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### Keywords:

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

### Abstract

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 , a lower bound of order on the maximum number of vertices in an arc-transitive graph of degree and diameter two.

### 1. INTRODUCTION

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 , a lower bound on the maximum order (number of vertices) of an arc-transitive graph of degree 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

• (1)

the Frobenius map for the Galois field , where p is a prime and is a power of p. We postpone the definition of the unitary graph and the -invariant partition 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 be a prime power and a divisor of 2e. Let be such that belongs to the -orbit on containing λ, and let denote the size of this -orbit. Then the unitary graph is connected of diameter two and girth three. Moreover, the following hold for :

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

Given integers , the well-known degree-diameter problem [11] asks for finding the maximum order of a graph of maximum degree Δ and diameter at most D together with the corresponding extremal graphs. Denote by 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 .

Theorem 2. For any prime power ,

• (2)

In particular, for ,

• (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 (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 , and in this case (2) gives . 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 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 .

### 2. UNITARY GRAPHS

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 with p a prime. The mapping is an automorphism of the Galois field . The Galois field is then the fixed field of this automorphism. Let be a three-dimensional vector space over and a nondegenerate σ-Hermitian form (that is, β is sesquilinear such that and ). The full unitary group consists of those semilinear transformations of that induce a collineation of PG(2, q2) which commutes with β. The general unitary group is the group of nonsingular linear transformations of leaving β invariant. The projective unitary group PGU(3, q) is the quotient group , where is the center of GU(3, q) and I the identity transformation. The special projective unitary group PSU(3, q) is the quotient group , 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 , 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 , where ψ is the Frobenius map as defined in (1).

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

Thus, for ,

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

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

If u1 and u2 are isotropic, then the vector subspace of spanned by them contains exactly absolute points. The Hermitian unital 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 . It is well known [12, 13] that is a linear space with points, lines, 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 . Thus, for every G with , is a G-doubly point-transitive linear space. This implies that G is also block-transitive and flag-transitive on , where a flag is an incident point-line pair.

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

Denote

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

• (4)
• (5)

for an integer and a nonisotropic orthogonal to both and .

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

Hence, the adjacency relation of is symmetric.

Define

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

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

• (6)

is a partition of into blocks each with size q2.

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

Then and .

An arc of a graph is an ordered pair of adjacent vertices. A graph Γ is G-arc transitive if 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 of the vertex set of Γ is G-invariant if for any block and the image of P under g, , is a block of , where is the image of σ under g. The quotient graph is the graph with vertex set such that are adjacent if and only if there is at least one edge of Γ between P and Q. If for any two adjacent , all vertices of P except only one have neighbors in Q in the graph Γ, then Γ is called an almost multicover [6] of . (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 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]) is a -arc transitive graph of degree (where ) that admits as a -invariant partition such that the quotient graph is a complete graph and is an almost multicover of . Moreover, for each pair of distinct points of , is the only vertex in that has no neighbor in .

### 3. PROOF OF THEOREMS 1 AND 2

Throughout this section, we denote

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

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

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

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

Lemma 6.

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

For , denote

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

Corollary 7. We have

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

The last statement follows from the fact that , G is 2-transitive on X, and is transitive on . Here and in the following, denotes the point-wise stabilizer of 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. , for any distinct and any ;
2. , for any and with .

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

Case 1. .

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

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

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

• (11)

Since , 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 , and , the equation about b has solutions, there are at least values of b that satisfy and . Each such tuple determines a unique c via (11) and hence a unique common neighbor of and . Moreover, since , for different pairs the vertices belong to different blocks of and so are distinct. Therefore, .

Case 2. but .

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

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

The remaining proof is similar to Case 2 above.

Case 3. .

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

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

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

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

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

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

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

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

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

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

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

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

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

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

### 4. REMARKS

In the case when , the well-known Moore bound [11] gives for any Δ. The equality holds only when and possibly 57, and for all other Δ we have (see [11]). It is known [4] that for every Δ such that is a prime. It is proved in [10] that the counterpart of for vertex-transitive graphs satisfies if , 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 (now well known as the McKay–Miller–Širáň graphs) with degree and order . 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 .

In view of (3) and the comments above, it is natural to ask whether there exist infinitely many such that for some constant . One may also ask whether there exists a constant such that for all . However, this would not make much sense unless the same question for 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 . See [S. Zhou, A note on the degree-diameter problem for arc-transitive graphs, Bulletin of the ICA, to appear].

### ACKNOWLEDGMENTS

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.

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