EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS 2025, Vol. 18, Issue 2, Article Number 6059 ISSN 1307-5543 – ejpam.com Published by New York Business Global Lower and Upper Acyclicities on Unitary Cayley Graphs of Finite Commutative Rings Denpong Pongpipat1, Nuttawoot Nupo1,∗ 1 Department of Mathematics, Faculty of Science, Khon Kaen University, Khon Kaen, 40002, Thailand Abstract. A unitary Cayley graph Γn of a finite cyclic ring Zn is a graph with vertex set Zn and two vertices x and y are adjacent if and only if x−y is a unit in Zn or equivalently, gcd(x−y, n) = 1. A nonempty subset A of Zn is called an acyclic set of Γn if a subgraph of Γn induced by A contains no cycles. The maximum cardinality among the acyclic sets of Γn is called the upper acyclic number of Γn and is denoted by Λ(Γn). Moreover, the maximum number k of vertices of Γn in which every subgraph of Γn induced by k vertices contains no cycles is called the lower acyclic number of Γn and denoted by λ(Γn). In this paper, we determine the lower and upper acyclic numbers for unitary Cayley graphs of Zn and their complements. 2020 Mathematics Subject Classifications: 05C25, 05C69, 05C99 Key Words and Phrases: Acyclic sets, Lower acyclic numbers, Upper acyclic numbers, Unitary Cayley graphs 1. Introduction In algebraic graph theory, the structure of algebraic methods are studied and then applied to problems about graphs. An interesting topic is to study properties of graphs in connection to algebraic systems. A well-known connection between graphs and algebraic system is the construction of graphs from algebras. Algebraic tools can be used to give elegant proofs of graph theoretic facts. For each n ≥ 2, let Γn denote the unitary Cayley graph of a ring Zn, the ring of integers modulo n, whose vertex set is Zn itself and two vertices x and y are joined by edge if x − y is a unit in the ring Zn. Let us denote all elements in Zn by integers 0, 1, 2, . . . , n − 1. It is well known that all units in the ring Zn are the integers a in which gcd(a, n) = 1. Therefore, the edge set of Γn can be ex- pressed as E(Γn) = {{x, y} : x, y ∈ Zn and gcd(x − y, n) = 1}. Clearly, if p is prime, then Γp is a complete graph. Moreover, all unitary Cayley graphs of order greater than 2 always contain cycles and their structures are highly symmetric. Moreover, there are ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v18i2.6059 Email addresses: denpong p@kkumail.com (D. Pongpipat), nuttanu@kku.ac.th (N. Nupo) https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0) D. Pongpipat, N. Nupo / Eur. J. Pure Appl. Math, 18 (2) (2025), 6059 2 of 11 some remarkable properties between algebraic graph theory and number theory. Some prominent results of unitary Cayley graphs were studied by several researchers. In 1995, Dejter and Giudici [1] showed that unitary Cayley graphs are unions of disjoint hamil- tonian cycles and presented the sufficient condition for being bipartite graphs. In 2007, Klotz and Sander [2] determined some invariant properties of unitary Cayley graphs and studied their perfectness. In 2012, Kiani and Aghaei [3] provided isomorphism theorems for unitary Cayley graphs of rings associated with Jacobson radicals. In 2014, Naghipour [4] considered some properties of induced subgraphs of unitary Cayley graphs of commu- tative rings. One of interesting parameters of graphs is the acyclic number which is