The isoperimetric number of the incidence graph of PG(n,q)

Let $\Gamma_{n,q}$ be the point-hyperplane incidence graph of the projective space $\operatorname{PG}(n,q)$, where $n \ge 2$ is an integer and $q$ a prime power. We determine the order of magnitude of $1-i_V(\Gamma_{n,q})$, where $i_V(\Gamma_{n,q})$ is the vertex-isoperimetric number of $\Gamma_{n,q}$. We also obtain the exact values of $i_V(\Gamma_{2,q})$ and the related incidence-free number of $\Gamma_{2,q}$ for $q \le 16$.


Introduction
A fundamental problem in graph theory is to understand various expansion properties of graphs. The expansion of a graph is commonly measured by its isoperimetric number, also known as the Cheeger constant, or its vertex-isoperimetric number. These two parameters have been studied extensively, especially in the study of expanders, and a number of results on them exist in the literature (see for example [13]). A major concern is to produce good (sharp) lower bounds for these isoperimetric numbers and related invariants. Such isoperimetric inequalities are closely related to problems in probabilistic combinatorics, theoretical computer science, spectral graph theory, etc.
Inspired by a conjecture of Babai and Szegedy, in this paper we study the vertexisoperimetric number of the point-hyperplane incidence graph of the projective space PG(n, q).
Let Γ = (V, E) be a graph. The vertex-boundary N(X) of a subset X ⊆ V is the set of vertices in V \ X that are adjacent to at least one vertex in X. The vertex-isoperimetric number of Γ is defined [13] as The problem of determining the vertex-isoperimetric number of a graph is known to be NP-complete. There are very few families of graphs whose vertex-isoperimetric numbers have been computed exactly (see e.g. [10]). The reader is referred to [11] for the history of this problem and related results. The related problem of determining min{|N(X)| : ∅ = X ⊆ V } has also been studied extensively ( [11,15]); see, for example, [9] for Harper's classical result on this problem for hypercubes and [3] for an isoperimetric inequality for the discrete torus. As pointed out in [8], many isoperimetric problems can be put into the form for bipartite graphs. In this case a closely related parameter is as follows. Let Γ be a bipartite graph with bipartition {V 1 , V 2 } such that |V 1 | = |V 2 |. If S ⊆ V 1 and T ⊆ V 2 are such that |S| = |T | and there is no edge of Γ between S and T , then (S, T ) is called an incidence-free pair. The incidence-free number of Γ, first introduced in [5] and denoted byᾱ(Γ), is defined to be the maximum size of S among all incidence-free pairs (S, T ).
This parameter is particularly useful for bounding i V (Γ) for bipartite graphs Γ. Indeed, for any incidence-free pair (S, T ) with |S| =ᾱ(Γ), by setting X = S ∪ (V 2 \ T ) in (1) we obtain The incidence graph (or Levi graph) [2] of a 2-(v, k, λ) design D is the bipartite graph with one part of the bipartition consisting of the points of D and the other part the blocks of D such that a point is adjacent to a block if and only if they are incident in D. Obviously, if D is a symmetric design, then its incidence graph is a k-regular bipartite graph with v vertices in each part such that any two vertices in the same part have exactly λ common neighbours in the other part. Conversely, any k-regular bipartite graph with these properties is isomorphic to the incidence graph of a symmetric 2-(v, k, λ) design. Such a graph Γ is called a (v, k, λ)-graph and its bipartition is denoted by We require k to be a positive integer but we allow the degenerate case λ = 0 for which the graph is a perfect matching. It is well known [2] that the parameters (v, k, λ) for a symmetric design satisfy Throughout the paper we use Γ n,q to denote the incidence graph of the point-hyperplane design of the projective space PG(n, q), where n is a positive integer and q a prime power. More explicitly, let V 1 and V 2 be the sets of 1-dimensional and n-dimensional subspaces of F n+1 q respectively, where F q is the finite field of order q. Γ n,q is the bipartite graph with bipartition {V 1 , V 2 } and adjacency relation giving by subspace containment. Alternatively, we can write , it follows that Γ n,q is a It is readily seen that Γ n,q has diameter 2.
