The isoperimetric number of the incidence graph of PG(n,q)
aa r X i v : . [ m a t h . C O ] D ec The isoperimetric number of the incidence graph ofPG( n, q ) Andrew Elvey-Price ∗ , Muhammad Adib Surani † and Sanming Zhou ‡ School of Mathematics and StatisticsThe University of MelbourneParkville, VIC 3010, AustraliaOctober 11, 2018
Abstract
Let Γ n,q be the point-hyperplane incidence graph of the projective space PG( n, q ),where n ≥ q a prime power. We determine the order of mag-nitude of 1 − i V (Γ n,q ), where i V (Γ n,q ) is the vertex-isoperimetric number of Γ n,q .We also obtain the exact values of i V (Γ ,q ) and the related incidence-free numberof Γ ,q for q ≤ Keywords: isoperimetric number; vertex-isoperimetric number; incidence-freenumber; projective plane; projective space
AMS subject classification (2010):
A fundamental problem in graph theory is to understand various expansion properties ofgraphs. The expansion of a graph is commonly measured by its isoperimetric number, alsoknown as the Cheeger constant, or its vertex-isoperimetric number. These two parametershave been studied extensively, especially in the study of expanders, and a number of resultson them exist in the literature (see for example [13]). A major concern is to producegood (sharp) lower bounds for these isoperimetric numbers and related invariants. Suchisoperimetric 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 vertex-isoperimetric number of the point-hyperplane incidence graph of the projective spacePG( n, q ). ∗ E-mail: [email protected] † E-mail: [email protected] ‡ E-mail: [email protected]
V, E ) be a graph. The vertex-boundary N ( X ) of a subset X ⊆ V is the set ofvertices in V \ X that are adjacent to at least one vertex in X . The vertex-isoperimetricnumber of Γ is defined [13] as i V (Γ) = min (cid:26) | N ( X ) || X | : ∅ 6 = X ⊆ V, | X | ≤ | V | (cid:27) . (1)The problem of determining the vertex-isoperimetric number of a graph is known to beNP-complete. There are very few families of graphs whose vertex-isoperimetric numbershave been computed exactly (see e.g. [10]). The reader is referred to [11] for the historyof this problem and related results. The related problem of determining min {| N ( X ) | : ∅ 6 = X ⊆ V } has also been studied extensively ([11, 15]); see, for example, [9] for Harper’sclassical result on this problem for hypercubes and [3] for an isoperimetric inequality forthe discrete torus.As pointed out in [8], many isoperimetric problems can be put into the form forbipartite graphs. In this case a closely related parameter is as follows. Let Γ be abipartite graph with bipartition { V , V } such that | V | = | V | . If S ⊆ V and T ⊆ V aresuch 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 denotedby ¯ α (Γ), is defined to be the maximum size of S among all incidence-free pairs ( S, T ).That is, ¯ α (Γ) = max S ⊆ V min {| S | , | V \ N ( S ) |} . This parameter is particularly useful for bounding i V (Γ) for bipartite graphs Γ. Indeed,for any incidence-free pair ( S, T ) with | S | = ¯ α (Γ), by setting X = S ∪ ( V \ T ) in (1) weobtain i V (Γ) ≤ − ¯ α (Γ) | V | . (2)The incidence graph (or Levi graph) [2] of a 2-( v, k, λ ) design D is the bipartite graphwith one part of the bipartition consisting of the points of D and the other part theblocks 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 bipartitegraph 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 withthese 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 { V (Γ) , V (Γ) } .We require k to be a positive integer but we allow the degenerate case λ = 0 for whichthe graph is a perfect matching. It is well known [2] that the parameters ( v, k, λ ) for asymmetric design satisfy λ ( v −
