CC BY
14URAL MATHEMATICAL JOURNAL, Vol. 7, No. 2, 2021, pp. 43-50
DOI: 10.15826/umj.2021.2.003
UNIT AND UNITARY CAYLEY GRAPHS FOR THE RING OF EISENSTEIN INTEGERS MODULO n
Reza Jahani-Nezhad", Ali Bahrami""
Department of Pure Mathematics, Faculty of Mathematical Sciences,
University of Kashan, Kashan 87317-53153, I. R. Iran
"jahanian@kashanu.ac.ir, ""alibahrami1972@gmail.com
Abstract: Let En be the ring of Eisenstein integers modulo n. We denote by G(En) and Ggn, the unit graph and the unitary Cayley graph of En, respectively. In this paper, we obtain the value of the diameter, the girth, the clique number and the chromatic number of these graphs. We also prove that for each n > 1, the graphs G(En) and Ge„ are Hamiltonian.
Keywords: Unit graph, Unitary Cayley graph, Eisenstein integers, Hamiltonian graph.
1. Introduction
Associating a graph with an algebraic object is an active research subject in algebraic graph theory, an area of mathematics in which methods of abstract algebra are employed in studying various graph invariants and tools in graph theory are used in studying various properties of the associated algebraic structure. The Cayley graph of a finite group was first considered in 1878 by Arthur Cayley [12]. Research on graphs associated with rings was started in 1988 by I. Beck [10]. In the literature, there are some other graphs associated with rings, such as the Cayley graph of a commutative ring [1], the unitary Cayley graph of a ring [3], the total graph of a ring [5], the zero divisor graph of a ring [6], the unit graph of a ring [7] and the comaximal graph of a ring [16].
Let R be a commutative ring with non-zero identity. We denote by U(R), J(R) and Z(R) the group of units of R, the Jacobson radical of R and the set of zero divisors of R, respectively. The unitary Cayley graph of a ring R, denoted by Or, is the graph whose vertex set is R, and in which {a, b} is an edge if and only if a — b € U(R). In 1995 this graph was initially introduced by Dejter and Giudici [14] for Zn, the ring of integers modulo n. In 2009, Akhtar et al. [3] generalized the unitary Cayley graph GZn to Or for a finite ring R. The unit graph of a ring R, denoted by G(R), is a graph whose vertices are elements of R and two distinct vertices a and b are adjacent if and only if a + b in U(R). In 1990, the unit graph was first investigated by Chung [13] and Grimaldi [16] for Zn. In 2010, Ashrafi, et al. [7] generalized the unit graph G(Zn) to G(R) for an arbitrary ring R. Numerous results about unit and unitary Cayley graphs were obtained, see for examples [3, 7, 18, 19, 21, 22].
The following facts are well known, see for example [4] and [17]. Let u be a primitive third root of unity. Then the set of all complex numbers a+bu, where a and b are integers, forms an Euclidean domain with the usual complex number operations and Euclidean norm N (a + bu) = a2 + b2 — ab. This domain will be denoted by E and will be called the ring of Eisenstein integers. The units of E are ±1, ±w and ±uJ. The primes of E (up to a unit multiple) are the usual prime integers that are congruent to 2 modulo 3 and Eisenstein integers whose norm is a usual prime integer. Let n be a natural number and let (n) be the principal ideal generated by n in E. Then the factor ring
E/(n) is isomorphic to the ring n = (a + bw|a, b € Zn}, where Zn is the ring of integers modulo n. Thus En is a principal ideal ring. This ring is called the ring of Eisenstein integers modulo n. In [4] this ring is studied and its properties are investigated, its units are characterized and counted. It is easy to see that a + bw is a unit in En if and only if N (a + bw) is a unit in Zn. Recall that a ring is local if it has a unique maximal ideal. It is shown that
(1) if p is a prime integer, then the ring Epk is local if and only if p = 3 or p = 2 (mod 3);
(2) let <^(R) denote the number of units in a ring R, then ^>(E3k) = 2 x 32k-1 and
' p2fc-2(p2 - i) if p _ 2 (mod 3),
(pfc - pfc-1)2 if p = i (mod 3).