Considerable interest in Γ n,q arises from algebraic graph theory and finite geometry. For example, it is known [7] that these graphs form a major subfamily of the family of 2-arc transitive Cayley graphs of dihedral groups. (A graph is 2-arc transitive if its automorphism group is transitive on the set of oriented paths of length 2.) In [1], Babai and Szegedy conjectured that there is a positive absolute constant c such that any finite 2-arc transitive graph with diameter d has vertex-isoperimetric number at least c/ √ d. They wrote further that "it would be interesting to find reasonable symmetry conditions which would imply an expansion rate of Ω(1/ √ d)". The main result in the present paper (Theorem 1 below) is in line with this conjecture and provides a new family of symmetric graphs with expansion rate at least Ω(1/ √ d). As noted in [19], i V (Γ 2,q ) is closely related to arcs in the projective plane PG(2, q). Given integers k, d > 1, a (k; d)-arc in PG(2, q) is a set of k points, of which no d + 1 are collinear. It is known that k ≤ (d − 1)(q + 1) + 1 for any (k; d)-arc in PG(2, q); a (k; d)-arc is maximal if equality holds. A (k; 2)-arc is usually called a k-arc.
Several results on i V (Γ n,q ) and related problems exist in the literature. Harper and Hergert [8] and Ure [19] studied the related problem of finding the minimum |N(X)| for a subset X of points with a given size in the projective plane PG(2, q). In [14], Lanphier et al. studied the isoperimetric number of Γ n,q . In [16], Mubayi and Williford studied the independence number of the quotient of Γ n,q with respect to the partition each of whose part consists of a point of PG(n, q) and its dual hyperplane. De Winter et al. [5] and Stinson [18] studied the incidence-free number of Γ n,q .
Determining the precise value of i V (Γ n,q ) turns out to be a very challenging problem, even in the case when n = 2 and q is small. In this paper we will first prove the following bounds for i V (Γ n,q ) and thus determine the order of magnitude of 1 − i V (Γ n,q ). We will then determine the exact values of i V (Γ 2,q ) for all prime powers q ≤ 16. Theorem 1. Let n ≥ 2 be an integer, q = p e a prime power and ǫ > 0 a real number with 0 < ǫ < 1 4 . Then for some real number c n,q with 1 The upper bound c n,q < 1 is best possible. Indeed, due to the existence of Denniston maximal arcs [6] in PG(2, 2 2k ), we necessarily have thatᾱ(Γ 2,2 2k ) = 2 3k − 2 2k + 2 k , so can be forced arbitrarily close to 1 for sufficiently large k. We suspect that the lower bound for c n,q can be improved to 1 2 without the need for an error term. Theorem 1 will be proved in the next two sections: In section 2 we give a lower bound for i V (Γ) for any (v, k, λ)-graph Γ and use it to prove the lower bound for i V (Γ n,q ) as given in (4). In section 3 we obtain a lower bound forᾱ(Γ n,q ) and thus the required upper bound in (4) by using (2).
Theorem 2. Let q ≤ 16 be a prime power. Then the values of i V (Γ 2,q ) andᾱ(Γ 2,q ) are as given in Table 1. Moreover, the equality in (2) holds for Γ 2,q . That is, In Table 1 new results from this paper are highlighted in bold. We include c 2,q in order to estimate how close it is to the given bounds 0.5 c 2,q < 1.
Proof Since the roles of V 1 (Γ) and V 2 (Γ) are symmetric, we may assume S ⊆ V 1 (Γ) without loss of generality. Let T i denote the number of vertices in V 2 (Γ) adjacent to exactly i vertices in S. By counting the number of edges from S to N(S), as well as the number of paths of length 2 from S to S, in two different ways, we obtain as required.

It follows that
We now use Theorem 4 to prove the following lower bound for any (v, k, λ)-graph.