the maximum number of vertices that the subgraph induced by such vertices contains no cycles. There are many discussions of acyclic numbers, for instance, in 2009, Samodivkin [5] investigated the acyclic number of graphs with cut-vertices. Later in 2017, Petrusevski and Skrekovski [6] proposed a conjecture on this parameter. Moreover, they provided some conditions that make such the conjecture weaker for planar graphs. Throughout the paper, the notation Γn stands for the unitary Cayley graph of a finite commutative ring Zn. In addition, let us denote by Γn the complement of Γn which is a simple graph obtained from Γn by deleting all edges in E(Γn) and then adding all edges outside E(Γn). In this paper, we study some properties on unitary Cayley graphs of finite commutative rings. Furthermore, we present the construction of cycles of lengths 3 and 4 which is useful for finding the maximum cardinality among acyclic sets of Γn and Γn, such the number is called the upper acyclic number. Moreover, we provide some charac- terization of the structure of Γn and Γn, where n is a power of a prime number,to find the maximum number k of vertices of Γn in which every subgraph of Γn induced by k vertices contains no cycles. This cardinality is called the lower acyclic number, and all sets mentioned in this research are considered to be finite sets. 2. Preliminaries A graph G is a pair (V (G), E(G)) where V (G) is the vertex set of G and E(G) is an edge set of G. An edge of G joining between vertices u, v ∈ V (G) is written as {u, v}, that is, {u, v} ∈ E(G) means that u is adjacent to v in G. Let v ∈ V (G). The degree of v, denoted by deg(v), is the number of vertices adjacent to v in G. Furthermore, let C be a sequence v1, v2, . . . , vk of distinct k vertices of G where k ≥ 3. If vi and vi+1 are adjacent in G for all i = 1, 2, . . . , k − 1 and v1 is adjacent to vk, then C is called a cycle in G with length k and denoted by Ck. Moreover, if k is odd (even), then Ck is said to be an odd (even) cycle. In particular, if k = 3, then the cycle C3 is sometimes called a triangle. A graph G is said to be complete if {u, v} ∈ E(G) for all u, v ∈ V (G). In addition. a graph G will be called a complete p-partite graph if V (G) is partitioned into p sets, called a partite set, provided that vertices in the same partite set are not mutually adjacent and every two vertices from different partite sets must be adjacent. Let H = (V (H), E(H)) be a graph. The graph H is called a subgraph of G if V (H) ⊆ V (G) and E(H) ⊆ E(G). A decomposition of G is a collection of edge disjoint subgraphs of G in which every edge D. Pongpipat, N. Nupo / Eur. J. Pure Appl. Math, 18 (2) (2025), 6059 3 of 11 of G belongs to exactly one subgraph. Next, let W be a nonempty subset of V (G). A subgraph of G induced by W , simply called an induced subgraph and denoted by G[W ], is a subgraph of G satisfying the condition that if u, v ∈ W and {u, v} ∈ E(G), then {u, v} ∈ E(G[W ]), as well. Throughout the paper, all sets are finite sets and all graphs are simple. More information about graph theory can be found in [7]. We now provide some definitions and prominent facts which play a crucial role in the paper. Definition 1. [8] Let G be a group. A nonempty subset H of G is called a subgroup of G if H itself is a group under the group operation of G restricted to H. Moreover, a left coset of a subgroup H of G is a set of