On graphs with complete multipartite -graphs

Graphs with complete multipartite µ-graphs Akihiro Munemasa1 1 Graduate School of Information Sciences Tohoku University joint work with A. Jurišić and Y. Tagami Discrete Math. 310 (2010), 1812–1819 February 6, 2012 Ghent University Akihiro Munemasa Graphs Distance-Regular Graphs Brouwer–Cohen–Neumaier (1988). Examples: Dual polar spaces = {max. totally isotropic subsp.} and their subconstituent: eg. alternating forms graph. Main Problem: Classify distance-regular graphs. classification of feasible parameters characterization by parameters characterization by local structure A local characterization of the graphs of alternating forms and the graphs of quadratic forms graphs over GF(2) A. Munemasa, D.V. Pasechnik, S.V. Shpectorov Akihiro Munemasa Graphs Definition of a distance-regular graph Γi−1 (x) Γi (x) Γi+1 (x) ai x ci bi y Γi (x): the set of vertices at distance i from x the numbers ai , bi , ci are independent of x and y ∈ Γi (x). ai , bi , ci are called the parameters of a distance-regular graph Γ. Akihiro Munemasa Graphs 1-Homogeneity Nomura (1987) obtained inequalities among ai , bi , ci requiring constant number of edges between cells is an additional condition (1-homogeneity). Akihiro Munemasa Graphs Generalized Quadrangle of order (s, t) 1 x s−1 1 st t(s − 1) t 1 1 1 st(s − 1) s − 1 st t 1 y s−1 1 t t t(s − 1) st On Graphs with Complete Multipartite µ-Graphs – p.3/5 Akihiro Munemasa Graphs Local Characterization of Alternating Forms Graph Alt(n, 2) over GF(2) Local Graph = Γ(x) = neighborhood of x. Assume that a distance-regular graph Γ has the same local graph as Alt(n, 2), i.e., Grassmann graph (= line graph of PG(n − 1, 2)), and the same parameters (in particular c2 = µ = 20). Then Γ∼ = Alt(n, 2) or Quad(n − 1, 2) (M.–Shpectorov–Pasechnik). Key idea: “µ local = local µ”, where “µ = Γ(x) ∩ Γ(y)” with y ∈ Γ2 (x). Taking z ∈ Γ(x) ∩ Γ(y), µ of local of Γ = µ of Γ(z) = (Γ(x) ∩ Γ(y)) ∩ Γ(z) = Γ(z) ∩ (Γ(x) ∩ Γ(y)) = local of Γ(x) ∩ Γ(y) = local of µ of Γ. Akihiro Munemasa Graphs Local Characterization of Alternating Forms Graph Alt(n, 2) over GF(2) If local graphs of Γ are Grassmann (line graph of PG(n − 1, 2)), then “µ local = local µ” implies µ of Grassmann = local of µ hence 3 × 3 grid = local of µ µ-graphs of Γ are locally 3 × 3-grid, and µ = c2 = 20 = =⇒ J(6, 3). Akihiro Munemasa Graphs 6 3
Jurišić–Koolen, 2003 From now on, a µ-graph of a graph is the subgraph induced on the set of common neighbors of two vertices at distance 2. Cocktail party graph = complete graph K2p minus a matching = complete multipartite graph Kp×2 (p parts of size 2) Classified 1-homogeneous distance-regular graphs with cocktail party µ-graph Kp×2 with p ≥ 2. Akihiro Munemasa Graphs Examples Kp×2 .. . local ↓ K6×2 K5×2 K4×2 K3×2 K2×2 µ-graph Gosset Schläfli 1 1 5-cube n-cube 2 2 J(5, 2) J(n, 2) J(n, k) 2 × 3 2 × (n − 2) k × (n − k) K5×2 K4×2 K3×2 K2×2 K1×2 The bottom rows are all grids. Jurišić–Koolen (2007): 1-homogeneous distance-regular graphs with cocktail party µ-graph Kp×2 with p ≥ 2 are contained in those shown above and some of their quotients. Akihiro Munemasa Graphs Jurišić–Koolen, 2007 Complete multipartite graph Kp×n is a generalization of cocktail