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 define the Product Cube Graph PC(R) over a finite commutative ring R as a graph on nonzero elements and adjacency occurs when their product is a cube in

R. We investigate its structural properties, including connectivity, diameter, clique structure, and bipartiteness. For finite fields F , PC(F ) is completely characterized.

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

MSC (2020): 05C25, 05C75.

Introduction

The interaction between algebraic structures and graph theory has generated a substantial body of research over the past two decades. A foundational construction in this area is the zero-divisor graph of a commutative ring introduced by Anderson and Livingston [3], in which adjacency is defined via zero products. Since then, numerous variations of graphs associated with rings have been developed, including the total graph [4], the unitary Cayley graph [5], the comaximal graph [2], and the annihilator graph [6]. These constructions capture structural properties of rings through additive, multiplicative, or annihilation-based adjacency relations.

More recent studies have explored graphs defined through power-related or residue-based conditions. For example, graphs determined by quadratic or higher power residues in finite fields have been investigated in connection with number-theoretic properties [1, 8]. Unity product graphs, in which adjacency is defined when the product of two elements is a unit, have also been studied from spectral and structural perspectives [9]. Furthermore, Cayley graphs on finite commutative rings continue to provide a rich source of algebraically defined graph models [10].

Gupta [7] introduced the Product Square Graph over a finite commutative ring, where adjacency is determined by products that are squares. Building on this idea, we define the Product Cube Graph on the non-zero elements of a finite commutative ring, with two vertices adjacent if their product is a cube element. This construction extends the family of algebraically defined graphs by incorporating multiplicative cube relations, providing a new framework to investigate higher power structures in finite commutative rings.

Definition 1. Let R be a finite commutative ring. Define

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

The Product Cube Graph of R, denoted by PC(R), is the simple undirected graph with vertex set V = R \ {0}, where distinct vertices x, y V are adjacent if and only if xy C.

An element x R is called a cube element if x C, that is, if x = t3 for some t R \{0}.

For distinct vertices a, b V , we write a b if ab C.

Example 1. Let R = Z9. The set of cube elements is C = {t3 : t Z9 \ {0}}. A direct computation modulo 9 gives

13 43 73 1, 23 53 83 8, 33 63 0.

Hence, C = {0, 1, 8}.

The vertex set of PC(Z9) is

V = Z9 \ {0} = {1, 2, 3, 4, 5, 6, 7, 8}.

Two distinct vertices x, y V are adjacent if and only if xy {0, 1, 8}. The resulting graph structure is illustrated in Figure 1

‌Figure 1: Product Cube Grpah of the ring Z9.

Properties of Product Cube Graph

‌Theorem 2. Let R be a commutative ring. Then there exists a non-zero element x R

such that x3 = 0 if and only if R contains a non-zero nilpotent element.

Proof. Suppose there exists x R \ {0} such that x3 = 0. Then x is nilpotent by definition. Conversely, assume that R contains a non-zero nilpotent element y. Then yn = 0 for some integer n 2. Consider the element z = y n1. Since y = 0 and n is minimal, we have

z = 0. Moreover,

z3 = (y n1)3 = y3(n1).

Because 3(n 1) n for all n 2, it follows that y3(n1) = 0. Hence z3 = 0, and therefore

R contains a non-zero element whose cube is zero.

‌Theorem 3. Let R be a finite commutative r ing containing a n on-zero n ilpotent element. Then the zero-divisor graph (R) is a subgraph of the product cube graph P C(R).

Proof. Since R contains a non-zero nilpotent element, by Theorem 2 there exists y R \{0}

such that y3 = 0. Hence 0 is a cube element, that is, 0 C.

In the zero-divisor graph (R), two distinct non-zero zero-divisors u and v are adjacent if and only if uv = 0. Since 0 C, it follows that whenever uv = 0, we have uv C, and therefore u is adjacent to v in P C(R).

Thus every edge of (R) is also an edge of P C(R), and hence (R) is a subgraph of

P C(R).

Theorem 4. Let R be a commutative ring containing a non-zero nilpotent element. Then any two vertices of P C(R) corresponding to non-zero zero-divisors of R are connected by a path in P C(R), i.e., all non-zero zero-divisors lie in one connected component of P C(R).

Proof. Since R contains a non-zero nilpotent element, by the Theorem 3 the zero-divisor graph (R) is a subgraph of P C(R).

