, Lemma 2.1], although the assumptions here are weaker. Lemma 6. Let Γ be an amply regular graph with parameters (v, k, λ, µ), which is locally co-edge-regular with parameters (k, λ, µ′ ). If the intersection number α(Γ ) exists, then α(Γ ) |Γ2 (x) ∩ Γ (y)| = µ (λ − µ′ ) whenever x, y are vertices at distance 2. A. Jurišić et al. / Discrete Mathematics 310 (2010) 1812–1819 1816 Proof. The result follows by a two-way counting of the edges between the sets Γ (x, y) and Γ (y) ∩ Γ2 (x): X z ∈Γ2 (x)∩Γ (y) |Γ (x, y, z )| = X w∈Γ (x,y) |Γ2 (x) ∩ Γ (y, w)| = X w∈Γ (x,y) (|Γ (y, w)| − |Γ (x, y, w)|).  Lemma 7. For an integer n ≥ 2 let Γ be a graph in which every µ-graph is isomorphic to Kn,n . If the intersection number α(Γ ) exists and α(Γ ) = 1, then Γ has diameter at least 3. Proof. Suppose the intersection number α(Γ ) exists and α(Γ ) = 1. Hence the diameter of Γ is at least 2. Let ∆ be a local graph of Γ . Then, by Lemma 4, the graph ∆ is regular, it has diameter 2, every µ-graph in ∆ is isomorphic to Kn , and the intersection number α(∆) exists with α(∆) = 0. In particular, ∆ is co-edge-regular, and ∆ contains no K1,1,2 as an induced subgraph. By Lemma 5, ∆ is triangle-free, so ∆ is edge-regular, hence strongly regular with parameters (v ′ , k′ , 0, n), where v ′ = 1 + k′ + k′ (k′ − 1)/n. By Lemma 4, Γ is regular. Since ∆ is regular, Γ is edge-regular. Now suppose that Γ has diameter 2. Then Γ is strongly regular with parameters (v, k, λ, µ), where k = v ′ = 1 + k′ + k′ (k′ − 1)/n, λ = k′ , and µ = 2n. By Lemma 6 and µ′ = n, we have 1 + k′ + k′ (k′ − 1) n − 2n = 2n(k′ − n). Therefore, λ = 2n2 − 2n + 1 and k = 2n(2n2 − 3n + 2). Then, by [1, p. 8], the nontrivial eigenvalues of Γ are: r = 2n2 − 2n, s = −2n + 1 and the multiplicity f of the eigenvalue r is f = 120n − 84 (s + 1)k(k − s) = 8n4 − 32n3 + 66n2 − 86n + 80 − . µ(s − r ) 2n2 − 1 However, (120n−84)/(2n2 −1) = 12(10n−7)/(2n2 −1) is not an integer for any integer n ≥ 2 (for note gcd(2n2 −1, 12) = 1 and 2n2 − 1 > 10n − 7 for n ≥ 5). This contradiction proves that Γ has diameter at least 3.  Theorem 8. For integers t , n ≥ 2 let Γ be a connected graph in which every µ-graph is isomorphic to Kt ×n . If the intersection number α(Γ ) exists with α(Γ ) ≥ 2, then α(Γ ) = t. Proof. Let us assume the intersection number α(Γ ) exists. Then there exist vertices x, y, z of Γ such that d(x, y) = 1, d(x, z ) = d(y, z ) = 2. By Lemma 2, we have α(Γ ) ∈ {t − 1, t }. Suppose α(Γ ) = t − 1. Then t ≥ 3 by α(Γ ) ≥ 2. Taking local graphs successively and using Lemma 4, we obtain a graph ∆ of diameter 2, in which every µ-graph is isomorphic to Kn,n and α(∆) = 1. This contradicts Lemma 7.  Theorem 8 gives an affirmative answer to Problem 1. 