the form gH := {gh : h ∈ H}. The set of all cosets of H is denoted by G/H, that is, G/H := {gH : g ∈ G}. Definition 2. [2] Let n ≥ 2 be a positive integer. The unitary Cayley graph Γn = (Zn, Un) is defined by the additive group of the ring Zn and the multiplicative group Un of its units such that V (Γn) = Zn and E(Γn) = {{a, b} : a, b ∈ Zn and a − b ∈ Un} or equivalently, E(Γn) = {{a, b} : a, b ∈ Zn and gcd(a− b, n) = 1}. Definition 3. [9] In number theory, the Euler’s totient function counts the positive in- tegers up to a given integer n that are relatively prime to n. Generally, a well-known notation written for the Euler’s totient function is φ(n) and may also be called the Euler’s phi function. In other words, it is defined as the number of integers k such that 1 ≤ k ≤ n and gcd(k, n) = 1. Theorem 1. [10] Let n be an even positive integer such that n ≥ 8. Then φ(n) ≥ 4. Remark 1. [2] Let Γn be the unitary Cayley graph with n vertices. Then |E(G)| = n(φ(n)) 2 and the degree of a vertex v ∈ V (Γn) is given by deg(v) = φ(n). Theorem 2. [2] Let n = pk be such that p is prime and k ∈ N \ {1}. Then Γn is a complete p-partite graph where each partite set has size pk−1. Corollary 1. [1] If n is an even positive integer, then the unitary Cayley graph Γn has no odd cycles. In particular, Γn has no triangles. Definition 4. Let Γn be the unitary Cayley graph of Zn. The complement Γn of Γn is the graph in which V (Γn) = Zn and E(Γn) = {{a, b} : a, b ∈ Zn and gcd(a− b, n) ̸= 1}. Theorem 3. [11] Let n = pk be such that p is prime and k ∈ N \ {1}. Then Γn is decomposed into p complete graphs of order pk−1. Lemma 1. [7] If every vertex of a finite simple graph G has degree at least 2, then G contains a cycle. Definition 5. [7] A nonempty subset X of V (Γn) is an independent set of Γn if every pair of vertices in X is not adjacent in Γn. D. Pongpipat, N. Nupo / Eur. J. Pure Appl. Math, 18 (2) (2025), 6059 4 of 11 3. Lower acyclic numbers of Γn and Γn This section begins with the definition of a lower acyclic number of the graph G. Definition 6. A lower acyclic number of G, denoted by λ(G), is the maximum number of k vertices in which every induced subgraph of k vertices contains no cycles. Example 1. Let G be a graph with V (G) = {a, b, c, d, e, f, g, h, i} and let E(G) be defined as the following diagram: Figure 1: A graph G We observe that an induced subgraph G[{a, d, g}] forms a cycle of length 3. Therefore, λ(G) = 2. Lemma 2. Let n be an even positive integer such that n ≥ 8. Then the unitary Cayley graph Γn contains a cycle C4 of length 4 as a subgraph. Proof. Let Φn = {k ∈ N : 1 ≤ k ≤ n and gcd(k, n) = 1}. Then |Φn| = φ(n). For convenience, we write Φn = {1, a1, a2, . . . , aφ(n)−1}, where ai < aj and 1 ≤ i < j ≤ φ(n) − 1. We claim that C4 is contained in Γn. Firstly, since gcd(1 − 0, n) = gcd(1, n) = 1, there exists an edge between vertices 0 and 1 in Γn. Consider a1 ∈ Φn. We have gcd(a1 − 0, n) = gcd(a1, n) = 1. Then, there exists an edge between vertices 0 and a1 in Γn. Next, by Theorem 1, we get that a1 + 1 < n. Then a1 + 1 ∈ Zn and gcd((a1 + 1) − a1, n) = gcd(1, n) = 1. Hence, there is an edge between vertices a1 and a1 + 1 in Γn. Finally, we have gcd((a1 + 1) − 1, n) = gcd(a1, n) = 1. Then there exists an edge between vertices a1 + 1 and 1 in Γn. Hence, 0, 1, a1, and a1 + 1 form a cycle of length 4 in Γn. Before we present the lower acyclic number of Γn where n is