1) = k ( k − . (3)Throughout the paper we use Γ n,q to denote the incidence graph of the point-hyperplanedesign of the projective space PG( n, q ), where n is a positive integer and q a prime power.More explicitly, let V and V be the sets of 1-dimensional and n -dimensional subspaces of2 n +1 q respectively, where F q is the finite field of order q . Γ n,q is the bipartite graph with bi-partition { V , V } and adjacency relation giving by subspace containment. Alternatively,we can write V = {h u i : u ∈ F n +1 q } , V = {h v i ⊥ : v ∈ F n +1 q }h u i and h v i ⊥ are adjacent in Γ n,q if and only if u · v = 0 . Since PG( n, q ) is a symmetric 2- (cid:16) q n +1 − q − , q n − q − , q n − − q − (cid:17) design [2], it follows that Γ n,q is a (cid:16) q n +1 − q − , q n − q − , q n − − q − (cid:17) -graph. 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 familyof 2-arc transitive Cayley graphs of dihedral groups. (A graph is 2-arc transitive if itsautomorphism group is transitive on the set of oriented paths of length 2.) In [1], Babaiand Szegedy conjectured that there is a positive absolute constant c such that any finite2-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 conditionswhich 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 symmetricgraphs with expansion rate at least Ω(1 / √ d ).As noted in [19], i V (Γ ,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 arecollinear. It is known that k ≤ ( d − q + 1) + 1 for any ( k ; d )-arc in PG(2 , q ); a ( k ; d )-arcis 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 andHergert [8] and Ure [19] studied the related problem of finding the minimum | N ( X ) | fora subset X of points with a given size in the projective plane PG(2 , q ). In [14], Lanphieret al. studied the isoperimetric number of Γ n,q . In [16], Mubayi and Williford studied theindependence number of the quotient of Γ n,q with respect to the partition each of whosepart consists of a point of PG( n, q ) and its dual hyperplane. De Winter et al. [5] andStinson [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 followingbounds for i V (Γ n,q ) and thus determine the order of magnitude of 1 − i V (Γ n,q ). We willthen determine the exact values of i V (Γ ,q ) for all prime powers q ≤ Theorem 1.
Let n ≥ be an integer, q = p e a prime power and ǫ > a real number with < ǫ < . Then i V (Γ n,q ) = 1 − c n,q q n +12 ( q − q n +1 − for some real number c n,q with − O ( p ǫ − ) ≤ c n,q < . The upper bound c n,q < , k ), we necessarily have that ¯ α (Γ , k ) = 2 k − k + 2 k , so3able 1: The cases when q ≤ q ¯ α (Γ ,q ) i V (Γ ,q ) c ,q (3 decimal places)2 2 / / / / / / / / / / c , k can be forced arbitrarily close to 1 for sufficiently large k . We suspect that the lowerbound for c n,q can be improved to without the need for an error term.Theorem 1 will be proved in the next two sections: In section 2 we give a lower boundfor i V (Γ) for any ( v, k, λ )-graph Γ and use it to prove the lower bound for i V (Γ n,q ) asgiven in (4). In section 3 we obtain a lower bound for ¯ α (Γ n,q ) and thus the required upperbound in (4) by using (2).As far as we know, no exact value of i V (Γ ,q ) is known even for small q , and the exactvalue of ¯ α (Γ ,q ) is known [19] only for q ∈ { , , , , } . The next result gives the exactvalues of ¯ α (Γ ,q ) for q ∈ { , , , , } and i V (Γ ,q ) for all prime powers q ≤
16. Wewill prove this result in section 4.
Theorem 2.
Let q ≤ be a prime power. Then the values of i V (Γ ,q ) and ¯ α (Γ ,q ) areas given in Table 1. Moreover, the equality in (2) holds for Γ ,q . That is, i V (Γ ,q ) = 1 − ¯ α (Γ ,q ) q + q + 1 . (5)In Table 1 new results from this paper are highlighted in bold. We include c ,q in orderto estimate how close it is to the given bounds 0 . . c ,q < i V (Γ n,q ) Let Γ be a ( v, k, λ )-graph. It is known [17] that | N ( S ) | ≥ k | S | k + λ ( | S |− for any S ⊆ V (Γ)or S ⊆ V (Γ). We will prove a stronger bound in the following lemma. This was alreadyknown in [19] for projective planes, and here we generalise it to all symmetric 2-designs. Lemma 3.