It is known that for a commutative ring with non-zero zero-divisors, the zero-divisor graph (R) is connected (see [3]). Hence any two non-zero zero-divisors are joined by a path in (R).

Because (R) is a subgraph of P C(R), the same path exists in P C(R). Therefore the corresponding vertices are connected in P C(R).

Proposition 5. Let R be a commutative ring with unity, and let

C = {a3 : a R}

be the set of cube elements of R. If R is a ring of characteristic 3 and 0 C, then C is always a subring of R.

Proof. Closure under multiplication and additive inverses always hold in any commutative ring. For addition, note that

a3 + b3 = (a + b)3 3ab(a + b).

If char(R) = 3, the second term vanishes, and the sum of cubes is again a cube. Therefore

C is a subring in this case.

Theorem 6. Let R be a commutative ring with unity, and let C R be a subring. Then C

has the same characteristic as R.

Proof. Since 1R C, it serves as the multiplicative identity of C. Hence 1C = 1R. Let char(R) = n. Then n · 1R = 0, and therefore

n · 1C = n · 1R = 0,

so char(C) | n.

Conversely, let char(C) = m. Then m · 1C = 0, and since 1C = 1R, we obtain m · 1R = 0 in R. By minimality of n, it follows that n | m. Hence m = n, and therefore char(C) = char(R).

Lemma 7. Let R be a finite commutative ring with unity. Then a non-zero element a R

is a zero-divisor if and only if it is not a unit.

Proof. If a is a zero-divisor, then ab = 0 for some b = 0. If a is a unit, multiplying by a1

gives b = 0, a contradiction. Hence a can not be a unit.

Conversely, suppose a is not a unit. Consider the map : R R given by (x) = ax. If a were not a zero-divisor, then would be injective, and since R is finite, injective implies surjective. Thus 1 = ax for some x, making a a unit, a contradiction. Hence a is a zero- divisor.

‌Theorem 8. Let R be a finite commutative ring with unity. If R contains a non-zero nilpotent element and a zero-divisor z such that z3 = 0, then PC(R) is connected and

diam(PC(R)) 5.

Proof. Since R contains a non-zero nilpotent element, hence the zero-divisor graph (R) is a subgraph of PC(R). It is known that (R) is connected with diameter at most 3 (see [3]). Thus any two non-zero zero-divisors are at distance at most 3 in PC(R).

Let u be a unit. Since z3 C and z3 = 0, define w = u1z3. Then

uw = z3 C,

so u w. Because z is a zero-divisor, w is also a zer-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. Thus d(u, v) 5.

Therefore P C(R) is connected and diam(P C(R)) 5.

Corollary 9. Let R be a finite commutative r ing w ith u nity a nd char(R) = n , w here n is composite. Let R contains a non-zero nilpotent element, and there exists a prime divisor p of n such that n p3. Then P C(R) is connected and

diam(P C(R)) 5.

Proof. Since char(R) = n, the subring generated by 1 in R is canonically isomorphic to Zn. We identify Zn with this subring of R.

Let p be a prime divisor of n such that n p3. In Zn, we have

n

p 0 (mod n),

p

so p is a non-zero zero-divisor in Zn. Since n p3, we also have p3 0 (mod n), and hence

p3 = 0 in Zn.

Viewing p as an element of R, it follows that z = p R is a zero-divisor and satisfies z3 = 0 in R. Since R also contains a non-zero nilpotent element, all hypotheses of Theorem 8 are satisfied. Therefore PC(R) is connected and

diam(PC(R)) 5.

‌Theorem 10. Let R be a commutative ring with unity. The subgraph of PC(R) induced by the non-zero cube elements is a clique if and only if either 0 C or R is an integral domain.

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

xy = (ab)3.

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

If ab = 0, then xy = 0. Thus adjacency holds if and only if 0 C. Therefore the induced subgraph is complete precisely when either 0 C or no non-zero elements multiply to zero, that is, when R is an integral domain.

Theorem 11. Let R be a commutative ring with unity such that either 0 C or R is an integral domain. Let

C = {a3 : a R}

be the set of cube elements of R. Then PC(R) is not bipartite if |C| 3.

Proof. By the theorem 10, the subgraph of PC(R) induced by the non-zero cube elements

C is a clique (provided either 0 C or R is an integral domain).

A clique with at least three vertices contains a triangle. Since |C| 3, the subgraph induced by C contains a triangle.

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

PC(R) is not bipartite.

