The Distant Graph of the Projective Line Over a Finite Ring with Unity

We discuss the projective line P(R)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {P}(R)$$\end{document} over a finite associative ring with unity. P(R)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {P}(R)$$\end{document} is naturally endowed with the symmetric and anti-reflexive relation “distant”. We study the graph of this relation on P(R)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {P}(R)$$\end{document} and classify up to isomorphism all distant graphs G(R,Δ)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$G(R, \Delta )$$\end{document} for rings R up to order p5\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p^5$$\end{document}, p prime.


Introduction
The aim of this paper is to characterize the distant graph G(R, Δ) of the projective line over any finite ring R. It is an undirected, connected graph with the degree of a vertex equal to |R|. The starting point of our investigation is showing the connection between this graph and the distant graph G(R/J, Δ J ) of the projective line over the factor ring R/J, where J is the Jacobson radical of R. To this end we use, introduced by Blunck and Havlicek in [9], an equivalence relation, called radical parallelism, on the set of points of the projective line, which determines the interdependence between P(R) and P(R/J). Next we describe the graph G(R/J, Δ J ). Using structures theorems [2] on finite rings with unity we get that the graph G(R/J, Δ J ) is isomorphic to the tensor product of the distant graphs arising from projective lines whose underlying rings are full matrix rings over finite fields. The projective line over any full matrix ring M n (q), i.e. the ring of n × n matrices over the finite field F (q) of order q, is in bijective correspondence with the Grassmannian G (n, 2n, q) of n-dimensional subspaces of a 2n-dimensional vector space over F (q). Then we describe G(M 2 (q), Δ) for only if, they are distant. This graph is connected and its diameter is less or equal 2 [16, 1.4

.2. Proposition].
One of the basic concepts of graph theory is that of a clique. A clique in an undirected graph G is a subset of the vertices such that any two distinct vertices comprise an edge, i.e. the subgraph of G induced by these vertices is complete. A maximum clique of a graph G is a clique, such that there is no clique in G with more vertices. A maximal clique is a clique which is not properly contained in any clique.
All maximal cliques in the distant graph G(R, Δ) induce its isomorphic subgraphs, which is due to the fact that the group GL 2 (R) acts transitively on P(R). Hence all maximal cliques in the distant graph G(R, Δ) have the same number of vertices, denoted by ω G(R, Δ) , and at the same time, they are maximum cliques. To describe maximal cliques of the distant graph we make use of the following definitions.
Definition 3. An (n − 1)-spread in the (2n − 1)-dimensional projective space P G(2n − 1, q) over the finite field with q elements F (q) is a set of (n − 1)dimensional subspaces such that each point of P G(2n − 1, q) is contained in exactly one element of this set.
In our considerations we will often use the following fact about the direct product of projective lines.
In [1] was pointed out that in another way to state this is to say where Δ i stands for the distant relation on P(R i ). It means that G(R, Δ) is the tensor product of the graphs G(R 1 , Δ 1 ), G(R 2 , Δ 2 ), . . . , G(R n , Δ n ), i.e., the vertex set of G(R, Δ) is the Cartesian product of vertex sets of the graphs G(R 1 , Δ 1 ), G(R 2 , Δ 2 ), . . . , G(R n , Δ n ), and for all elements (x 1 , x 2 , . . . , x n ), (x 1 , x 2 , . . . , x n ) ∈ G(R, Δ) holds If a ring R cannot be written as R R 1 × R 2 , where R 1 , R 2 are nonzero rings, then it is called indecomposable ring.

Construction of the Distant Graph on the Projective Line
In order to describe the distant graph G(R, Δ) of the projective line over a ring R we show the connection between this graph and the distant graph G(R/J, Δ J ) of the projective line over the factor ring R/J. Next we find the graph G(R/J, Δ J ). The points of P(R/J) are in one-one correspondence with the equivalence classes of the radical parallelism relation on P(R). Each of these comprises |J| elements. See [9] for more details. Write Proof. This follows from Remark 1 and the fact that Φ −1 R/J(a, b) contains exactly |J| points for any R/J(a, b).