Let Γ be a ( v, k, λ ) -graph and let m = j λ ( | S |− k k + 1 . Then for any non-emptysubset S of V (Γ) or V (Γ) we have | N ( S ) || S | ≥ km − λ ( | S | − m ( m + 1) ≥ k k + λ ( | S | − . (6)4 roof Since the roles of V (Γ) and V (Γ) are symmetric, we may assume S ⊆ V (Γ)without loss of generality. Let T i denote the number of vertices in V (Γ) adjacent to exactly i vertices in S . By counting the number of edges from S to N ( S ), as well as the numberof paths of length 2 from S to S , in two different ways, we obtain P ki =1 T i = | N ( S ) | , P ki =1 iT i = k | S | and P ki =1 i ( i − T i = λ | S | ( | S | − n ≥ n for all integers n , wehave 0 ≤ k X i =1 (cid:0) ( i − m ) − ( i − m ) (cid:1) T i = k X i =1 ( i ( i − − mi + m ( m + 1)) T i = λ | S | ( | S | − − mk | S | + m ( m + 1) | N ( S ) | . Hence | N ( S ) || S | ≥ km − λ ( | S | − m ( m + 1) . Since λ ( | S |− k ≤ m ≤ λ ( | S |− k + 1 by the definition of m , we have (cid:18) λ ( | S | − k + 1 − m (cid:19) (cid:18) m − λ ( | S | − k (cid:19) ≥ , that is, (cid:18) λ ( | S | − k + 1 (cid:19) (cid:18) m − λ ( | S | − k (cid:19) ≥ m ( m + 1) . This yields 2 km − λ ( | S | − m ( m + 1) ≥ k k + λ ( | S | − (cid:3) Define f : [0 , v ] → [0 , v ] by f ( x ) = k xk + λ ( x −
1) + x. (7) Theorem 4.
Let Γ be a ( v, k, λ ) -graph. Then for any ∅ 6 = X ⊆ V (Γ) we have | N ( X ) | ≥ f ( | X | − f − ( | X | )) − | X | . Proof
By taking the first and second derivatives, one can see that f is increasing, concaveand bijective. So f − exists and is increasing and convex.Denote S = X ∩ V (Γ) and T = X ∩ V (Γ). We may assume | S | ≤ | T | without loss ofgenerality. By Lemma 3, | N ( X ) | = | N ( S ) \ T | + | N ( T ) \ S | max {| N ( S ) | − | T | , } + max {| N ( T ) | − | S | , }≥ max { f ( | S | ) − | S | − | T | , } + max { f ( | T | ) − | T | − | S | , } . If | S | ≤ f − ( | X | ), then | N ( X ) | ≥ f ( | X | − | S | ) − | X | ≥ f ( | X | − f − ( | X | )) − | X | , asclaimed.Otherwise assume | S | > f − ( | X | ). Then | X | − f − ( | X | ) > | X | − | S | = | T | ≥ | S | . Bythe concavity of f , we have f ( | S | ) + f ( | X | − | S | ) ≥ f (cid:0) f − ( | X | ) (cid:1) + f (cid:0) | X | − f − ( | X | ) (cid:1) = | X | + f (cid:0) | X | − f − ( | X | ) (cid:1) . It follows that | N ( X ) | ≥ f ( | S | ) − | X | + f ( | X | − | S | ) − | X |≥ f (cid:0) | X | − f − ( | X | ) (cid:1) − | X | . (cid:3) We now use Theorem 4 to prove the following lower bound for any ( v, k, λ )-graph.
Theorem 5.
Let Γ be a ( v, k, λ ) -graph and µ = √ k − λ . Then i V (Γ) ≥ ( k − µ ) k + µ k + µ . Proof
