Product Cube Graphs over Finite Commutative Rings: Structural Properties

Product Cube Graphs over Finite Commutative Rings: Structural Properties

Nidhi Khandelwal, Pravin Garg, Surekha Jain *

Department of Mathematics University of Rajasthan, Jaipur-302004 Rajasthan, India

Abstract

We introduce the Product Cube Graph PC(R) associated with a finite commutative ring R, whose vertices are the non-zero elements of R, and two distinct vertices are adjacent whenever their product is the cube of a non-zero element of the ring. We study fundamental structural properties of PC(R), including connectivity, diameter, clique structure, and bipartiteness. For finite fields F, we completely characterize the graph PC(F), showing that its structure is determined by the divisibility of |F*| by 3.

Keywords: Product cube graph, finite commutative ring, cube elements, zero-divisor graph, connectivity, finite fields.

MSC (2020): 05C25, 05C75.

1. Introduction

   The interaction between algebraic structures and graph theory has generated extensive research over the past two decades. In recent years, considerable attention has been devoted to associating graphs with commutative rings and other algebraic structures and studying the interplay between ring-theoretic and graph-theoretic properties, see the book

   [6] by Anderson et al. A foundational construction in this area is the zero-divisor graph of a commutative ring introduced by Anderson and Livingston [2], in which adjacency is defined via zero products. Since then, numerous variations of graphs associated with rings have been developed, including the total graph [3, 4, 5, 10], the comaximal graph [11, 14], the maximal graph [12], the annihilator graph [8, 9], the unit graph [7], Cayley graphs on finite commutative rings [1]. These studies provide a natural framework for investigating algebraic structures and motivate the search for new graph constructions associated with ring elements.

   Gupta [13] introduced the Product Square Graph over a finite commutative ring , whose vertices are the non-zero elements of R, with adjacency defined whenever the product of two vertices is a non-zero square element. They investigated several structural properties of this graph, including connectedness, planarity, domination number, girth, and characterizations of rings for which the graph realizes important graph classes. Motivated by this construction, we introduce the Product Cube Graph (), where adjacency is determined by cube elements instead of square elements. This graph extends the family of

   algebraically defined graphs by incorporating multiplicative cube relations and provides a new framework for studying higher-power structures in finite commutative rings.

   The main objective of this paper is to investigate the structural properties of the Product Cube Graph. We establish several results concerning its connectivity, diameter, clique structure, and bipartiteness. In addition, we derive explicit degree formulas for unit vertices and obtain a complete characterization of PC(F) when F is a finite field. These results demonstrate how cube-element relations influence the graph-theoretic structure associated with finite commutative rings and further enrich the interplay between algebraic structures and graph theory.

   Definition 1.1. Let R be a finite commutative ring. Consider the set

   C = {t3: t R \ {0}}.

   The Product Cube Graph of R, denoted by PC(R), is the simple undirected graph whose vertex set is

   V(PC(R)) = R \ {0},

   where distinct vertices x,y V(PC(R)) are adjacent if and only if xy C.

   An element x R is called a cube element if x C, i.e., if x = t3 for some t R\{0}. For distinct vertices x, y V(PC(R)), the notation x y is used to indicate that x and y are adjacent.

   Example 1.2. Consider the ring of integers modulo 9, namely 9. Since

   13 43 73 1, 23 53 83 8, 33 63 0 (mod 9), The set of cube elements is C = {0, 1, 8}. Hence, the vertex set of PC(9) is

   V(PC(9)) = 9 \ {0} = {1, 2, 3, 4, 5, 6, 7, 8},

   where two distinct vertices x and y are adjacent if and only if xy {0, 1, 8}. The graph PC(9) is shown in Figure 1.

   ‌Figure 1: Product Cube Graph of the ring 9.

2. Properties of Product Cube Graph

   Theorem 2.1. Let R be a ring. Then R contain a non-zero element x satisfying x3 = 0 if and only if R contain a non-zero nilpotent element.

   Proof. Suppose there exists x R\{0} such that x3 = 0. Then, clearly x is nilpotent element. Conversely, let R contain a non-zero nilpotent element y. Then 1 = 0 for some minimal integer n 2. Set z = yn1. By minimality of n , we have 0. Further,

   3 = (1)3 = 3(1).

   Since, 3(n 1) n for all n 2. It follows that y3(n1) = 0. Hence z3 = 0, proving that R

   contain a non-zero element whose cube is zero.

   Theorem 2.2. Let R be a ring containing a non-zero nilpotent element. Then the zero-divisor graph

   (R) is a subgraph of the product cube graph PC(R).

   Proof. Since R contain a non-zero nilpotent element, by Theorem 2.1 there exists z R\{0} such that z3 = 0. Hence, 0 C.

   Let u and v are adjacent vertices in PC(R). Then, uv = 0. Since 0 C, we obtain uv C.

   Therefore, u and v are adjacent in PC(R).

   Thus, every edge of (R) is also an edge of PC(R), proving that (R) is a subgraph of

   PC(R).

