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ON LOCALLY GQ(S, T) GRAPHS with STRONGLY REGULAR Μ-SUBGRAPHS

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ON LOCALLY GQ(S, T) GRAPHS with STRONGLY REGULAR Μ-SUBGRAPHS Algebra i analiz St. Petersburg Math. J. Tom. 17 (2005), vyp. 3 Vol. 17 (2006), No. 3, Pages 443–452 S 1061-0022(06)00913-7 Article electronically published on March 9, 2006 ON LOCALLY GQ(s, t) GRAPHS WITH STRONGLY REGULAR µ-SUBGRAPHS V. I. KAZARINA AND A. A. MAKHNEV Abstract. The connected locally GQ(s, t) graphs are studied in which every µ- subgraph is a known strongly regular graph (i.e., Km,m for a positive integer m,the Moore graph with parameters (k2 +1,k,0, 1), k =2, 3, or 7, the Clebsch graph, the Gewirtz graph, the Higman–Sims graph, or the second neighborhood (with param- eters (77, 16, 0, 4)) of a vertex in the Higman–Sims graph). It is proved that if Γ is a strongly regular locally GQ(s, t) graph in which every µ-subgraph is isomorphic to a known strongly regular graph ∆, then one of the following statements is true: (1) ∆ = Kt+1,t+1 and either s =1andΓ=K3×(t+1),ors =4,t =1,andΓisa quotient of the Johnson graph J(10, 5), or s = t =1, 2, 3, 8, 13; (2) ∆ is a Petersen graph and Γ is a unique locally GQ(2, 2) graph with parameters (28, 15, 6, 10); (3) ∆ is the Gewirtz graph and Γ is the McLaughlin graph. Introduction We consider nonoriented graphs with no loops and no multiple edges. For a vertex a of a graph Γ, we denote by Γi(a)thei-neighborhood of a, i.e., the subgraph induced by Γ on the set of all vertices lying at a distance of i from a. We put Γ(a)=Γ1(a) and a⊥ = {a}∪Γ(a). If the graph Γ is fixed, then we write [a] instead of Γ(a). Unless otherwise specified, by a subgraph we mean an induced subgraph. Largely, we use the same notation as in [1]. Let F be a class of graphs. We say that a graph Γ is a locally F graph if [a] lies in F for each vertex a of Γ. If the class F consists of graphs isomorphic to a graph ∆, then we say that Γ is a locally ∆ graph. Let Γ be a graph, and let a, b ∈ Γ. The number of vertices in [a] ∩ [b] is denoted by µ(a, b) if the distance between a and b in Γ is 2 and by λ(a, b)ifa and b are adjacent in Γ. In the sequel, the induced subgraph [a] ∩ [b] is called a µ-subgraph (respectively, a λ-subgraph). By the degree of a vertex we mean the number of vertices in the neighborhood of that vertex. A graph Γ is said to be regular of degree k if the degree of every vertex a in Γisequaltok. A graph Γ is edge regular with parameters (v, k, λ)ifithasv vertices, is regular of degree k, and each edge of it lies in λ triangles. A graph Γ is completely regular with parameters (v, k, λ, µ) if it is edge regular with the corresponding parameters and [a] ∩ [b]containsµ vertices for any two vertices a and b that are 2 apart in Γ. A completely regular graph is said to be strongly regular if it has diameter 2. By Km1,...,mn we denote a complete multipartite graph {M1,...,Mn} with parts Mi of order mi.If m1 = ··· = mn = m, then this graph is denoted by Kn×m. For a subgraph ∆ of Γ, we 2000 Mathematics Subject Classification. Primary 05C75. Key words and phrases. Strongly regular graphs, locally F graphs, geometry of rank 2. This work was supported by RFBR (grant 02–01–00772). c 2006 American Mathematical Society 443 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 444 V. I. KAZARINA AND A. A. MAKHNEV denote by Xi(∆) the set of all vertices in Γ − ∆ that are adjacent to exactly i vertices of ∆. A geometry G of rank 2 is an incidence system with a set P of points and set B of blocks, and without multiple blocks. Here, each block can be identified with the set of points incident to it, and incidence becomes the usual inclusion. Two points in P are collinear if they lie in the same block. The point graph of a geometry G is the graph on the set P in which two points are adjacent if they are distinct and collinear. The block graph is defined similarly. We say that a geometry is connected if its point graph is connected. For a ∈ P , we define the residue Ga as the geometry with the set of points Pa