Figure 1. The connection between the distant graphs
Proof. Suppose that the above assumptions are satisfied. We have We first give the proof for the case n = 2. Without loss of generality, assume that By the definition of the tensor product of graphs G 1 , G 2 we get that vertices (k 1 l1 , k 2 l2 ), (k 1 ) in G 1 × G 2 are joined by an edge if, and only if, are elements of the clique K i ti , and so they are joined by an edge if, and only if, l i = l i . Therefore vertices written down in rows of the above sets forming edges with all elements of some vertex-disjoint maximal clique of K 1 Applying the induction we get the claim.
Proof. " ⇒" This is straightforward from [7, Corollary 6.8]. " ⇐" An isomorphism or an anti-isomorphism α i : Any finite ring with identity is semiperfect. By the structure theorem of such rings [2] R/J is artinian semisimple and idempotents lift modulo J. Hence it has a unique decomposition into a direct product of simple rings: According to Theorem 1 we get Any simple ring R i is isomorphic to a full matrix ring M ni (q i ) over the finite field with q i elements: It follows then that the description of the projective line over any finite ring can be based on the projective line over the full matrix ring P(M n (q)). There is a bijection between P(M n (q)) and the Grassmannian G (n, 2n, q), i.e. the set of all n-dimensional subspaces of V (2n, q) [4,2.4 Theorem.]. Consequently, any point of P(M n (q)) can be expressed by using of a basis of the corresponding n-dimensional subspace of V (2n, q). .
The distant graph of the projective line over the full matrix ring G(M n (q), Δ), is isomorphic to the graph on G (n, 2n, q) whose vertex set is G (n, 2n, q) and whose edges are pairs of complementary subspaces X, Y ∈ G (n, 2n, q): Another graph on G (n, 2n, q) is the well known Grassmann graph, which has the same set of vertices as the distant graph but X, Y ∈ G (n, 2n, q) form an edge, whenever both X and Y have codimension 1 in X + Y , i.e. they are adjacent (in symbols: ∼ ): G(M n (q), Δ) can be described using the notion of the Grassmann graph [6, Theorem 3.2]. These graphs have been thoroughly investigated by many authors, for example in [5,6,20]. In [19] finite Grassmann graphs are uniquely determined as distance-regular graphs, however, this special case of the Grassmann graph G (n, 2n, q) is not characterized. We can give the number of vertices of G(M n (q), Δ) (cf. [15, p. 920]), i.e. the number of n-dimensional subspaces of V (2n, q): The degree of a vertex v ∈ G(M n (q), Δ) is equal to the number of n-dimensional subspaces of V (2n, q) that are disjoint to any given n-dimensional subspace: Vol. 72 (2017) The Distant Graph of the Projective Line 1951 It means that deg(v) = |M n (q)| and generally if v ∈ G(R, Δ) then deg(υ) = |R|, which is also due to the fact that GL 2 (R) acts transitively on P(R). Maximal cliques in G(M n (q), Δ) correspond to n-spreads in the 2n-dimensional vector space over F (q). It is known that such an n-spread contains q n + 1 ndimensional vector subspaces. Any partition of the vertex set V G(M n (q), Δ) into maximal cliques corresponds to an n-parallelism of the vector space V (2n, q). Therefore and on account of [3, Theorem 1.], which has been also proved (independently) by Denniston [13], there exists a partition of the set V G(M 2 (q), Δ) into q 2 +q +1 maximal cliques with ω G(M 2 (q), Δ) = q 2 +1 for any q.
We pay attention now to the distant graph G(M 2 (2), Δ) which has 35 vertices. We can identify the graph G M n (q), Δ and the corresponding Grassmannian G (n, 2n, q). Then all automorphisms of the distant graph G(M 2 (2), Δ) are linear or superpositions of linear with the automorphisms defined by duality and annihilator mapping; see [20]. Automorphisms of the first type fix the two orbits of partitions and those of the second type exchange them. Below we write down one partition from each orbit. In both tables (Tables 1 and  2) the seven members of the partition are maximal cliques of size five, which are labelled as I, II, . . . , VII. Thereby each point of the graph G(M 2 (2), Δ) is described in terms of two basis vectors of its corresponding subspace in G (2, 4, 2).
We study now cliques formed by vertices of any two maximal cliques of the first partition (Table 1). We are interested only in cliques containing vertices of both maximal cliques. We see that there exists exactly one such maximum clique with four elements for any two different maximal cliques. As an example, we show edges formed by vertices of cliques I and II (Fig. 2).  So, for any two of three vertex-disjoint maximal cliques we have one maximum clique and we checked that three such maximum cliques are of two distinct kinds: either any two of them have one common vertex (Fig. 3) or they are pairwise disjoint (Fig. 4). The same result can be drawn for the second partition.
By direct verification we found that the maximal cliques I, II, . . . , VII can be seen as the points of the projective plane of order 2, where three points form a line of this plane if, and only if, the three maximal cliques are of the second kind (Fig. 5).
Thus we get a simple alternative construction of the Fano plane described by Hirschfeld in [17,Theorem 17.5.6] in projective geometry language.
It is not known whether there exists a partition of the vertex set of any graph G(M n (q), Δ). But this problem is well known as the problem about the existence of an n-parallelism in combinatorial designs. Sarmiento in [21] described the partition of the design corresponding to that of V G(M 3 (2), Δ) .

The Classification of Distant Graphs
We start with a characterization of the distant graph of the projective line over a finite local ring.  Any finite commutative ring is the direct product of local rings [18,VI.2]. Thus the distant graph of the projective line over any finite commutative ring is known by the above and Theorem 1. Every finite ring is isomorphic to the direct product of rings of prime power order [18, I.1]. Hence the distant graph of the projective line over a finite ring can be also described as the tensor product of the distant graphs of the projective lines over rings of prime power order. We classify below distant graphs G(R, Δ), where R is an indecomposable ring up to order p 5 , p prime. We use some facts and the notations that were established in [12]. Namely, any finite ring can be represented as R = S ⊕ M , where S = m i=1 R i , R i are primary rings and M is a bimodule over the ring S. M is also an additive subgroup of J, so M ⊆ J and we thus get R/J S/J. In the case of p, p 2 for any p we get complete graphs of order p + 1, p 2 + 1 and the graph G(R, Δ), where R is local and |R| = p 2 , |J| = p. If |R| = p 3 then we have three graphs G(R, Δ) for any p: the complete graph of order p 3 + 1, the graph where R is local and |J| = p 2 and the graph of the projective line over the ring of lower triangular 2 × 2-matrices over a field F (p), which is a sum of p 2 + p vertex-disjoint maximal cliques with ω G(R, Δ) = p + 1.