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Keywords:

  • symmetric graph;
  • arc-transitive graph;
  • Hermitian unital;
  • unitary graph;
  • degree-diameter problem

Abstract

  1. Top of page
  2. Abstract
  3. 1. INTRODUCTION
  4. 2. UNITARY GRAPHS
  5. 3. PROOF OF THEOREMS AND
  6. 4. REMARKS
  7. ACKNOWLEDGMENTS
  8. REFERENCES

Unitary graphs are arc-transitive graphs with vertices the flags of Hermitian unitals and edges defined by certain elements of the underlying finite fields. They played a significant role in a recent classification of a class of arc-transitive graphs that admit an automorphism group acting imprimitively on the vertices. In this article, we prove that all unitary graphs are connected of diameter two and girth three. Based on this, we obtain, for any prime power inline image, a lower bound of order inline image on the maximum number of vertices in an arc-transitive graph of degree inline image and diameter two.

1. INTRODUCTION

  1. Top of page
  2. Abstract
  3. 1. INTRODUCTION
  4. 2. UNITARY GRAPHS
  5. 3. PROOF OF THEOREMS AND
  6. 4. REMARKS
  7. ACKNOWLEDGMENTS
  8. REFERENCES

We study a family of arc-transitive graphs [3] associated with Hermitian unitals. Such graphs are called unitary graphs [6] due to their connections with unitary groups of degree three over a Galois field. The vertices of a unitary graph are the flags of a Hermitian unital, and the adjacency relation is determined by two linear equations defining the line-components of the flags involved. Unitary graphs played an important role in a recent classification [6] of a class of arc-transitive graphs that admit an automorphism group acting imprimitively on the vertices. (A graph is arc-transitive if its automorphism group is transitive on the set of ordered pairs of adjacent vertices.) With focus on combinatorial aspects of unitary graphs, in the present article, we prove that all unitary graphs are connected with large order (compared with their degrees), small diameter and small girth. Based on this, we then obtain, for any prime power inline image, a lower bound on the maximum order (number of vertices) of an arc-transitive graph of degree inline image and diameter two.

The distance between two vertices in a graph is the length of a shortest path joining them, and ∞ if there is no path between the two vertices. The diameter of a graph is the maximum distance between two vertices in the graph. The girth of a graph is the length of a shortest cycle, and ∞ if the graph contains no cycle at all. Two vertices are neighbors of each other if they are adjacent in the graph.

Denote by

  • display math(1)

the Frobenius map for the Galois field inline image, where p is a prime and inline image is a power of p. We postpone the definition of the unitary graph inline image and the inline image-invariant partition inline image of its vertex set to the next section (see Definition 3 and (6), respectively). The following is the first main result of this article.

Theorem 1. Let inline image be a prime power and inline image a divisor of 2e. Let inline image be such that inline image belongs to the inline image-orbit on inline image containing λ, and let inline image denote the size of this inline image-orbit. Then the unitary graph inline image is connected of diameter two and girth three. Moreover, the following hold for inline image:

  1. any two vertices in different blocks of inline image have at least inline image common neighbors;
  2. any two vertices in the same block of inline image have exactly inline image common neighbors.

Given integers inline image, the well-known degree-diameter problem [11] asks for finding the maximum order inline image of a graph of maximum degree Δ and diameter at most D together with the corresponding extremal graphs. Denote by inline image the maximum order of an arc-transitive graph of degree Δ and diameter at most D. Based on Theorem 1, we obtain the following lower bound on inline image.

Theorem 2. For any prime power inline image,

  • display math(2)

In particular, for inline image,

  • display math(3)

As far as we know, these bounds are the first general lower bounds for the arc-transitive version of the degree-diameter problem, despite the fact that a huge amount of work has been done [11] on this problem for general graphs and its restrictions to several other graph classes (e.g., bipartite graphs, vertex-transitive graphs, Cayley graphs). The reader may compare (3) with the well-known Moore bound inline image (for general graphs) and consult [11] for the state-of-the-art of the degree-diameter problem.

The extremal graphs that prove (2) form a subfamily of the family of unitary graphs as we will see in the proof of Theorem 2. The smallest unitary graphs arise when inline image, and in this case (2) gives inline image. Our graphs are constructed from Hermitian unitals, which are well-known doubly point-transitive linear spaces. In this regard, we would like to mention that some efforts have been made to construct graphs using certain finite geometries that give good bounds for the vertex-transitive version of the degree-diameter problem; see [1, 9] for example.