party graph Kp×2 . Examples .. . µ-graph K6×n K5×n K4×n K5×n 3.O7 (3) + K4×n O6 (3) Meixner K3×n K2×n K3×n O5 (3) U5 (2) Patterson 3.O6− (3) K2×n GQ(2, 2) GQ(3, 3) GQ(9, 3) GQ(4, 2) K1×n = Kn n=t+1 They assumed distance-regularity, but having Kp×n as µ-graphs turns out to be a very strong restriction already. Akihiro Munemasa Graphs “local µ = µ local” In local characterization, local of ↑ known µ-graph = µ of local ↑ ↑ ↑ derive known assume In µ characterization, local of ↑ known µ-graph = µ of local ↑ ↑ ↑ assume known derive Example µ of local local of µ = Kp×n = ↑ ↑ ↑ ↑ K(p−1)×n assume K(p−1)×n derive Akihiro Munemasa Graphs Taking local, µ = Kp×n → µ = K(p−1)×n Assume every µ-graph of Γ is Kp×n . Taking local graph (p − 1) times, one obtains a graph ∆ whose µ-graphs are K1×n = Kn : equivalently, 6 ∃K1,1,2 , ∀ edge⊂ ∃!maximal clique Such graphs always come from a geometric graph such as GQ? .. . K6×n K5×n K4×n K3×n K2×n µ-graph K5×n K4×n 3.O7 (3) O6+ (3) Meixner K3×n − O5 (3) U5 (2) Patterson 3.O6 (3) K2×n GQ(2, 2) GQ(3, 3) GQ(9, 3) GQ(4, 2) K1×n = Kn n=t+1 Akihiro Munemasa Graphs The parameter α For a graph Γ, we say the parameter α exists if ∃x, y, z, d(x, y) = 1, d(x, z) = d(y, z) = 2 and |Γ(x) ∩ Γ(y) ∩ Γ(z)| = α(Γ) for all such x, y, z. Example: α(GQ(s, t)) = 1 if s, t ≥ 2. Akihiro Munemasa Graphs α-graph is a clique, hence α ≤ p Suppose every µ-graph of Γ is Kp×n ,and α exists. Claim: Γ(x) ∩ Γ(y) ∩ Γ(z) is a clique. Indeed, if nonadjacent u, v ∈ Γ(x) ∩ Γ(y) ∩ Γ(z), then x, y, z ∈ Γ(u) ∩ Γ(v) ∼ = Kp×n , but d(x, y) = 1, d(x, z) = d(y, z) = 2 : contradiction. α(Γ) is bounded by the clique size in Γ(x) ∩ Γ(z) ∼ = Kp×n which is p. Akihiro Munemasa Graphs The parameter α We have shown α(Γ) ≤ p. One can also shows α(Γ) ≥ p − 1. If ∆ is a local graph, then α(∆) exists and α(∆) = α(Γ) − 1. Akihiro Munemasa Graphs Regularity Lemma Let Γ be a connected graph, M a non-complete graph. Assume every µ-graph of Γ is M . Then Γ is regular. Proof. By two-way counting (BCN, p.4, Proposition 1.1.2.) Lemma Let Γ be graph, M a graph without isolated vertex. Assume every µ-graph of Γ is M . Then every local graph of M has diameter 2. Akihiro Munemasa Graphs Reduction Lemma Let Γ be a connected graph. Assume every µ-graph of Γ is Kp×n , and α exists. Let ∆ be a local graph of Γ. Then Γ is regular, ∆ has diameter 2, every µ-graph of ∆ is K(p−1)×n . α(∆) exists and α(∆) = α(Γ) − 1. We know α(Γ) = p or p − 1. Suggests that the reverse procedure of taking a local graph does not seem possible so many times, meaning p cannot be too large. Akihiro Munemasa Graphs Main Result Theorem Let Γ be a connected graph. Assume every µ-graph of Γ is Kp×n , where p, n ≥ 2, and α exists in Γ. Then (i) p = α(Γ) unless (p, α(Γ)) = (2, 1) and diameter ≥ 3. (ii) If n ≥ 3, then p = α(Γ) = 2 =⇒ Γ locally GQ(s, n − 1), p = α(Γ) = 3 =⇒ Γ locally2 GQ(n − 1, n − 1), p = α(Γ) = 4 =⇒ Γ locally3 GQ(2, 2), p ≥ 5: impossible. proof of (i). Rule out (p, α(Γ)) = (2, 1) when diameter= 2 (strongly regular). Akihiro Munemasa Graphs Open Problem Rule out (p, α(Γ)) = (2, 1) when diameter≥ 3. This might occur even when n = 2: µ-graph of Γ is cocktail party graph K2×2 = C4 . Nonexistence was conjectured by Jurišić–Koolen (2003). Akihiro Munemasa Graphs