5. Extensions of generalized quadrangles Let Γ be a graph. A clique C of Γ is called regular if every vertex outside C is adjacent to the same number e > 0 of vertices in C . We call e the nexus of C . Lemma 9. Let Γ be a connected regular graph. Assume that Γ contains a triangle but no K1,1,2 as an induced subgraph, and that the intersection number α(Γ ) exists. If Γ is co-edge-regular with parameters (v, k, µ), then one of the following statements holds: (i) for every vertex x, the local graph Γ (x) consists of µ cliques of sizek/µ, each of which  is a regular clique with nexus 1, when x is adjoined. In particular, Γ is strongly regular with parameters v, k, µk − 1, µ ; or (ii) µ = 2, and for every vertex x, the local graph Γ (x) consists of two cliques of size s+ and s− , each of which is a regular clique with nexus 1, when x is adjoined, where s± = k± p k2 − 4(v − k − 1) 2 . Proof. Suppose the graph Γ is co-edge-regular with parameters (v, k, µ). Lemma 5 implies α(Γ ) = 1. Since Γ contains no induced subgraphs K1,1,2 , every edge is contained in a unique maximal clique. Thus Γ is locally a disjoint union of cliques. Let x be a vertex of Γ , M a maximal clique of Γ containing x. We claim |M ∩ Γ (z )| = 1 for all z ∈ Γ2 (x). (2) Clearly |M ∩Γ (z )| ≤ 1 since otherwise Γ (x, z ) would have an edge, producing K1,1,2 as an induced subgraph. If M ∩Γ (z ) = ∅, then pick w ∈ M. Since Γ has diameter 2, we have z ∈ Γ2 (w), hence |Γ (x, w, z )| = α(Γ ) = 1. The unique vertex u of Γ (x, w, z ) belongs to {x, w} ∪ Γ (x, w) = M, and so u ∈ M ∩ Γ (z ), which contradicts M ∩ Γ (z ) = ∅. Therefore, the claim is proved. A. Jurišić et al. / Discrete Mathematics 310 (2010) 1812–1819 1817 Next, let M be a maximal clique of Γ containing x. We claim M ∩ Γ (z ) = {x} for all z ∈ Γ (x) \ M . (3) Since M is a maximal clique and z 6∈ M, there exists y ∈ M such that y and z are nonadjacent. If z is adjacent to some vertex w ∈ M \ {x}, then {x, y, z , w} ∼ = K1,1,2 , which is a contradiction. This proves (3). By (2) and (3), M is a regular clique with nexus 1. For w ∈ M \ {x} we have |Γ (w) ∩ Γ2 (x)| = |Γ (w)| − |M \ {w}| = k − (|M | − 1) and, by (2), v−k−1= X z ∈Γ2 (x) |M ∩ Γ (z )| = X w∈M
|Γ (w) ∩ Γ2 (x)| = (|M | − 1) k − (|M | − 1) . Thus, s = |M | − 1 satisfies the quadratic equation s2 − ks + v − k − 1 = 0. Let s± denote the two solutions of this equation. Then, by |M | − 1 ∈ {s+ , s− } and v − k − 1 > 0, we have s+ + s− = k and s± > 0. (4) Let m± be the number of maximal cliques of size s± + 1 containing x. Then obviously we have m+ s+ + m− s− = k. This, together with (4) implies that either m+ m− = 0 or m+ = m− = 1. In the former case, all maximal cliques containing x have the same size s + 1 = 1 + k/µ, because µ = m+ + m− by (2). In particular, Γ is strongly regular with the desired parameters. This gives (i). The latter case leads to (ii).  Lemma 10. Let t ≥ 2, n ≥ 2 be integers, and assume (t , n) 6= (2, 2). Let Γ be a connected k-regular graph on v vertices in which every µ-graph is isomorphic to Kt ×n . Assume that the intersection number α(Γ ) exists. Then Γ is amply regular with parameters (v, k, λ, nt ), where and
(λ + n) (n − 1)λ + n(t − 1) k= n2 (t − 1) (5)