We will give the definition of the unitary graph inline image and related concepts in the next section. The proof of Theorems 1 and 2 together with some preparatory results will be given in Section 'PROOF OF THEOREMS 1 AND 2'. We conclude the article with remarks on Theorem 2 and related questions on the order of inline image.

2. UNITARY GRAPHS

  1. Top of page
  2. Abstract
  3. 1. INTRODUCTION
  4. 2. UNITARY GRAPHS
  5. 3. PROOF OF THEOREMS AND
  6. 4. REMARKS
  7. ACKNOWLEDGMENTS
  8. REFERENCES

In order to make this article reasonably self-contained, we first gather basic definitions and results on unitary groups and Hermitian unitals. After this we will give the definition of a unitary graph. The reader is referred to [5, 8, 12, 13] for more information on unitary groups and Hermitian unitals, and to [5] for undefined terminology on permutation groups.

Let inline image with p a prime. The mapping inline image is an automorphism of the Galois field inline image. The Galois field inline image is then the fixed field of this automorphism. Let inline image be a three-dimensional vector space over inline image and inline image a nondegenerate σ-Hermitian form (that is, β is sesquilinear such that inline image and inline image). The full unitary group inline image consists of those semilinear transformations of inline image that induce a collineation of PG(2, q2) which commutes with β. The general unitary group inline image is the group of nonsingular linear transformations of inline image leaving β invariant. The projective unitary group PGU(3, q) is the quotient group inline image, where inline image is the center of GU(3, q) and I the identity transformation. The special projective unitary group PSU(3, q) is the quotient group inline image, where SU(3, q) is the subgroup of GU(3, q) consisting of linear transformations of unit determinant. PSU(3, q) is equal to PGU(3, q) if 3 is not a divisor of inline image, and is a subgroup of PGU(3, q) of index 3 otherwise. It is well known that the automorphism group of PSU(3, q) is given by the semidirect product inline image, where ψ is the Frobenius map as defined in (1).

Choosing an appropriate basis for inline image allows us to identify vectors of inline image with their coordinates and express the corresponding Hermitian matrix of β by

  • display math

Thus, for inline image,

  • display math

If inline image, then u1 and u2 are called orthogonal (with respect to β). A vector inline image is called isotropic if it is orthogonal to itself, that is, inline image, and nonisotropic otherwise. Let

  • display math

be the set of one-dimensional (1D) subspaces of inline image spanned by its isotropic vectors. Hereinafter inline image denotes the 1D subspace of inline image spanned by inline image. The elements of X are called the absolute points. It is well known that inline image, PSU(3, q) is 2-transitive on X, and inline image leaves X invariant.

If u1 and u2 are isotropic, then the vector subspace inline image of inline image spanned by them contains exactly inline image absolute points. The Hermitian unital inline image is the block design [8] with point set X in which a subset of X is a block (called a line) precisely when it is the set of absolute points contained in some inline image. It is well known [12, 13] that inline image is a linear space with inline image points, inline image lines, inline image points in each line, and q2 lines meeting at a point. (A linear space [2] is an incidence structure of points and lines such that any point is incident with at least two lines, any line with at least two points, and any two points are incident with exactly one line.) It was proved in [12, 13] that inline image. Thus, for every G with inline image, inline image is a G-doubly point-transitive linear space. This implies that G is also block-transitive and flag-transitive on inline image, where a flag is an incident point-line pair.

A line of PG(2, q2) contains either one absolute point or inline image absolute points. In the latter case, the set of such inline image absolute points is a line of inline image; all lines of inline image are of this form. So we can represent a line of inline image by the homogenous equation of the corresponding line of PG(2, q2).

Denote

  • display math

Definition 3. ([6]) Let inline image be a prime power and inline image a divisor of 2e. Suppose inline image is such that inline image belongs to the inline image-orbit on inline image containing λ. The unitary graph inline image is defined to be the graph with vertex set inline image such that inline image, inline image are adjacent if and only if L1 and L2 are given by:

  • display math(4)
  • display math(5)

for an integer inline image and a nonisotropic inline image orthogonal to both inline image and inline image.

The requirement on λ is equivalent to that inline image for at least one integer inline image. (But inline image is independent of the choice of t.) This ensures that inline image is well defined as an undirected graph. In fact, since r is a divisor of 2e, we have inline image for some integer j. Since inline image, the equations of L1 and L2 can be rewritten as

  • display math

Hence, the adjacency relation of inline image is symmetric.