even, the following example is needed for n = 4, 6. Further results for n > 6 will be proved in Theorem 5. D. Pongpipat, N. Nupo / Eur. J. Pure Appl. Math, 18 (2) (2025), 6059 5 of 11 Example 2. The unitary Cayley graphs Γ4 and Γ6 are shown as follows. Figure 2: The unitary Cayley graphs Γ4 and Γ6 We see that Γ4 is a cycle of length 4 and Γ6 is a cycle of length 6. Therefore, λ(Γ4) = 3 and λ(Γ6) = 5. Lemma 3. Let n be an odd positive integer such that n ≥ 3. Then the unitary Cayley graph Γn contains a cycle C3 of length 3 as a subgraph. Proof. Let n be an odd positive integer. We claim that C3 is contained in Γn. Since gcd(1 − 0, n) = 1 and gcd(2 − 1, n) = 1 , there exist an edge between vertices 0 and 1 in Γn, and an edge between vertices 1 and 2 in Γn. Next, we show that gcd(2, n) = 1. As the fact that n is an odd positive integer, then there exists an integer k such that n = 2k+1. This means that n is not divisible by 2. Since the only positive divisors of 2 are 1 and 2, and n is odd, 2 does not divide n. So the only common positive divisor of 2 and n is 1. It follows that 1 = gcd(2, n) = gcd(2 − 0, n). Then there is an edge between vertices 0 and 2 in Γn. Hence, 0, 1 and 2 form a cycle of length 3 in Γn. In order to complete our results of the part of acyclic numbers of Γn, we present such the numbers as follows. Theorem 4. Let n be an odd positive integer such that n ≥ 3. Then λ(Γn) = 2. Proof. By Lemma 3, the statement holds. Theorem 5. Let n be an even positive integer such that n ≥ 8 and n is not prime. Then λ(Γn) = 3. Proof. By Corollary 1, Γn does not contain a triangle. It follows that λ(Γn) ≥ 3. By Lemma 2, we can construct C4 which is contained in Γn. Then λ(Γn) ≤ 3. Therefore, λ(Γn) = 3. For more results related to the acyclic numbers of the complement Γn, we present these numbers in the following discussion. In particular, we explore their properties, characteris- tics, and the conditions under which they arise. This allows for a deeper understanding of D. Pongpipat, N. Nupo / Eur. J. Pure Appl. Math, 18 (2) (2025), 6059 6 of 11 how acyclic numbers behave in the context of the complement graph and their significance in graph theory. The results presented here provide a foundation for further investigation and possible extensions of this concept. Example 3. The complements Γ4 and Γ6 of the unitary Cayley graphs Γ4 and Γ6 are shown as follows. Figure 3: The complements Γ4 and Γ6 We observe that Γ4 does not contain any cycle. For Γ6, it is obvious that Γ6 contains triangles. Therefore, λ(Γ4) = 4 and λ(Γ6) = 2. Lemma 4. Let n be a positive integer such that n ≥ 8 and n is not prime. Then the com- plement Γn of the unitary Cayley graph Γn contains a cycle C3 of length 3 as a subgraph. Proof. Let n be a positive integer such that n ≥ 8 and n is not prime. Case 1 : n is an even integer. Then there exists an integer k such that n = 2k. Since gcd(4 − 2, 2k) = gcd(2, 2k) ̸= 1, gcd(6 − 4, 2k) = gcd(2, 2k) ̸= 1 and gcd(6 − 2, 2k) = gcd(2(2), 2k) ̸= 1, there exist an edge between vertices 2 and 4 in Γn, an edge between vertices 4 and 6 in Γn, and an edge between vertices 2 and 6 in Γn. So 2, 4 and 6 induce a cycle C3 in Γn as a subgraph. Case 2 : n is an odd integer. Assume that n = pm1 1 · pm2 2 · · · pmt t where pi and mi are prime and positive, respectively such that i = 1, 2, . . . , t and pi < pj for i < j. Since gcd(p1−0, n) = gcd(p1, p m1 1 ·pm2 2 · · · pmt