) =1 , k i-h2
In this article, some properties of the graphs G(En) and GEn are studied. The diameter, the girth, chromatic number and clique number, in terms of n, are found. Also, we prove that for each n > 1, the graphs G(En) and GEn are Hamiltonian and the independence number of GEn is calculated. An earlier study was carried out for the unit and unitary graphs for the ring of Gaussian integers modulo n, see [9].
Throughout the article, by a graph G we mean a fnite undirected graph without loops or multiple edges. If the degree of each vertex in G is equal to k, where k is a positive integer, then G is called k-regular graph. For a graph G and for any two vertices a and b of G, we recall that a walk between a and b is an alternating sequence a = v0, ei, vi, e2, ..., ek, vk = b of vertices and edges of G, denoted by
ei e® ek , ,
a = v0-> v1-> ...-> vk = b,
such that for every i with 1 < i < k, the edge ei has endpoints vi-1 and v^ Also, a path between a and b is a walk between a and b without repeated vertices. A cycle of a graph is a path such that the start and end vertices are the same. The number of edges (counting repeats) in a walk, path or a cycle, is called its length. A Hamiltonian path (cycle) in G is a path (cycle) in G that visits every vertex of G exactly once. A graph is called Hamitonian if it contains a Hamiltonian cycle. For vertices a and b of G, we define d(a, b) to be the length of a shortest path from a to b (d(a, a) = 0 and d(a, b) = to if there is no such path). The diameter of G is
diam (G) = sup{d(a, b) | a, b € V(G)}.
The girth of G, denoted by gr(G) is the length of a shortest cycle in G, (gr(G) = to if G contains no cycle). For a positive integer r, a graph is called r-partite if the vertex set admits a partition into r classes such that vertices in the same partition class are not adjacent. A r-partite graph is called complete if every two vertices in different parts are adjacent. The complete 2-partite graph (also called the complete bipartite graph) with exactly two partitions of size n and m, is denoted by A complete graph on the n vertices, denoted by is a graph such that every two of
distinct vertices are adjacent. A clique in G is a set of pairwise adjacent vertices of G. A clique of the maximum size is called a maximum clique. The clique number of G, denoted by w(G), is the number of vertices of a maximum clique in G. We color the vertices of G so that no two joined vertices have the same color. If we color the vertices, we call it a coloring of G. The chromatic number x(G) of the graph G is the minimum number of colors of colorings of G. The tensor product or Kronecker product G ® H of two graphs G and H is the graph with vertex set V(G) x V(H), in which (a, b) is adjacent to (c, d) if and only if a is adjacent to c in G and b is adjacent to d in H. For other notions not mentioned in this introduction, one can refer to [11, 15].
Throughout this article, the integers p and pi are used implicitly to denote primes congruent to 2 modulo 3, while q and qj likewise denote prime integers congruent to 1 modulo 3. For classical theorems and notations in commutative algebra, the interested reader is referred to [8].
2. The Unit and Unitary Cayley Graphs of En
In this section, we determine diameter and girth of the unit and unitary Cayley graphs of En. The case when n is a power of a prime is considered first. Then the general case is considered.
2.1. The Unit and Unitary Cayley Graphs of Etn
For the sake of compeleteness, we mention here two important results that will be kept throghout the paper.
Proposition 1 [3, Proposition 2.2].
(a) Let R be a ring. Then GR is a regular graph of degree | U(R)|.
(b) Let R be a local ring with the maximal ideal m. Then Gr is a complete mutipartite graph whose partite sets are the cosets of m in R. In paticular, Gr is a compelete graph if and only if R is a field.
Theorem 1 [2, Theorem 3.1]. Let R be a ring. Then G(R) is a complete r-partite graph if and only if R is a local ring with the maximal ideal m and r = \R/m\ = 2'\ for some n € N or R is a finite field.
Theorem 2. Let n be a positive integer. Then the following statements hold
(1) diam(G(Esn))=diam(Ge3n) = 2;
(2) gr(G(Ean))=gr(GE3„ ) = 3.
Proof. For each positive integer n, E3n is a local ring with the maximal ideal (2 + u), see [4]. Since ^(E3n ) = 2 x 32n-1, we have
= 3.