Define

  • display math

where inline image is the stabilizer of λ in inline image. Then inline image is the size of the inline image-orbit on inline image containing λ, or the least integer inline image such that inline image. Of course inline image is a divisor of inline image.

Denote by inline image the set of flags of inline image with point-entry inline image. Then

  • display math(6)

is a partition of inline image into inline image blocks each with size q2.

Denote by inline image the unique line of inline image through two distinct points inline image. Denote

  • display math
  • display math

Then inline image and inline image.

An arc of a graph is an ordered pair of adjacent vertices. A graph Γ is G-arc transitive if inline image is transitive on the set of vertices of Γ and also transitive on the set of arcs of Γ. This is to say that any arc of Γ can be mapped to any other arc of Γ by an element of G, and the same statement holds for vertices. A partition inline image of the vertex set of Γ is G-invariant if for any block inline image and inline image the image of P under g, inline image, is a block of inline image, where inline image is the image of σ under g. The quotient graph inline image is the graph with vertex set inline image such that inline image are adjacent if and only if there is at least one edge of Γ between P and Q. If for any two adjacent inline image, all vertices of P except only one have neighbors in Q in the graph Γ, then Γ is called an almost multicover [6] of inline image. (Since Γ is G-arc transitive, if all vertices of P except one have neighbors in Q, then all vertices of Q except one have neighbors in P, and the subgraph of Γ induced by inline image with these two exceptional vertices deleted, is a regular bipartite graph.)

Unitary graphs were introduced in [6] during the classification of a class of imprimitive arc-transitive graphs. A major step toward this classification is the following result which will be used in our proof of Theorem 1.

Theorem 4. ([6]) inline image is a inline image-arc transitive graph of degree inline image (where inline image) that admits inline image as a inline image-invariant partition such that the quotient graph inline image is a complete graph and inline image is an almost multicover of inline image. Moreover, for each pair of distinct points inline image of inline image, inline image is the only vertex in inline image that has no neighbor in inline image.

3. PROOF OF THEOREMS 1 AND 2

  1. Top of page
  2. Abstract
  3. 1. INTRODUCTION
  4. 2. UNITARY GRAPHS
  5. 3. PROOF OF THEOREMS AND
  6. 4. REMARKS
  7. ACKNOWLEDGMENTS
  8. REFERENCES

Throughout this section, we denote

  • display math

We need the following two lemmas in the proof of Theorem 1.

Lemma 5.
  1. inline image is adjacent to inline image in Γ if and only if there exist inline image, inline image, inline image, and inline image with inline image, such that

    1. inline image satisfies inline image, and inline image;
    2. L2 is given by
      • display math(7)
  1. inline image is adjacent to inline image if and only if there exist inline image and inline image with inline image such that
    1. inline image satisfies inline image, and inline image;
    2. L2 is given by
      • display math(8)
Proof.
  1. Denote inline image. Then inline image is adjacent to inline image if and only if there exist an integer inline image and a nonisotropic inline image orthogonal to both u1 and u2 such that L and L2 are given by (4) and (5), respectively. It is clear that (4) gives inline image if and only if inline image. Since inline image are orthogonal, we have inline image and so inline image. Using this and the assumption that u0 is nonisotropic, we obtain inline image. Since inline image are orthogonal, we then have inline image. Since u2 is isotropic, we have inline image. Setting inline image, inline image and inline image, we have inline image, inline image, inline image, inline image and inline image. One can check that L2 given by (5) is exactly inline image as shown in (7). Conversely, if these conditions are satisfied, then inline image is adjacent to inline image.
  2. Let inline image. Then inline image is adjacent to inline image if and only if there exist an integer inline image and a nonisotropic inline image orthogonal to both u1 and u2 such that inline image and L2 are given by (4) and (5), respectively. Since u0 and u1 are orthogonal, we have inline image. Since u0 is nonisotropic, we then have inline image. One can see that (4) becomes inline image, which gives inline image if and only if inline image and inline image. Since u0 and u2 are orthogonal, we have inline image and so inline image. Since u2 is isotropic, we have inline image. Set inline image, inline image, and inline image. Then inline image, inline image, inline image, and (5) can be simplified to give (8).

It is known that every line of inline image through 0 other than inline image is of the form:

  • display math

Lemma 6.