Let f be as defined in (7). Using (3) it can be verified that f − ( v ) = µvk + µ . Notethat, for any x ∈ [0 , v ] and α ∈ [0 , f ( αx ) = f ( αx + (1 − α )0) ≥ αf ( x ) + (1 − α ) f (0) = αf ( x ). Similarly, f − ( αx ) ≤ αf − ( x ). Thus, for any ∅ 6 = X ⊆ V (Γ) with | X | ≤ | V (Γ) | = v ,by Theorem 4, | N ( X ) || X | ≥ − | X | f (cid:0) | X | − f − ( | X | ) (cid:1) ≥ − | X | f (cid:18) | X | v (cid:0) v − f − ( v ) (cid:1)(cid:19) ≥ − v f (cid:0) v − f − ( v ) (cid:1) = − v f (cid:18) v − µvk + µ (cid:19) = ( k − µ ) k + µ k + µ , where the last equality is obtained by a straightforward evaluation of f at kvk + µ by using(3). (cid:3) i V (Γ n,q ) asstated in Theorem 1. Proof of Theorem 1 (lower bound)
Let n ≥ q a prime power.It suffices to prove i V (Γ n,q ) > − q n +12 ( q − q n +1 − . Let ( v, k, λ ) = (cid:16) q n +1 − q − , q n − q − , q n − − q − (cid:17) be the parameters of Γ n,q , and let µ = √ k − λ = q n − . Case 1: ( n, q ) = (2 , ∅ 6 = X ⊂ V (Γ , ) be such that | X | ≤ v = 7 and i V (Γ , ) = | N ( X ) || X | . Let f be as defined in (7). If | X | ≤
6, then by the same reasoning asin Theorem 5 we obtain | N ( X ) || X | ≥ − | X | f (cid:0) | X | − f − ( | X | ) (cid:1) ≥ − f (cid:0) − f − (6) (cid:1) = 11 √ (cid:16) − √ (cid:17) > − √ . If | X | = 7, then | N ( X ) | ≥ ⌈ f (7 − f − (7)) − ⌉ = 5 and hence | N ( X ) || X | ≥ > − √ . Case 2: ( n, q ) = (2 , k = 4 and µ = √ i V (Γ , ) ≥ (cid:16) − √ (cid:17)
16 + 364 + 3 √ > − √ . Case 3: n ≥ q ≥
4. In this case we have k − λ = µ = q n − ≥ k ≥ k (4 + λ ) > k − λ = 4 µ . Thus k − µ > µ ≥
2. Therefore, k − kµ + 2 µ µ ( k − µ ) = 1 µ + 1 k − µ + 2 µ ( k − µ ) < ≤ q. This together with Theorem 5 implies i V (Γ n,q ) ≥ ( k − µ ) k + µ k + µ = 1 − µk + µ ( k − µ ) k − kµ +2 µ > − µk + q − q n +12 ( q − q n +1 − . (cid:3) i V (Γ n,q ) The following results are taken from [5, Corollary 10], [5, Corollary 14] and [16, Theorem5]. We will use them in the proof of the upper bound for i V (Γ n,q ) as stated in Theorem 1. Lemma 6.
Let p be a prime and q a prime power. Then (a) ¯ α (Γ ,q ) ≥ q ; (b) ¯ α (Γ ,p ) ≥ √ p ; (c) ¯ α (Γ ,p k ) ≥ p k for all positive integers k ; (d) ¯ α (Γ ,p k +1 ) ≥ p k ¯ α (Γ ,p ) for all positive integers k ; and (e) ¯ α (Γ n +2 ,q ) ≥ q ¯ α (Γ n,q ) for all positive integers n . To establish the upper bound in Theorem 1 we will also use some known results on thewell-known circle problem and its primitive version. For any real number r >
0, define C ( r ) = (cid:8) ( x, y ) ∈ Z : x + y ≤ r (cid:9) and C ′ ( r ) = (cid:8) ( x, y ) ∈ Z : x + y ≤ r, ( x, y ) = 1 (cid:9) . Lemma 7.
Let r > and ǫ > be real numbers. Then | C ( r ) | = πr + O ( r ); | C ′ ( r ) | = 6 π r + O (cid:16) r + ǫ (cid:17) ; X ( x,y ) ∈ C ′ ( r ) p x + y = 4 π r √ r + O (cid:0) r ǫ (cid:1) . Proof