(λ − n(t − 1)) λ − n(nt − n − 1) ≥ 0. (6) Equality holds in (6) if and only if Γ has diameter 2. Proof. Let ∆ = Γ (x) be the local graph of Γ with respect to a vertex x. Observe that, by Lemma 4, ∆ is co-edge-regular with parameters (k, λx , µ′ ) for some integer λx , where µ′ = n(t − 1). We claim that ∆ is strongly regular with parameters (k, λx , λ′x , µ′ ) for some integer λ′x . Indeed, if ∆ is triangle-free, then the claim holds with λ′x = 0. If t ≥ 3, then the claim follows from Lemma 4(iii). Now suppose that ∆ contains a triangle and t = 2. In this case, Lemma 4(ii) implies that ∆ contains no K1,1,2 as an induced subgraph, and Lemma 4(iv) implies that the intersection number α(∆) exists. As µ′ = n ≥ 3, Lemma 9 implies that ∆ is strongly regular. Therefore, we have proved the claim. Since every µ-graph of ∆ is isomorphic to K(t −1)×n , the graph ∆ is locally co-edge-regular with parameters (λx , λ′x , µ′′ ), where µ′′ = n(t − 2). By Lemma 4(iv), we have α(∆) = α(Γ ) − 1. If α(Γ ) = 1, then t = 2 by Lemma 2. Then Lemma 7 implies that ∆ has diameter at least 3, while Lemma 4(ii) implies ∆ has diameter 2. This contradiction shows that α(Γ ) ≥ 2. Now by Theorem 8, we have α(Γ ) = t. Therefore, Lemma 6 implies (t − 1)(λx − µ′ ) = µ′ (λ′x − µ′′ ). Thus λ′x = λx n + n(t − 2) − t + 1. (7) Since µ′ (k − 1 − λx ) = λx (λx − 1 − λ′x ), we have k=1+ λx (λx − 1 − λ′x + µ′ ). µ′ Substituting (7) into the above equality, we see that λx is a solution to (5) regarded as an equation in λ. Since (5) is a quadratic equation in λ with exactly one nonnegative solution, we conclude that λx is independent of the choice of x, and so we denote it by λ for the rest of the proof. Now, for y ∈ Γ2 (x), we have, by Lemma 6, 0 ≤ Γ (x) \ Γ (y) ∪ Γ2 (y) = n−1 n2 ( t − 1 )
= k − µ − |Γ2 (y) ∩ Γ (x)| = k − µ −

λ − n(t − 1) λ − n(nt − n − 1) . µ(λ − µ′ ) α(Γ ) (8) Equality holds in (8) if and only if Γ (x) ⊂ Γ (y) ∪ Γ2 (y). Since x and y are arbitrary vertices at distance 2, equality holds if and only if Γ has diameter 2.  A. Jurišić et al. / Discrete Mathematics 310 (2010) 1812–1819 1818 Theorem 11. For an integer n ≥ 3 let Γ be a connected graph in which every µ-graph is isomorphic to Kn,n . If the intersection number α(Γ ) exists and α(Γ ) = 2, then Γ is locally GQ(λ/n, n − 1). In particular, Γ has diameter 2 if and only if Γ is locally GQ(n − 1, n − 1). Proof. Suppose the intersection number α(Γ ) exists and α(Γ ) = 2. Let ∆ = Γ (x) be the local graph of Γ with respect to a vertex x. Then by Lemma 4, Γ is regular, and by Lemma 4(i), ∆ is also a regular graph in which every µ-graph is isomorphic to Kn . In particular, Γ is amply regular with parameters (v, k, λ, 2n), and ∆ contains no induced subgraphs K1,1,2 . By Lemma 4(iv), the intersection number α(∆) exists with α(∆) = 1. This means that ∆ is not triangle-free. Therefore, we can apply Lemma 9 to ∆. As µ(∆) = n ≥ 3, the graph ∆ is locally nKλ/n and every maximal clique is a regular clique with nexus 1. Hence, ∆ is the collinearity graph of a generalized quadrangle GQ(λ/n, n − 1), where λ/n 6= 1. By Lemma 10, Γ has diameter 2 if and only if