t ) ̸= 1, gcd(2p1−p1, n) = gcd(p1, p m1 1 ·pm2 2 · · · pmt t ) ̸= 1, and gcd(2p1 − 0, n) = gcd(2p1, p m1 1 · pm2 2 · · · pmt t ) ̸= 1, so there exist an edge between vertices 0 and p1 in Γn, an edge between vertices p1 and 2p1 in Γn, and an edge between vertices 0 and 2p1 in Γn, respectively. Thus, 0, p1 and 2p1 induce a cycle C3 in Γn. Theorem 6. Let n be a positive integer such that n ≥ 8. Then λ(Γn) = { 2 if n is not prime ; n if n is prime. Proof. We consider the following two cases. Case 1 : n is not prime. By Lemma 4, It follows that λ(Γn) = 2. Case 2 : n is prime. Then Γn is a complete graph which implies that Γn is an empty graph. Then λ(Γn) = n, immediately. D. Pongpipat, N. Nupo / Eur. J. Pure Appl. Math, 18 (2) (2025), 6059 7 of 11 4. Upper acyclic numbers of Γn and Γn This section begins with the definition of an upper acyclic number of the graph G. Definition 7. A nonempty subset A of the vertex set V (G) of a graph G is called an acyclic set of G induced by A contains no cycles. An upper acyclic number of a graph G, denoted by Λ(G), is the maximum cardinality among acyclic sets of G, that is, Λ(G) = max{|A| : A is an acyclic set of G}. Example 4. In Figure 1, we observe that the set A = {a, b, c, d, e, f} is an acyclic set of G with the largest cardinality. Thus, Λ(G) = 6. Next, we find upper acyclic numbers of unitary Cayley graphs and their complements. The following two examples illustrate results for Γn and Γn where n = 4, 6. Other results will be presented in the sequel. Example 5. The set {0, 1, 2} of vertices in the left diagram of Figure 2 and the set {0, 1, 2, 3} of vertices in the left diagram of Figure 3 are acyclic sets of Γ4 and Γ4, respec- tively. Therefore, it is not hard to conclude that Λ(Γ4) = 3 and Λ(Γ4) = 4. Example 6. It is easy to see that the set {0, 1, 2, 3, 4} of vertices in the right diagram of Figure 2 is an acyclic set of Γ6 with maximum cardinality. Furthermore, the set {0, 1, 2, 3} of vertices in the right diagram of Figure 3 is an acyclic set of Γ6 with maximum cardi- nality. Therefore, Λ(Γ6) = 5 and Λ(Γ6) = 4. Theorem 7. Let p be a prime number such that p ≥ 3. Then Λ(Γp) = 2 and Λ(Γp) = p. Proof. Let p be a prime number such that p ≥ 3. Then Γp is a complete graph and Γp is an empty graph. It follows that Λ(Γp) = 2 and Λ(Γp) = p, respectively. Theorem 8. Let n = pk be such that p is prime and k ∈ N\{1}. Then Λ(Γn) = pk−1+1. Proof. Let n = pk be such that p is prime and k ∈ N \ {1}. By Theorem 2, we obtain that Γn is a complete p-partite graph such that each partite set has size pk−1. Let P be a partite set of Γn. Thus the induced subgraph Γn[P ] is an empty graph. Let v ∈ V (Γn) \ P . Hence Γn[P ∪ {v}] is a tree. It follows that P ∪ {v} is an acyclic set of Γn which leads to Λ(Γn) ≥ |P ∪ {v}| = pk−1 + 1. We now suppose that there is an acyclic set, say A, of Γn in which |A| > pk−1 + 1. Then there exist x, y, z ∈ A such that x ∈ P1 and y, z /∈ P1 where P1 is a partite set of Γn. We now consider the following two cases. Case 1 : y, z ∈ P2 for some a partite set P2 in which P1 ̸= P2. Since |A| > pk−1 + 1 where p is prime and k ≥ 2, there exists u ∈ A \ {x, y, z} such that u /∈ P2. If u ∈ P1, then the sequence of edges uy, yx, xz, zu forms a cycle of length 4 in Γn[A] (see Figure 4.) since Γn is a complete p-partite graph. This contradicts to the acyclicity of A. On the other hand, if u ∈ P3 where P3 ̸= P1, then the