   Lemma 2.3 ([2]). Let R be a commutative ring. Then, the zero-divisor graph (R) is connected and

   (()) 3.

   Theorem 2.4. Let R be a commutative ring containing a non-zero nilpotent element. Then., all non- zero zero-divisors of R belong to the same connected component of PC(R).

   Proof. By Lemma 2.3, the graph (R) is connected. Hence, any two non-zero zero-divisors are joined by a path in (R).

   Since, (R) is a subgraph of PC(R) by Theorem 2.2, the same path exists in PC(R). Therefore, all non-zero zero-divisors of R belong to a single connected component of PC(R).

   Theorem 2.5. Let R be a commutative ring of characteristic 3, and let

   C = {t3 : t R \ {0}}

   If 0 C, then C is a subring of R.

   Proof. Since 0 C, the set C contains the additive identity. For x³, y³ C, we have

   x³y³ = (xy)³ C. So, C is closed under multiplication. Also,

   x³ + y³ = (x + y)³ 3xy(x + y).

   As char(R) = 3,

   x³ + y³ = (x + y)³ C,

   even when x + y = 0, and hence C is closed under addition. Further,

   x³ = (x)³ C.

   So, C is closed under additive inverses. Therefore, C is a subring of R.

   Theorem 2.6. Let R be a finite commutative ring with unity 1R, and let C be a subring of R. Then

   char(C) = char(R).

   Proof. Since 1R C, we have 1C = 1R. Let char(R) = n. Then

   n · 1C = n · 1R = 0,

   so, char(C) / n.

   Conversely, let char(C) = m. Then Therefore, n | m. Consequently, thus,

   m · 1C = m · 1R = 0, m = n,

   char(C) = char(R).

   Remark 2.7. The above theorem also holds when char(R)=0. Since 1C =1R implies

   . 1 = 1 0

   for every positive integer .

   Lemma 2.8. Let R be a finite commutative ring with unity. Then a non-zero element x R is a zero- divisor if and only if it is not a unit.

   Proof. Suppose that x is a zero-divisor. Then there exists , 0, such that

   xy = 0.

   If x is a unit, then multiplying by x1 gives

   y = 0,

   a contradiction. Hence x is not a unit.

   Conversely, suppose x is not a unit. Consider the map : R R such that (r) = xr.

   If x is not a zero-divisor, then is injective. Since R is finite, is surjective. Therefore, there exists uch that

   xr = 1,

   which implies that x is a unit, a contradiction. Hence x is a zero-divisor.

   Theorem 2.9. Let R be a finite commutative ring with unity. If R contain a non-zero nilpotent element and a zero-divisor z such that 3 0, then PC(R) is connected and

   diam(PC(R)) 5.

   Proof. Since R contain a non-zero nilpotent element, Theorem 2.2 implies that (R) is a

   subgraph of PC(R). By Lemma 2.3,

   diam((R)) 3.

   Hence, any two non-zero zero-divisors are at distance at most 3 in PC(R).

   Let u be a unit. D efine w = u1z3. Then

   uw = z3 C,

   so u w. Because z is a zero-divisor, w is also a zero-divisor. Hence every unit is adjacent to a zero-divisor.

   If u and v are units, then

   u u1z3 P v1z3 v,

   where P is a path of length at most 3 between zero-divisors u1z3 and v1z3. Thus d(u, v)

   5.

   Therefore PC(R) is connected and diam(PC(R)) 5.

   Corollary 2.10. Let R be a finite commutative ring with unity and char(R) = n , where n is composite. Let R contain a non-zero nilpotent element, and there exists a prime divisor p of n such that n p3. Then PC(R) is connected and

   diam(PC(R)) 5.

   Proof. Since char(R) = n, the subring generated by 1R is isomorphic to n. Let p be a prime divisor of n such that n p3. Then,

   n

   p 0 (mod

   n), p

   so, p is a non-zero zero-divisor in n. Further, since n p3,

   3 0 ( ).

   Hence, viewing p as an element of R, we obtain a zero-divisor z = p satisfying 3

   1. Since R also contains a non-zero nilpotent element, all hypotheses of Theorem 2.9 are satisfied. Therefore PC(R) is connected and

      diam(PC(R)) 5.

      Theorem 2.11. Let R be a commutative ring with unity. The subgraph of PC(R) induced by the non-zero cube elements is a clique if 0 C.

      Proof. Let x = a3 and y = b3 be distinct non-zero cube elements. Then

      xy = (ab)3.

      If 0, then xy C, so x and y are adjacent. If ab = 0, then xy = 0. Since 0 C, x and

      y are adjacent.

      Remark 2.12. The preceding theorem shows that the condition 0 guarantees completeness of the induced subgraph on the non-zero cube elements. The following result establishes a necessary condition for this induced subgraph to be complete.

      Theorem 2.13. Let R be a commutative ring with unity. The subgraph of PC(R) induced by the non-zero cube elements is a clique only if either 0 or every non-zero cube element is not a zero-divisor.