Since C ( ⌊ r ⌋ ) ⊆ C ( r ) ⊆ C ( ⌈ r ⌉ ) and C ′ ( ⌊ r ⌋ ) ⊆ C ′ ( r ) ⊆ C ′ ( ⌈ r ⌉ ), it suffices to provethese equalities for positive integers r .Let r > X ( x,y ) ∈ C ′ ( r ) p x + y = r X i =1 √ i ( | C ′ ( i ) | − | C ′ ( i − | )8 r X i =1 π √ i + r − X i =1 (cid:16) √ i − √ i + 1 (cid:17) (cid:18) | C ′ ( i ) | − π i (cid:19) + √ r (cid:18) | C ′ ( r ) | − π r (cid:19) = 4 π r √ r + O ( r ǫ ) , where the last line follows from the fact that √ i − √ i + 1 = O ( √ i ). (cid:3) Proof of Theorem 1 (upper bound)
In view of (2), in order to prove the upperbound in (4) it suffices to prove¯ α (Γ n,q ) ≥ (cid:18) − O ( p ǫ − ) (cid:19) q n +12 (8)for any integer n ≥
1, prime power q = p e and real number ǫ >
0. It turns out that thekey 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 to¯ α (Γ ,p ) ≥ p √ p − O (cid:16) p + ǫ (cid:17) . (9)We prove this by construction. Let S = (cid:26) h ( x, y, i : ( x, y ) ∈ C (cid:18) p √ p π (cid:19)(cid:27) and T = ( h ( a, b, c ) i ⊥ : a, b, c ∈ Z , (cid:12)(cid:12)(cid:12) c − p (cid:12)(cid:12)(cid:12) < p − p √ π √ a + b ) . We first claim that S ∪ T is an independent set of Γ ,p . Indeed, for any combination of x, y, a, b, c as above, we have 0 < ( x, y, · ( a, b, c ) < p , because (cid:12)(cid:12)(cid:12) ax + by + c − p (cid:12)(cid:12)(cid:12) ≤ | ax + by | + (cid:12)(cid:12)(cid:12) c − p (cid:12)(cid:12)(cid:12) < p x + y √ a + b + p − p √ π √ a + b ≤ p . Thus S ∪ T is an independent set of Γ ,p .It follows directly from Lemma 7 that | S | = p √ p + O (cid:16) p (cid:17) . We can get a lowerbound 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 | T | ≥ X ( x,y ) ∈ C ′ ( π √ p ) p − − p √ π p x + y ! p − (cid:12)(cid:12)(cid:12) C ′ (cid:16) π √ p (cid:17)(cid:12)(cid:12)(cid:12) − p √ π X ( x,y ) ∈ C ′ ( π √ p ) p x + y = p − (cid:16) √ p + O (cid:16) p + ǫ (cid:17)(cid:17) − p √ π (cid:16) √ πp + O (cid:16) p + ǫ (cid:17)(cid:17) = p √ p − O (cid:16) p + ǫ (cid:17) . From this and the definition of ¯ α we obtain (9) immediately.We now deal with the general case by using Lemma 6 and what we proved in Case 1. Case 2: n ≥ q = p e a prime power.By (e) in Lemma 6, ¯ α (Γ n,q ) ≥ q n − ¯ α (Γ ,q ) for odd n ≥
1. This together with (a)in Lemma 6 implies that (8) holds for any odd integer n ≥
1. Again, by (e) in Lemma6, ¯ α (Γ n,q ) ≥ q n − ¯ α (Γ ,q ) for even n ≥
2. Hence it suffices to prove (8) for n = 2.This has been proved in (9) when e = 1. In general, if e = 2 k + 1 ≥ α (Γ ,q ) ≥ p k ¯ α (Γ ,p ) ≥ (cid:16) − O ( p ǫ − ) (cid:17) p k +1)2 = (cid:16) − O ( p ǫ − ) (cid:17) q as required. If e = 2 k ≥ α (Γ ,q ) ≥ p k ≥ (cid:16) − O ( p ǫ − ) (cid:17) q . (cid:3) In this section we will first prove that the values of ¯ α (Γ ,q ) in Table 1 are correct. When q ≤
7, the exact value of ¯ α (Γ ,q ) is given in [19]. We determine the values of ¯ α (Γ ,q ) for q = 8 , , , ,
16 by proving matching upper and lower bounds.
Lemma 8.