Proof Let f be as defined in (7). Using (3) it can be verified that f −1 (v) = µv k+µ . Note that, for any where the last equality is obtained by a straightforward evaluation of f at kv k+µ by using (3).
Equipped with Theorem 5, we are now ready to prove the lower bound for i V (Γ n,q ) as stated in Theorem 1.
To establish the upper bound in Theorem 1 we will also use some known results on the well-known circle problem and its primitive version. For any real number r > 0, define C(r) = (x, y) ∈ Z 2 : x 2 + y 2 ≤ r and C ′ (r) = (x, y) ∈ Z 2 : x 2 + y 2 ≤ r, (x, y) = 1 . (x,y)∈C ′ (r) Proof Since C(⌊r⌋) ⊆ C(r) ⊆ C(⌈r⌉) and C ′ (⌊r⌋) ⊆ C ′ (r) ⊆ C ′ (⌈r⌉), it suffices to prove these equalities for positive integers r. Let r > 0 be an integer. The first two equalities are well-known in the literature as the Gauss circle problem and the primitive Gauss circle problem respectively; see [12] and [20]. The third one follows from the first two because (x,y)∈C ′ (r) where the last line follows from the fact that Proof of Theorem 1 (upper bound) In view of (2), in order to prove the upper bound in (4) it suffices to provē for any integer n ≥ 1, prime power q = p e and real number ǫ > 0. It turns out that the key step is to handle the special case when n = 2 and q is a prime.
Case 1: n = 2 and q = p is a prime. In this case (8) is equivalent tō We prove this by construction. Let We first claim that S ∪ T is an independent set of Γ 2,p . Indeed, for any combination of x, y, a, b, c as above, we have 0 < (x, y, 1) · (a, b, c) < p, because Thus S ∪ T is an independent set of Γ 2,p .
It follows directly from Lemma 7 that |S| =  . We can get a lower bound for |T | by only picking the points where (a, b) = 1 and identifying (a, b, c) with (−a, −b, p − c). Using this and Lemma 7, we obtain From this and the definition ofᾱ we obtain (9) immediately. We now deal with the general case by using Lemma 6 and what we proved in Case 1.

Proof of Theorem 2
In this section we will first prove that the values ofᾱ(Γ 2,q ) in Table 1 are correct. When q ≤ 7, the exact value ofᾱ(Γ 2,q ) is given in [19]. We determine the values ofᾱ(Γ 2,q ) for q = 8, 9, 11, 13, 16 by proving matching upper and lower bounds.
Proof In [4] it is shown that there are only four (17; 3)-arcs in PG (2,9) up to isomorphism. These can be given as coordinates on the affine plane, where i denotes an element satisfying i 2 + 1 = 0:  Proof Suppose otherwise. Then there exists S ∈ V 1 (Γ 2,q ) such that |S| = 20 and |N(S)| ≤ 71. Let T i denote the number of lines of PG(2, q) that are incident to exactly i points in S. Using the same notation and technique as in the proof of Lemma 3, we obtain which implies that (T 4 ≤ 3 and T 5 = 0) or (T 4 = 0 and T 5 = 1) and T i = 0 for all i ≥ 6. Removing these (at most) three points from S gives us a new (17; 3)-arc S ′ with |N(S) ′ | ≤ 71, contradicting Lemma 9.
So far we have determined the values ofᾱ(Γ 2,q ) for q ∈ {8, 9, 11, 13, 16}. Combining these with the values ofᾱ(Γ 2,q ) for q ∈ {2, 3, 4, 5, 7} given in [19], we obtain the results in the second column of Table 1. In light of (2), these give us upper bounds for i V (Γ 2,q ) as needed in the third column of Table 1. The matching lower bound for i V (Γ 2,q ) seems to be difficult to obtain analytically, and so we run a program to achieve this. Since testing all subsets takes exponential time, we weaken some of the constraints and give a polynomial time program for the relaxed problem. Theorem 11. Let q be a prime power and v = q 2 + q + 1. Then the optimal value of the following program is a lower bound for i V (Γ 2,q ). Furthermore, this problem can be solved in polynomial time with respect to q.