  1. inline image is adjacent to inline image if and only if there exist inline image, inline image, and inline image with inline image, such that
    1. inline image satisfies inline image, and inline image;
    2. L2 is given by
      • display math(9)
  1. inline image is adjacent to inline image if and only if there exist inline image and inline image with inline image such that
    1. inline image satisfies inline image, and inline image;
    2. L2 is given by
      • display math(10)
Proof.
  1. Denote inline image. Then inline image is adjacent to inline image if and only if there exist an integer inline image and a nonisotropic inline image orthogonal to both u1 and u2 such that inline image and L2 are given by (4) and (5), respectively. Since inline image are orthogonal, we have inline image. Using this and the fact that u0 is nonisotropic, we get inline image. One can see that (4) becomes inline image, which gives inline image if and only if inline image and inline image. Since inline image are orthogonal, we have inline image and hence inline image. Since u2 is isotropic, we have inline image. Setting inline image, inline image, and inline image, we have inline image, inline image, inline image, inline image, inline image, inline image, and L2 given by (5) is exactly inline image in (9).
  2. Let u0 and u1 be as above. As in (a), we have inline image and inline image. One can see that (4) becomes inline image, which gives inline image if and only if inline image and inline image. Since inline image are orthogonal, we have inline image. Set inline image, inline image, and inline image. Then inline image and inline image since inline image is isotropic. Now L2 given by (5) is exactly inline image in (10).

For inline image, denote

  • display math

In other words, inline image is the set of vertices of Γ adjacent to inline image. Note that inline image, inline image and in general inline image. Lemmas 5(a) and 6(a) imply:

Corollary 7. We have

  • display math
  • display math

In particular, inline image and (0, N) are adjacent in Γ. Moreover, for distinct inline image, any vertex inline image other than inline image has exactly k neighbors in inline image.

The last statement follows from the fact that inline image, G is 2-transitive on X, and inline image is transitive on inline image. Here and in the following, inline image denotes the point-wise stabilizer of inline image in G, that is, the subgroup of G consisting of those elements of G which fix both ∞ and 0.

Proof of Theorem 1. The statements in (a)–(b) can be restated as follows.

  1. inline image, for any distinct inline image and any inline image;
  2. inline image, for any inline image and inline image with inline image.

Proof of (a). Since G is 2-transitive on X, it suffices to prove (a) for inline image and inline image. Noting that inline image, we have three possibilities to consider.

Case 1. inline image.

Since Γ is G-arc transitive and inline image is the only vertex of inline image not adjacent to any vertex of B(0) (Theorem 4), inline image is transitive on inline image. So it suffices to prove inline image for any inline image in this case.

By Lemmas 5(a) and 6(a), a vertex inline image is adjacent to both inline image and inline image if and only if there exist inline image, inline image with inline image and inline image such that inline image, inline image, and inline image. From these relations, we have inline image, inline image, inline image. Thus, the equation of inline image as given in (9) becomes

  • display math

This equation gives inline image (see (7)) if and only if inline image (which implies inline image as inline image) and inline image, or equivalently

  • display math(11)

Since inline image, the coefficient of c here is equal to zero if and only if b satisfies a quadratic equation, which has at most two solutions. Since for any inline image, and inline image, the equation inline image about b has inline image solutions, there are at least inline image values of b that satisfy inline image and inline image. Each such tuple inline image determines a unique c via (11) and hence a unique common neighbor of inline image and inline image. Moreover, since inline image, for different pairs inline image the vertices inline image belong to different blocks of inline image and so are distinct. Therefore, inline image.

Case 2. inline image but inline image.

It suffices to prove inline image for any inline image. By Lemmas 5(b) and 6(a), a vertex inline image is adjacent to both inline image and inline image if and only if there exist inline image, inline image with inline image and inline image with inline image such that inline image, inline image, inline image, and inline image. From these relations, we have inline image (which implies inline image as inline image), inline image, and inline image. Plugging these into (9), the equation of inline image becomes

  • display math

This equation gives inline image (see (8)) if and only if inline image and inline image, that is,

  • display math

The remaining proof is similar to Case 2 above.

Case 3. inline image.

In this case, we are required to prove inline image. By Lemmas 5(b) and 6(b), a vertex inline image is adjacent to both inline image and inline image if and only if there exist inline image, inline image with inline image and inline image such that inline image, inline image, inline image, and inline image. From these relations, we have inline image, and inline image. Plugging these into (10), the equation of inline image becomes

  • display math

This is identical to inline image (see (8)) if and only if inline image and inline image, that is,

  • display math

The rest of the proof is similar to Case 2 above.