      Proof: Assume that the subgraph induced by the non-zero cube elements is a clique and suppose that 0 C. Let x = a³ be a non-zero cube element. If x is a zero-divisor, then there exists a non-zero element b R such that

      a³b = 0.

      Since 0 C, we have b³ 0. Moreover,

      a³b³ = (ab)³ = 0 C.

      Hence, the vertices a³ and b³ are not adjacent, contradicting the assumption that the induced subgraph is a clique. Therefore, every non-zero cube element is not a zero-divisor.

      Theorem 2.14. Let R be a commutative ring such that 0 C or R is an integral domain. If |C| 4, then PC(R) is not bipartite.

      Proof. By the Theorem 2.11, the subgraph of PC(R) induced by the non-zero cube elements is a clique. Since there are at least three non-zero cube elements, this induced subgraph contains a triangle.

      A triangle is an odd cycle, and bipartite graphs cannot contain odd cycles. Hence,

      PC(R) is not bipartite.

      Theorem 2.15. Let R be a finite commutative ring with unity and let u be a unit of R. Then,

      || 2, 0 2 ,

      () = {|| 1, 0 2 ,

      ||, 0 2 .

      Proof. For y V(PC(R)),

      .

      Since u is a unit, multiplication by u is a bijective. For each c C the equation

      uy = c

      has a unique solution y = u1c. Hence there are exactly |C| elements y R such that uy

      C. If 0 C, then y = 0 satisfies uy = 0 C, but 0 V(PC(R)), so it must be excluded.

      Also, if 2 C, then = satisfies = 2 . However, since PC(R) is a simple graph, loops are not allowed. Hence = must be excluded. Therefore,

      || 2, 0 2 ,

      () = {|| 1, 0 2 ,

      ||, 0 2 .

      Proposition 2.16. Let F be a field and F = F \ {0} its multiplicative group. Then

      C = {x3 : x F}.

      is a subgroup of F.

      Proof. Define : by (x) = x3. Then for all , ,

      (xy) = (xy)3 = x3y3 = (x)(y).

      Thus, is a group homomorphism. Since

      C = Image()

      and the image of a homomorphism is a subgroup, it follows that C is a subgroup of F .

      Proposition 2.17. Let F be a finite field with |F| = q and let F = F \ {0}. If

      C = {x3 : x F},

      then

      || =

      1

      .

      (3, 1)

      Proof. Since F is cyclic group of order q 1. So, let F = t, where t is the generator of

      . We have C = t3k = t3. Since, the order of a cyclic subgroup equals the order of its generator,

      || = (3) = 1 .

      (3, 1)

      Theorem 2.18. Let F be a finite field with |F| = q. Then

      1, 3 ( 1)

      () {1 1 1 , 3 | ( 1)

      3 3 , 3

      Proof. Since F is a cyclic group of order q 1, let F = t for some . Then C =

      t3 and

      || =

      1

      .

      (3, 1)

      If 3 (q 1), then C = F, so xy C for all x, y F. Hence PC(F) = Kq1.

      If 3 | (q 1), then

      and

      || =

      1

      3

      For ti, tj

      F,

      F = C t C t2 C.

      t t t+ + 0 ( 3).

      Thus, C induces a clique of size1; tC and 2 are independent sets; and every vertex of

      3

      tC is adjacent to every vertex of 2. Hence

      () 1 1 1 .

      3 3 , 3

      =1

      Theorem 2.19. Let = be a finite commutative ring with 2 such that, for each i,

      there exists a non-zero element satisfying 3 = 0. Then PC(R) is connected and

      diam(PC(R)) 3.

      Proof. For each i, define

      = (0,0, . . . ,0, , 0, . . . ,0),

      where occurs in the ith coordinate. Then 0 and

      3 = (0, ,0, . . . ,0)

      Let = (1, , ) and = (1, , ) be distinct non-zero vertices in . So, there exists indices and such that

      Then,

      Since

      0 0 .

      y = (0,0, . . . ,0, , 0, . . . ,0), = (0,0, . . . ,0, , 0, . . . ,0). ()3 = 33 = 0,

      and

      We obtain,

      ()3 = 0,

      ,

      If = , we obtain . Hence, there exists a path from to of length 2. Otherwise, when . Then . Hence, there exists a path from to of length 3.

      So, () is connected and (() 3.

3. Conclusion

In this chapter, we introduced the Product Cube Graph () associated with a finite commutative ring R and studied its fundamental structural properties. We completely characterized () for finite fields, showing that the structure of the graph depends on the divisibility of || by 3. For finite product rings containing non-zero cube nilpotent elements, we established the connectivity of () and proved that its diameter is at most 3. In addition, explicit degree formulas for unit vertices were derived. These findings demonstrate that cube- element interactions give rise to rich graph-theoretic structures, thereby extending the scope of

ring-based graph constructions and suggesting several directions for future research in algebraic graph theory.

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