Theorem 12. Let R be a finite commutative ring with unity and let u be a unit of R. Then

deg(u) =

( |C| 1, if u2 C,

|C|, if u2 / C,

where C = {a3 : a R} is the set of cube elements of R. In particular, if u is a unit cube, then

deg(u) = |C| 1.

Proof. In PC(R), two vertices x and y are adjacent if and only if xy C.

Let u be a unit of R. Since multiplication by a unit is 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 PC(R) is simple, we

must exclude the case y = u. This occurs precisely when

uu = u2 C.

Therefore,

deg(u) =

|C|, if u2 / C,

(

|C| 1, if u2 C.

Finally, if u is a unit cube, say u = a3, then

u2 = a6 = (a2)3 C,

so deg(u) = |C| 1.

Proposition 13. Let F be a field and F = F \ {0} its multiplicative group. Define

C = {x3 : x F }.

Then C is a subgroup of F .

Proof. Define : F F by (x) = x3. Then

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

so is a homomorphism. Since C = Im() and the image of a homomorphism is a subgroup, it follows that C is a subgroup of F .

Proposition 14. Let F be a finite field with |F | = q and let F = F \ {0}. If

C = {x3 : x F },

then

|C| =

q 1

.

gcd(3, q 1)

Proof. Since F is cyclic of order q 1, say F = t, we have C = t3. Hence

|C| = ord(t3) = q 1 .

gcd(3, q 1)

Theorem 15. Let F be a finite field with |F | = q. Let PC(F ) be the product cube graph with vertex set F = F \ {0}, where x and y are adjacent if and only if xy C, with

C = {x3 : x F }.

Then

PC(F ) Kq1, if 3 (q 1),

=

3

3

3

Kq1 Kq1 , q1 , if 3 | (q 1).

Proof. Since F is cyclic of order q 1, let F = t. Then C = t3 and

q 1

|C| =

.

gcd(3, q 1)

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

3

If 3 | (q 1), then |C| = q1 and

F = C tC t2C.

For ti, tj F ,

ti tj ti+j C i + j 0 (mod 3).

3

Thus C induces a clique of size q1; tC and t2C are independent sets; and every vertex of

tC is adjacent to every vertex of t2C. Hence

PC(F ) = Kq1 Kq1 , q1 .

3 3 3

i=1

Theorem 16. Let R = Qt Ri be a finite commutative ring with t 2 such that for each

i

i there exists xi Ri \ {0} satisfying x3 = 0. Then the Product Cube Graph PC(R) is

connected and

diam(PC(R)) 3.

i

Proof. For each i, choose xi = 0 with x3 = 0. Then ei = (0, . . . , xi, . . . , 0) = 0 and

i

e3 = (0, . . . , 0),

so (0, . . . , 0) C. Hence xy = 0 implies x y.

Let a = (a1, . . . , at) and b = (b1, . . . , bt) be distinct nonzero vertices.

Case 1: Suppose there exist r = s with ar = 0 and bs = 0. Set

u = (0, . . . , xs, . . . , 0), v = (0, . . . , xr, . . . , 0).

Then

au = (0, . . . , asxs, . . . , 0), uv = 0, vb = (0, . . . , brxr, . . . , 0).

Since (asxs)3 = a3x3 = 0 and (brxr)3 = 0, we obtain

s s

a u, u v, v b,

so d(a, b) 3.

Case 2: Suppose there exists r such that ar = 0 and br = 0. Since t 2, choose k = r

and define

Then

u = (0, . . . , xk, . . . , 0).

au = (0, . . . , akxk, . . . , 0), ub = (0, . . . , bkxk, . . . , 0).

As (akxk)3 = a3x3 = 0 and (bkxk)3 = 0, we have

k k

a u b,

so d(a, b) 2.

Thus every pair of vertices has distance at most 3, and PC(R) is connected with

diam(PC(R)) 3.

Conclusion

In this paper, we introduced the Product Cube Graph PC(R) over a finite commutative ring R and studied its fundamental properties. We provided a complete characterization of PC(F ) for finite fields, showing how its structure depends on the divisibility of |F | by

3. For product rings with nilpotent cube elements, we established connectivity and a tight bound on the diameter. We also derived explicit formulas for the degrees of units in terms of cube elements. These results extend classical studies of power and zero-divisor graphs, revealing new structural patterns arising from cube operations and opening avenues for further exploration in ring-theoretic graph theory.

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