sequence of edges D. Pongpipat, N. Nupo / Eur. J. Pure Appl. Math, 18 (2) (2025), 6059 8 of 11 Figure 4: Two possible cases of the sequence of edges on cycles of length 4 and 3. ux, xy, yu forms a cycle of length 3 in Γn[A] (see Figure 4.) which is also a contradiction. Case 2 : y ∈ P2 and z ∈ P3 for some partite sets P2, P3 in which P1, P2, P3 are pairwise disjoint. We can observe that this case will generate a cycle of length 3 with the sequence of edges xy, yz, zx in Γn[A] similar to Figure 4. This also contradicts to the acyclicity of A. From the above two cases, we can conclude that Λ(Γn) = pk−1 + 1. Theorem 9. Let n be a positive integer such that n ≥ 8. If p is the least prime divisor of n, then n p + 1 ≤ Λ(Γn) ≤ n− ( φ(n) 2 + 1 ) . Proof. Let n be a positive integer such that n ≥ 8. Assume that p is the least prime divisor of n. To prove the lower bound of Λ(Γn), consider the set A = {0, p, 2p, . . . , n−p}. It is clear that |A| = n p . For each {x, y} ∈ A, we have x − y is the multiple of p which directly implies that gcd(x − y, n) ̸= 1, that is, {x, y} /∈ E(Γn). It follows that Γn[A] is an empty graph. Now, let v ∈ Zn \ A. It is not hard to verify that Γn[A ∪ {v}] contains no cycles. Therefore, A ∪ {v} is an acyclic set in Γn. Hence Λ(Γn) ≥ |A ∪ {v}| = n p + 1. For proving the upper bound of Λ(Γn), let B be any subset of Zn containing at least n − φ(n) 2 elements. By Remark 1, deg(u) = φ(n) for all u ∈ V (Γn), we obtain that deg Γn[B](w) ≥ φ(n) 2 for all w ∈ V (Γn[B]). Since n ≥ 8, we conclude by Theorem 1, that φ(n) ≥ 4 which leads to deg Γn[B](w) ≥ 2 for all w ∈ V (Γn[B]). By Lemma 1, we get that Γn[B] contains a cycle. It follows that B is not an acyclic set of Γn. Since B is arbitrary, we have that Λ(Γn) ≤ |B| − 1 ≤ ( n− φ(n) 2 ) − 1 = n− ( φ(n) 2 + 1 ) . Then the statement is proved, as required. D. Pongpipat, N. Nupo / Eur. J. Pure Appl. Math, 18 (2) (2025), 6059 9 of 11 We now consider the unitary Cayley graph Γ8 as shown in the following example. Actually, the lower bound and upper bound in Theorem 9 are sharp. Example 7. Consider the unitary Cayley graph Γ8 follows. Figure 5: The unitary Cayley graph Γ8 By Theorem 8, we obtain that Λ(Γ8) = 23−1 +1 = 5 = 8 2 + 1 which achieves the lower bound in Theorem 9. Moreover, we can observe that Λ(Γ8) = 5 = 8− ( φ(8) 2 + 1 ) which attains the upper bound mentioned in Theorem 9. Theorem 10. Let n = pk be such that p is prime and k ∈ N \ {1}. Then Λ(Γn) = 2p. Proof. By Theorem 3, we obtain that Γn is decomposed into p complete graphs of order pk−1. Choose two vertices of each complete graph. We get that those vertices form an acyclic set of Γn. Moreover, we can easily observe that there is no acyclic set containing more than two vertices from one complete graph. Therefore, Λ(Γn) = 2p. We mention a certain prominent fact that we will use in the next theorem. In com- binatorics,the Pigeonhole Principle states that if n items are put into m containers with n > m, then at least one container must contain more than one item. Theorem 11. Let n be an even positive integer such that n ≥ 4. Then Λ(Γn) = 4. Proof. Let n be an even positive integer such that n ≥ 4. Further, let A = {0, 1, 2, 3}. Clearly, Γn[A] contains no cycles. Hence, A is an acyclic set of Γn which implies that Λ(Γn) ≥ 4. Consider the set of any five vertices of Γn, say B = {v1, v2, v3, v4, v5}. By the Pigeonhole Principle, there exist at least