(2 + u)
Therefore, by Proposition 1, GE3n is a complete 3-partite graph and hence diam(GE3n ) = 2 and gr(GE3n ) = 3. Also, by Theorem 1, G(E3n/(2 + u)) is a complete bipartite graph. Thus diam (G(E3n )) = 2 and gr (G(E3n )) =3. □
Theorem 3. Let n be a positive integer and q be a prime integer congruent to 1 modulo 3. Then the following statements hold,
(1) diam (G(Eqn)) = diam (Oe9„ ) = 2;
(2) gr(G(Eqn))=gr(OEq„) = 3.
Proof. Since q is a prime integer congruent to 1 modulo 3, the ring Eqn is the product of the two local rings E/((a, + buj)n) and E/((a, + buJ)n) that have the same number of elements. The ideals (a + buj) and (a, + buj) are the only maximal ideals of Eqn, see [4]. Therefore, by [18, Theorem 3.5], we have
diam (G(Eqn)) = diam(0Eqn) = 2.
On the other hand, in view of the proof of [7, Proposition 5.10] and [3, Theorem 3.2], we obtain
gr(G(Eqn ))=gr(OE3n) = 3.
Lemma 1 [19, Lemma 4.1]. Let R be a finite ring and a € R. The following statements are equivalent:
(1) a € J(R);
(2) a + u € U(R) for any u € U(R).
Theorem 4 [9, Theorem 2.6]. Xei R = R\ x i?2 x •••• x a finite ring, where (Ri,nii) is
a local ring, for each i = 1, ■ ■ ■ , n. Then the following statements are equivalent:
(1) 2 € J(R);
(2) Gr = G(R);
(3) for every i = 1, ...,n, R| is even.
Theorem 5. Let n be a positive integer and p be a prime integer congruent to 2 modulo 3. Then the following statements hold
(1) diam (Gep„ ) =
1 if n = 1,
2 if n > 1;
(2) diam(G(Epn)) = 2;
(3) gr (G(Epn)) = gr (Gr„) = 3.
Proof. Since p is a prime integer congruent to 2 modulo 3, p is an Eisenstein prime integer. Hence Ep is a field. If n > 1, then the ring Epn is a local ring with the maximal ideal (p). Since
^(Epn)= p2fc-V - 1),
we obtain that (see [4]):
E„
<P>
= p2.
If p = 2, then it follows from Theorem 4 that G(Epn) = GEpn. In this case, by [3, Theorem 3.1] and [3, Theorem 3.2], we obtain gr(G(Epn)) = gr (GEpn) = 3 and
diam (G(Epn)) = diam (GEpn) =
1 if n = 1,
2 if n > 1.
We now assume that p = 2. Then GEpn is a complete p2-partite graph. Therefore, diam (GEpn) = 2 and gr(GEpn) = 3. Since G (Epn/(p)) is a complete (p2 + 1)/2-partite graph, we obtain that
diam (G(Epn)) = 2, gr (G(Epn)) = 3.
n
2.2. Diameter and Girth for the Graphs GEn and G(En)
The general case is now investigated. It is well known [8, Theorem 8.7] that every finite commutative ring can be expressed as a direct product of finite local rings, and this decomposition is unique up to permutations of such local rings. Throughout this section we assume that R = Ri x R2 x .... x Rt is a finite commutative ring, where each Ri is a finite commutative local ring with the maximal ideal m»- Since (u\, ...,ut) is a unit of R if and only if each m is a unit in Ri, we see immediately that Gr = GRl O Gr2... O GRt and G(R) = G(Ri) O G(R2)... O G(Rt). We denote by Ki the (finite) residue field Ri/rrii and fi = \Ki\. We also assume (after appropriate permutation of factors) that f1 < f2 < ... <
Theorem 6. Let n > 1 be an integer with at least two distinct prime factors. Then
diam (GBn) = diam (G(En)) = 2.
Proof. Let
m l
n = 3k x JJ pT x JJ j, i=i j=i
where pi and qj are prime integers such that pi = 2 (mod 3) and qj = 1 (mod 3), then
ml
En = x JJ Ep^i x JJ Ej , i=1 j=1
see [4]. This shows that, En is isomorphic to a direct product of finite local rings (Ri,nii)> such that for every i, \Ri/mi\ = 3 or pj or qj. By [3, Theorem 3.5 (b)], we conclude that
diam (GEn) = diam (G(En) = 2.