Let q be a prime power. Let x be a positive integer and m = j xq +1 k + 1 . If x + 1 m + 1 (cid:16) q + 1) − xm (cid:17) > q ( q + 1) − x, then ¯ α (Γ ,q ) ≤ x. Proof
Let S ⊆ V (Γ ,q ) be such that | S | = x + 1. Invoking Lemma 3 with parameters( v, k, λ ) = ( q + q + 1 , q + 1 ,
1) yields | S | + | N ( S ) | > q + q + 1. Moreover, if | S | > x + 1,then we can simply consider any subset of S of size x + 1 and the same result follows. (cid:3) By Lemma 8, we obtain ¯ α (Γ , ) ≤
16, ¯ α (Γ , ) ≤
21, ¯ α (Γ , ) ≤
28, ¯ α (Γ , ) ≤
36 and¯ α (Γ , ) ≤
52 immediately. As we will see shortly, all these bounds except the second oneare sharp.To obtain the sharp upper bound for ¯ α (Γ , ), we use the classification [4] of 3-arcs inPG(2 , S of V (Γ , ) with | S | = 17 such thatany vertex in V (Γ , ) is adjacent to at most three vertices in S .10 emma 9. Let S be a (17; 3)-arc of PG(2 , . Then | N ( S ) | ≥ . Proof
In [4] it is shown that there are only four (17; 3)-arcs in PG(2 ,
9) up to isomor-phism. These can be given as coordinates on the affine plane, where i denotes an elementsatisfying i + 1 = 0: • S = { (0 , , (0 , , (0 , , (1 , , (1 , , (1 , , (2 , i ) , (2 , i +1) , (2 , i ) , ( i, , ( i, i +2) , ( i +1 , , (2 i, , (2 i, i + 1) , (2 i + 2 , i ) , (2 i + 2 , i + 1) , (2 i + 2 , i ) }• S = { (0 , , (0 , , (0 , , (1 , , (1 , , (1 , , (2 , i ) , (2 , i +1) , (2 , i +2) , ( i, , ( i, , ( i, i +2) , ( i + 1 , i + 2) , ( i + 2 , , (2 i, i + 1) , (2 i, i ) , (2 i, i + 2) }• S = { (0 , , (0 , , (0 , , (1 , , (1 , , (1 , , (2 , i ) , (2 , i +1) , (2 , i +2) , ( i, , ( i, i ) , ( i, i +2) , ( i + 1 , i + 2) , ( i + 2 , , ( i + 2 , i + 1) , ( i + 2 , i + 1) , (2 i, }• S = { (0 , , (0 , , (0 , , (1 , , (1 , , (1 , i ) , (2 , , (2 , i ) , (2 , i + 1) , ( i, i ) , ( i + 1 , i +2) , (2 i, i + 1) , (2 i + 1 , , (2 i + 1 , i + 2) , (2 i + 2 , , (2 i + 2 , i + 1) , (2 i + 2 , i + 1) } It is straightforward to check that | N ( S i ) | evaluates to 72, 73, 73 and 74 respectively. (cid:3) Lemma 10. ¯ α (Γ , ) ≤ . Proof
Suppose otherwise. Then there exists S ∈ V (Γ ,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, weobtain X i =1 ( i − i − T i ≤ X i =1 ( i ( i − − i + 6) T i ≤ · (20 − − · ·
10 + 6 ·
71= 6 , which implies that ( T ≤ T = 0) or ( T = 0 and T = 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. (cid:3)
To prove the values of ¯ α (Γ ,q ) for q ∈ { , , , , } as shown in Table 1, it sufficesto show that ¯ α (Γ , ) ≥
16, ¯ α (Γ , ) ≥
19, ¯ α (Γ , ) ≥
28, ¯ α (Γ , ) ≥
36 and ¯ α (Γ , ) ≥ q = 8 we can use the construction in Lemma 6(e) to obtain ¯ α (Γ , ) ≥
16. When q = 16 we can construct a maximal arc [6] to obtain ¯ α (Γ , ) ≥
52. When q ∈ { , , } ,the following subset of V (Γ ,q ) yields ¯ α (Γ , ) ≥
19, ¯ α (Γ , ) ≥
28 and ¯ α (Γ , ) ≥ q = 9: { (0 , , (0 , , (0 , i ) , (0 , i + 1) , (1 , , (1 , , (1 , i + 1) , (2 , i ) , (2 , i + 1) , ( i, i ) , ( i, i +2) , ( i + 1 , , ( i + 1 , i + 2) , ( i + 1 , i + 2) , (2 i, i + 2) , (2 i, i + 1) , (2 i, i + 2) , (2 i + 2 , , (2 i +2 , i + 1) } , where i denotes an element satisfying i + 1 = 0.11 = 11: { (0,0), (0,8), (0,10), (1,3), (1,7), (1,8), (2,5), (2,7), (2,10), (3,3), (3,5), (3,7),(4,0), (4,8), (4,10), (5,1), (5,4), (5,9), (7,0), (7,4), (7,10), (9,1), (9,3), (9,9), (10,1), (10,4),(10,5), (10,9) } . q = 13: { (0,0), (0,3), (0,6), (0,10), (1,1), (1,4), (1,10), (3,0), (3,5), (3,10), (3,11), (4,1),(4,2), (4,6), (4,7), (6,2), (6,3), (6,5), (6,12), (7,1), (7,3), (7,6), (7,12), (8,0), (8,6), (8,7),(8,11), (9,2), (9,4), (9,7), (10,4), (10,5), (10,12), (11,3), (11,4), (11,11) } .So far we have determined the values of ¯ α (Γ ,q ) for q ∈ { , , , , } . Combiningthese with the values of ¯ α (Γ ,q ) for q ∈ { , , , , } given in [19], we obtain the results inthe second column of Table 1. In light of (2), these give us upper bounds for i V (Γ ,q ) asneeded in the third column of Table 1. The matching lower bound for i V (Γ ,q ) seems to bedifficult to obtain analytically, and so we run a program to achieve this. Since testing allsubsets takes exponential time, we weaken some of the constraints and give a polynomialtime program for the relaxed problem.Define h : [0 , v ] → [0 , v ] by h ( x ) = ( q + 1) xq + x + x and g : { , , . . . , v } → { , , . . . , v } by g ( x ) = ( x = 0 x (2 m ( x )( q +1) − x +1) m ( x )( m ( x )+1) x = 0 , where m ( x ) = j q + xq +1 k . Theorem 11.