Proof of (b): Since Γ is G-vertex transitive, it suffices to prove inline image for distinct inline image.

Consider inline image first, where inline image are distinct. By Lemma 6(a), a vertex inline image is in both inline image and inline image if and only if there exist inline image, inline image, and inline image with inline image such that inline image satisfies inline image, and inline image, for inline image, and inline image. Thus, inline image, inline image, and inline image. Note that inline image implies inline image. Using these relations, the equation of inline image (see (9)) can be simplified to

  • display math

This gives the equation of inline image (see (9)) if and only if inline image, or equivalently

  • display math

Here, we note that inline image for inline image. Since inline image, the right-hand side of this expression is neither 1 nor inline image. Thus there are inline image possible choices of f1, and each of them corresponds to exactly q values of g1 by inline image. It follows that inline image.

It remains to prove inline image for any inline image. By Lemma 6, a vertex inline image is in both inline image and inline image if and only if there exist inline image, inline image with inline image, and inline image, inline image with inline image, such that inline image satisfies inline image and inline image, and inline image. Thus inline image, inline image, inline image, and so inline image as inline image. Using these relations and (9), the equation of inline image can be simplified to

  • display math

One can see that this gives inline image (see (10)) if and only if

  • display math

Note that the right-hand side of this equation is neither 0 nor inline image. Similarly as in the previous paragraph, we obtain inline image.

So far we have completed the proof of (a) and (b).

Note that Γ is not a complete graph since, for example, inline image and inline image are not adjacent. Since inline image, by (a) the distance in Γ between any two nonadjacent vertices is equal to two. So Γ has diameter two. Since (0, N) and inline image are adjacent by Corollary 7 and they have at least one common neighbor by (a), Γ has girth three.

Proof of Theorem 2. Let inline image. Choose inline image and inline image. It is trivial that inline image (inline image) is in the inline image-orbit containing λ. Hence inline image is well-defined, and is connected of diameter two by Theorem 1. The assumption inline image ensures inline image and so inline image. Thus, by Theorem 4, inline image has order inline image and degree inline image. From this (2) follows immediately.

Now for inline image, we have inline image. Thus, inline image as claimed in (3).

4. REMARKS

  1. Top of page
  2. Abstract
  3. 1. INTRODUCTION
  4. 2. UNITARY GRAPHS
  5. 3. PROOF OF THEOREMS AND
  6. 4. REMARKS
  7. ACKNOWLEDGMENTS
  8. REFERENCES

In the case when inline image, the well-known Moore bound [11] gives inline image for any Δ. The equality holds only when inline image and possibly 57, and for all other Δ we have inline image (see [11]). It is known [4] that inline image for every Δ such that inline image is a prime. It is proved in [10] that the counterpart inline image of inline image for vertex-transitive graphs satisfies inline image if inline image, where q is a prime power congruent to 1 modulo 4. This bound came with the discovery [10] of an infinite family of vertex-transitive graphs inline image (now well known as the McKay–Miller–Širáň graphs) with degree inline image and order inline image. Since, as implied in [7], Definition 11, Lemma 17], such extremal graphs cannot be arc-transitive except for the Hoffman–Singleton graph H5, the same bound may not apply to inline image.

In view of (3) and the comments above, it is natural to ask whether there exist infinitely many inline image such that inline image for some constant inline image. One may also ask whether there exists a constant inline image such that inline image for all inline image. However, this would not make much sense unless the same question for inline image has an affirmative answer which, to the best of our knowledge, is unknown at present.

Note added in proof: Answering the first question above, the author noted the following recently: For every even integer inline image. See [S. Zhou, A note on the degree-diameter problem for arc-transitive graphs, Bulletin of the ICA, to appear].

ACKNOWLEDGMENTS

  1. Top of page
  2. Abstract
  3. 1. INTRODUCTION
  4. 2. UNITARY GRAPHS
  5. 3. PROOF OF THEOREMS AND
  6. 4. REMARKS
  7. ACKNOWLEDGMENTS
  8. REFERENCES

We appreciate Dr. Guillermo Pineda-Villavicencio for helpful discussions on the degree-diameter problem. The author was supported by a Future Fellowship (FT110100629) of the Australian Research Council.

REFERENCES

  1. Top of page
  2. Abstract
  3. 1. INTRODUCTION
  4. 2. UNITARY GRAPHS
  5. 3. PROOF OF THEOREMS AND
  6. 4. REMARKS
  7. ACKNOWLEDGMENTS
  8. REFERENCES
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