(λ − n) λ − n(n − 1) = 0, which in turn is equivalent to λ = n(n − 1). Therefore, Γ has diameter 2 if and only if Γ is locally GQ(n − 1, n − 1).  A graph which is locally GQ(p, q) is called a triangular extended generalized quadrangle [3]. Theorem 12. For integers t ≥ 1 and n ≥ 3 let Γ be a connected graph in which every µ-graph is isomorphic to Kt ×n . If the intersection number α(Γ ) exists, then t ≤ 4. Moreover, equality holds only if Γ is the unique distance-regular graph 3.O7 (3), which is locally locally locally GQ(2, 2). Proof. Suppose the intersection number α(Γ ) exists. We may assume t ≥ 4, as the assertion is trivial when t ≤ 3. Let Γ (s) denote a graph obtained from Γ by taking local graphs s times. By Lemma 4, the graph ∆ = Γ (t −2) has diameter 2 and satisfies the hypotheses of Theorem 11. Thus the graph ∆ is locally GQ(n − 1, n − 1). Then by [3], we have n = 3, 4, 5, 9 or 14. By (1), the parameters of Γ (t −1) , which is the point graph of GQ(n − 1, n − 1), are: v ′′′ = n(n2 − 2n + 2), k′′′ = (n − 1)n, λ′′′ = n − 2, µ′′′ = n ≥ 3. Since ∆ is strongly regular, its parameters are: k′′ = v ′′′ = n(n2 − 2n + 2), λ′′ = k′′′ = (n − 1)n, µ′′ = 2n, 1 k′′ (k′′ − 1 − λ′′ ) = (n2 − 2n + 2)(n3 − 3n2 + 5n − 1) + 1. v ′′ = 1 + k′′ + ′′ µ 2 Since we have assumed that t ≥ 4, the graph Γ (t −3) is strongly regular with parameters 1 k′ = v ′′ = (n2 − 2n + 2)(n3 − 3n2 + 5n − 1) + 1, 2 λ′ = k′′ = n(n2 − 2n + 2), µ′ = 3n, v ′ = 1 + k′ + = 1 12 k′ (k′ − 1 − λ′ ) µ′ (n9 − 10n8 + 49n7 − 150n6 + 319n5 − 488n4 + 545n3 − 420n2 + 202n − 12). Then the nontrivial eigenvalues of the graph Γ (t −3) are: λ′ − µ′ ± p (µ′ − λ′ )2 − 4(µ′ − k′ ) 2 = 1 2  p n3 − 2n2 − n ± (n − 1) n(n3 − 9n + 12) . (t −3) Since λ 6= µ − 1, Γ is not a conference graph, n(n3 − 9n + 12) is a perfect square (see, [1, p. 8]). For n ∈ {3, 4, 5, 9, 14}, this is the case only when n = 3. Finally we show that t cannot be larger than 4 even in the case n = 3. If t > 4, then the strongly regular graph Γ (t −4) is locally Γ (t −3) , and has parameters: ′ ′ k = v ′ = 117, λ = k′ = 36, √ µ = 12, v =1+k+ k(k − 1 − λ) µ = 898. Its eigenvalues are 117, 12 ± 249. Since this is not a conference graph, we must have integral eigenvalues. This is a contradiction. Now, when t = 4, we have shown that Γ is locally locally locally GQ(2, 2). Such a graph is known to be the unique distance-regular graph 3.O7 (3) (see [7]).  A. Jurišić et al. / Discrete Mathematics 310 (2010) 1812–1819 1819 Acknowledgement The authors would like to thank an anonymous referee for correcting several errors in an earlier version of the manuscript. References [1] [2] [3] [4] [5] [6] [7] [8] A.E. Brouwer, A.M. Cohen, A. Neumaier, Distance-Regular Graphs, Springer-Verlag, New York, 1989. H. Enomoto, Characterization of families of finite permutation groups by the subdegrees, II, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 20 (1973) 1–11. P.H. Fisher, S.A. Hobart, Triangular extended generalized quadrangles, Geom. Dedicata 37 (1991) 339–344. J.W.P. Hirschfeld, J.A. Thas, General Galois Geometries, Clarendon Press, Oxford, 1991. A. Jurišić, J. Koolen, 1-homogeneous graphs with Cocktail Party µ-graphs, J. Algebra Combin. 18 (2003) 79–98. A. Jurišić, J. Koolen, Distance-regular graphs with complete multipartite µ-graphs and the AT4 family, J. Algebra Combin. 25 (2007) 459–471. A. Jurišić, J. Koolen, Classification of the family AT4 (qs, q, q) of antipodal tight graphs (in press). S.E. Payne, J.A. Thas, Finite Generalized Quadrangles, in: Research Notes in Mathematics, Pitman, New York, 1984.