Let q be a prime power and v = q + q + 1 . Then the optimal value of thefollowing program is a lower bound for i V (Γ ,q ) . Furthermore, this problem can be solvedin polynomial time with respect to q .minimize c + da + b subject to a, b, c, d, e, f ∈ { , , . . . , v } (10) a + c + e = b + d + f = v (11) h ( a + b − h − ( a + b )) a + b − ≤ − ¯ α (Γ ,q ) v (12) a ≤ b (13) b + d ≥ g ( a ) (14) a + c ≥ g ( b ) (15) c + e ≥ g ( f ) (16) If e ≥ then qd ≥ a (17) If e = 2 then q ( d + 1) ≥ a + 1) (18) If q = 5 and a = 9 then b + d ≥
25 (19) If a > ¯ α (Γ ,p ) then b + d ≥ v − ¯ α (Γ ,p ) (20)12 roof Let S ⊂ V (Γ ,q ) be such that | S | ≤ v and | N ( S ) | / | S | = i V (Γ ,q ). Let A = S ∩ V , B = S ∩ V , C = N ( S ) ∩ V , D = N ( S ) ∩ V , E = V \ ( S ∪ N ( S )) and F = V \ ( S ∪ N ( S )). Without loss of generality we may assume that | A | ≤ | B | . Let a, b, c, d, e, f be the cardinalities of A, B, C, D, E, F respectively. It suffices to check that all the elevenconditions are satisfied.Conditions (10) and (11) are trivially true. Condition (12) follows from (2) and Theo-rem 4. Condition (13) follows from our assumption that | A | ≤ | B | . Condition (14) followsfrom Lemma 8 and the fact that N ( A ) ⊆ B ∪ D , and conditions (15) and (16) are similar.Note that if e ≥ D for each pair of points in A × E .Each such line in D then contains at most q points in A . So qd ≥ a and condition (17)follows.Condition (18) uses a similar combinatorial argument, but we also take into accountthe fact that the two points in E can have at most one line in D joining them. By countingthe number of 2-arcs from A to E (keeping in mind that all other lines in D contain atmost one point in E ), we obtain the inequality 2( q −
1) + q ( d − ≥ a .Condition (19) is covered in [19, Section 4.3.1]. Finally, condition (20) follows fromthe definition of ¯ α and the fact that N ( A ) ⊆ B ∪ D .We obtain the optimal value in polynomial time (with respect to q ) by enumeratingall ( q + q + 1) combinations of a, b, c, d, e, f . (cid:3) By running the program in Theorem 11 for each prime power q ≤
16 (see the appendixfor the MAGMA code), we obtain a lower bound for i V (Γ ,q ), which turns out to be exactlythe same as the upper bound obtained from ¯ α (Γ ,q ) via (2). Therefore, for such q thethird column of Table 1 gives the exact values of i V (Γ ,q ) and (5) holds. This completesthe proof of Theorem 2. A MAGMA Code
The following code solves the program in Theorem 11 by brute forcing through the entiresample space: for tup in [<2,2>, <3,3>, <4,6>, <5,7>, <7,13>,<8,16>, <9,19>, <11,28>, <13,36>, <16,52>] doq := tup[1];alph := tup[2];v := q^2 + q + 1;upperbound := 1 - alph/v;f := func
S. Zhou was supported by a Future Fellowship (FT110100629)of the Australian Research Council.
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