□
Theorem 7. Let n > 1 be an integer with at least two distinct prime factors. Then
gr (Ge„ )=gr(G(En)) = 3. Proof. By the argument similar to that above, we conclude that
ml
En = E3fc xj] Epai xj] E ßj
Uj ■
i=i j=i j
Thus, by [3, Theorem 3.2], we obtain gr(GEn) = 3. On the other hand, in view of the proof of [7, Theorem 5.10], we have gr(G(En)) € {3,4}.
Since n is an integer with at least two distinct prime factors, we can assume that n = ab with gcd (a, b) = 1. It is clear that
gcd (a2 + b2 - ab, n) = 1.
Thus, N (a + bw) is a unit in Zn, and so a + bw is a unit in En. This showes that x = a and y = bw are adjacent. Now, by taking z = b + aw, we have x + z = (a + b) + aw and y + z = b + (a + b)w. Clearly, N(x + z) = N(y + z) = a2 + b2 + ab is a unit in Zn which implies that x + z and y + z are unit elements of En. Therefore, we obtain the cycle
x —> y —> z —> x.
This implies that gr (G(En)) =3. □
2.3. Some Graph Invariants of Graphs GEn and G(En)
In the sequel, we obtain the clique number and the chromatic number for the graphs GEn and G(En).
Theorem 8. Let n > 1 be an integer and
m I
k TT ,v,®i w TT „Pi
3k xji pa ^n j.
i=1 j=1
Then the following statements hold
(1) if 3 | n, then x(Gen) = w(GEn) = 3 and a(GEn) = n2/3;
(2) if 3 f n, then
x(Gen) = w(GEn) = min {p2, q,| 1 < i < m, 1 < j < l, pi | n, q, | n}
and
n2
^ min {j92, qj| 1 < i < m, 1 < j < I, Pi\ n, qj | ??.}
Proof. 1. Let k be the biggest positive integer such that 3k | n. Then
m I
En = E3k x JJ Epai x JJ E .
i=l j=l j Since E3fc is a local ring with the maximal ideal (2 + w) and
E3k
= 3,
(2 + w)
it follows from [3, Proposition 6.1] that x(GEn) = w(GEn) = 3 and a(GEn) = n2/3.
2. If 3 f n, then it yields that En is isomorphic to a direct product of finite local rings (Ri,nii),
Jn 2
such that for every i, iRi/'m^ = pf or qj. Thus by [3, Proposition 6.1], we have x(Gen) = w(GBn) = k = min { p2, q, | 1 < i < m, 1 < j < l} and a(GEn) = n2/k. □
Theorem 9. Let n > 1 be an integer and
n = 3k xji pa ^n q
m l
k w TT ,v,a¿ w TT
j '
i=1 j=1
Then the following statements hold
(1) if 2 | n, then x(G(En)) = w(G(En)) = 4;
(2) if 2 f n, then
X(G(E„)) = uj(G(En)) = x f[(p2^ - p^~2) x " 9?'"1)2 + m + 21 + 1.
i=1 j=1
Proof. Since n = 3k x n 1=1 x IIj=1 j, we have
m l
= E3k x JJ Epai x JJ Eß
Pi 11 j
i=1 j=1 j
1. If 2 | n, then 2 / U(En). Hence, in view of the proof of [21, Theorem 2.2], we have
X(G(Era)) = W(G(Era)) = 4.
2. If 2 f n, Then 2 e U(En). By an argument similar to that above, we conclude that
ml
X(G(En)) = uj{G{En)) = x [] (pf* - pf5""2) x J](9?' - qf"1)2 + m + 21 + 1.
i=i j=i
□
We now state our final result.
Theorem 10. For each integer n > 1, the graphs G(En) and GEn are Hamitonian.
Proof. Let n > 1 be an integer. By Theorem 2, Theorem 3, Theorem 5 and Theorem 6, the graphs G(En) and GEn are connected. Thus G(En) is Hamiltonian graph, by [22, Theorem 2.1]. Also, it follows from [20, Lemma 4] that is Hamiltonian graph. □
3. Concluding Remarks
In this article, the diameter, the girth, the chromatic number and the clique number of G(En) and GEn are studied. We also prove that for each n > 1, the graphs G(En) and GEn are Hamiltonian and the independence number of GEn is calculated. We end our paper with the following two open questions:
Question 1. Is there any closed formula for a(G(En))? Question 2. When are G(En) and GEn Eulerian?
Acknowledgments
I would like to thank the referee for the valuable suggestions and comments.
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