three vertices vi, vj , vk ∈ B such that they are all even or odd. Hence the difference between any two vertices of vi, vj , vk must be even. Then {vl, vm} /∈ E(Γn) for all l,m ∈ {i, j, k}. Thus, {vl, vm} ∈ E(Γn) for all l,m ∈ {i, j, k}. It follows that Γn[{vi, vj , vk}] is a cycle of length 3 contained in Γn[B]. Therefore, B is not acyclic in Γn. Since B is arbitrary, we can conclude that Λ(Γn) < |B| = 5. Thus Λ(Γn) = 4, as required. D. Pongpipat, N. Nupo / Eur. J. Pure Appl. Math, 18 (2) (2025), 6059 10 of 11 Theorem 12. Let n be an odd positive integer such that n ≥ 5 and n is not prime. If p is the least prime divisor of n, then p+ 3 ≤ Λ(Γn) ≤ 2p. Proof. Let n be an odd positive integer such that n > 5 and n is not prime. Assume that n = pm1 1 · pm2 2 · · · pmt t where pi and mi are prime and positive numbers, respectively such that i = 1, 2, . . . , t and pi < pj for i < j. Let p be the least prime divisor of n and A = {0, 1, 2, . . . , p + 1, p + 2}. We show that Γn[A] contains no cycles. Suppose that Γn[A] contains a cycle Ck := {v1, v2, . . . , vk, v1} such that v1, v2, . . . , vk ∈ A and 3 ≤ k ≤ p+ 3. Without loss of generality, we can assume that v1 < vj for 2 ≤ j ≤ k. Let X := {x1, x2, . . . , xk} be such that x1 = 0 and xl = vl − v1 for all l = 2, 3, . . . , k. Then x1, x2, . . . , xk, x1 form a cycle in Γn[X]. We now consider the following two cases. Case 1 : (p+ 2)|n. Consider {x2, x3} ∈ E(Γn[X]), if x2 < x3, then x3 = 2p or x3 = 2p+ 2 and so x3 /∈ X, a contradiction. If x3 < x2, then x3 = 0 = x1, a contradiction as xi ̸= xj for i ̸= j. Case 2 : (p+ 2) ∤ n. Since {x1, xk} ∈ E(Γn[X]), it follows that xk = p = x2, a contradiction. Hence Γn[A] contains no cycles. That is, Λ(Γn) ≥ |A| = p + 3. Now, let B := {v0, v1, . . . , v2p} be arbitrary. Let Br be the set of elements in B having r as the remainder from dividing by p where 0 ≤ r ≤ p − 1. We show that Γn[B] contains a cycle. Now, we can place v0, v1, . . . , v2p in such p sets and by the Pigeonhole Principle, we obtain that one of those sets must contain at least three vertices x, y, z. Without loss of generality, assume that x, y, z ∈ Bs and x > y > z where 0 ≤ s ≤ p − 1. Then x = m1p + s, y = m2p + s and z = m3p+ s for some m1,m2,m3 ∈ Z. So x− y = (m1 −m2)p, y − z = (m2 −m3)p and x−z = (m1−m3)p, we obtain that gcd(x−y, n) ̸= 1, gcd(y−z, n) ̸= 1 and gcd(z−x, n) ̸= 1, respectively. Hence x, y, z, x form a cycle. It follows that Γn[B] contains a cycle.Therefore, Λ(Γn) ≤ |B| − 1 = 2p.Then the statement is proved, as required. 5. Conclusion In this paper, we have provided certain structural properties and some invariant proper- ties of unitary Cayley graphs Γn of finite commutative rings Zn. Such invariant properties consist of the lower acyclic number and the upper acyclic number. In the process of the D. Pongpipat, N. Nupo / Eur. J. Pure Appl. Math, 18 (2) (2025), 6059 11 of 11 study results, we have found that the decomposition of graphs is useful and plays an im- portant role for determining those invariant parameters. Finally, we have presented the results of invariant parameters in the unitary Cayley graphs and their complements. For further works, one can investigate those invariant parameters for various types of algebraic graphs and their complements. Acknowledgements The authors are grateful to the referee(s) for suggestions on the manuscript. 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