collinear to a and the set of blocks Ba = {B −{a}|a ∈ B ∈B}. Suppose a ∈ P and B ∈B. If a/∈ B (respectively, a ∈ B), then the pair (a, B) is called an antiflag (respectively, a flag). If any two blocks in B intersect at one point at most, then the set of blocks is called the set of lines and is denoted by L, and the geometry (P, L) is called a partial space of lines. A partial space of lines has order (s, t) if each line contains s + 1 points and each point lies on t + 1 lines. A partial space of lines of order (s, t) is called an α-partial geometry if for each an- tiflag (a, L) there exist exactly α lines passing through a and intersecting L (notation: pGα(s, t)). For α = 1, such a geometry is called a generalized quadrangle and is denoted by GQ(s, t). In the point graph for GQ(s, t), a coclique consisting of st + 1 points is called an ovoid.Byaspread in GQ(s, t)wemeanasetofst + 1 lines forming a partition of the set of points. The point graph of the geometry pGα(s, t) is strongly regular, with v =(s + 1)(1 + st/α), k = s(t +1),λ = s − 1+t(α − 1), and µ = α(t + 1). Any strongly regular graph with these parameters is called a pseudo-geometric graph for pGα(s, t). AgeometryG is called an extension of α-partial geometries if it is connected and the residue at each point coincides with pGα(s, t) for appropriate (s, t) (notation: EpGα). By connectedness, the order (s, t) of a geometry does not depend on the choice of a residue, and such an extension is denoted by EpGα(s, t). The geometry EpGα is said to be triangular if arbitrary three pairwise collinear points lie in the same block (which is necessarily unique). The study of triangular extensions of geometries G of class pGα is based on the study of locally connected graphs Γ(Ga) (and is equivalent to such a study if α =1). A subset Λ of a generalized triangle is called a hyperoval if each line intersects Λ at 0 or 2 points. In other words, a hyperoval in GQ(s, t) is a regular subgraph of degree t +1, without triangles, and having an even number of vertices. It is well known [2] that the µ-subgraphs in locally GQ(s, t) graphs are hyperovals. If a regular graph of degree k and diameter d has v vertices, then we have the following inequality proved by Moore: v ≤ 1+k + k(k − 1) + ···+ k(k − 1)d−1. A graph for which this inequality becomes equality is called a Moore graph.Asimple example of a Moore graph is a (2d + 1)-gon. Damerell [3] proved that a Moore graph of degree k ≥ 3 has diameter 2. In this case, we have v = k2 + 1, the graph is strongly regular with λ =0andµ =1,andthedegreek can be equal to 3 (the Petersen graph), to 7 (the Hoffman–Singleton graph), or to 57. Actually, it is not known whether a Moore graph of degree k = 57 really exists. The Clebsch graph is defined on the set of vectors of the four-dimensional linear space V over the field of two elements; in this case, two vectors are adjacent if the Hamming distance between them is 1 or 4. This is the only strongly regular graph with parameters License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use ON LOCALLY GQ(s, t) GRAPHS WITH STRONGLY REGULAR µ-SUBGRAPHS 445 (16, 5, 0, 2). The Gewirtz, Higman–Sims, and McLaughlin graphs are the graphs of rank 3 of the group L3(4), the Higman–Sims group, and the McLaughlin group on 56, 100, and 275 vertices, respectively. The Johnson graph J(n, m) is defined on the set of m-element subsets of a given n- element set X,andtwom-element subsets a and b are adjacent if |a ∩ b| = m − 1. A quotient Johnson graph J σ(2m, m) is the factor-graph the vertices of which are pairs (aσ,X − a), where σ is a permutation on X of order at most 2 that has at least 8 fixed points. If σ = 1, then the quotient is said to be standard and is denoted by J(2m, m). Let Γ be a connected locally GQ(s, t) graph, and let u and w be its vertices with d(u, w) = 2. It is well known that µ(u, w)isevenandmax{2(t +1), (s +1)(t +2− s)}≤ µ(u, w) ≤ 2(st +1). In the present paper, we study the connected locally GQ(s, t) graphs in which the µ- subgraphs are known strongly regular graphs.
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