Even though Cayley’s work dates back to the nineteenth century [1], the connection between group and graph theories became well known in the 1950s. Numerous graphs are defined, particularly on a finite group . Without going too far, we recall the following:

Numerous graphs, including the co-maximal ideal graph, co-prime graph, coprime order graph, commuting graph, intersection graph, Gruenberg-Kegel graph, zero-divisor graphs, and many more, are associated with algebraic structures in the literature. In [15], the concept of a -divisible graph was introduced.

The -divisible graph of a finite group , denoted by is a special type of simple undirected graph in which the set of vertices contains non-trivial subgroups of and two distinct vertices and are adjacent if and only if is a proper subgroup of such that or is a proper subgroup of such that .

The -divisible graph of a group has emerged as a significant topic of re search, with vertices representing subgroups and edges linking elements that exhibit a particular algebraic relationship, generally concerning the divisible order sum properties of the group’s subgroups. The connection between -divisibility and the square-free order property of finite groups was further investigated in [16]. Some results -divisible graph for finite cyclic groups were investigated in [17].

In a graph , if the neighborhood of a vertex induces a complete subgraph, then the vertex is said to be an extreme vertex [18]. Important research on the extreme vertices of the power graph for the finite abelian, dicyclic, and dihedral groups was carried out by AbuGhneim et al. [19]. Kumari et al. [18] extended this work to the power graph for the permutation group .

In this article, the extreme vertices of the -divisible graph of a finite cyclic group are examined. The rest of the paper is organized as follows. The fundamental findings of group theory and graph theory are presented in Section 2. Our study of the extreme vertices of the -divisible graph of a finite cyclic group is presented in Section 3.

This section provides an overview of the notation and fundamental results used throughout the paper. The following notations are used:

order of element ,

sum of elements orders of ,

-divisible graph of ,

vertices , are adjacent,

is subgroup of .

In addition, we make use of the following key results from group theory.

Lemma 2.1. [20]“Every subgroup of a cyclic group is cyclic. Moreover, If , then the order of any subgroup of is a divisor of ; and, for each positive divisor of , the group has exactly one subgroup of order ".

Lemma 2.2. [20]“If and are two subgroups of a finite cyclic group G and , then ."

Lemma 2.3.[20]“Group is always a cyclic group of order ."

Theorem 2.4.[17]“Let be a finite cyclic group of order where is a prime. If is a subgroup of order , then degree of vertex in -divisible graph of group is for all ."

Lemma 3.1.Let be a cyclic subgroup of order from the group where is prime, then .

Proof. Given that is a finite cyclic subgroup group of order , then contains exactly elements of order where . So, .

Lemma 3.2. Let and be cyclic subgroups of a finite p-group of orders and , respectively. Then

if and only if .

Proof. Recall that for a cyclic subgroup of order , we have . Hence .

We use the identity (valid for odd . Setting and , both odd, gives .

First, if , then the gcd equals the smaller number, so , which implies , hence .

Conversely, if , write with odd. Then . Since is odd, using gives for some integer . Thus , and dividing by yields .

Therefore .

We examine the conditions under which the nontrivial subgroups of a group cannot serve as extreme vertices in the -divisible graph of the finite group

Theorem 3.3. Let be a nontrivial cyclic subgroup of order of the group where is the prime number. If is divisible by two distinct primes, then the cyclic subgroup cannot be an extreme vertex in the -divisible graph of the finite group .

Proof. Let and be two diferent prime divisors of , so both must be odd. It is given that is a cyclic subgroup of order , so there exists a unique non-trivial subgroup of each order where .

Taking , there exists a unique subgroup of order from such that . Here , so . Hence, and are adjacent in the -divisible graph of the finite group .

Similarly, take , then there exists a unique subgroup of order from such that . Here , so . Hence, and are adjacent in the -divisible graph of the finite group .

Here, adjacent to both and however, they are not adjacent to each other because neither divides nor divides . Thus, cannot be an extreme vertex of the finite group .

According to Theorem 3.3, if a nontrivial cyclic subgroup of order of a finite group is an extreme vertex in the -divisible graph of the finite group , then must be a prime power.

Theorem 3.4. If is a non-trivial cyclic subgroup of order from a finite group and there exist two distinct cyclic subgroups and of order and that properly include with the condition that and . If , then cannot be an extreme vertex in the -divisible graph of the finite group .

Proof. The subgroup is a proper subgroup of and , then ~ k \end{document} ]]> and ~ k \end{document} ]]>. So, and are integers greater than 1. By the fundamental theorem of arithmetic, every integer greater than 1 can be decomposed into prime factors. Using the concept , there exist two distinct odd primes and such that and

.

It is given that is a cyclic subgroup of order , so there exists a unique nontrivial cyclic subgroup of each order where . In addition, the subgroup of order properly includes when .

Taking , there exists a unique subgroup of order from that properly includes such that . Hence, and are adjacent in the -divisible graph of the finite group .

It is given that is a cyclic subgroup of order , so there exists a unique nontrivial cyclic subgroup of each order where . In addition, the subgroup of order properly includes when .

Taking , there exists a unique subgroup of order from that properly includes such that . Hence, and are adjacent in the -divisible graph of the finite group .

Here, adjacent to both and nevertheless, they are not adjacent to each other because neither nor . Thus, cannot be an extreme vertex in the -divisible graph of the finite group .

Corollary 3.5. If is a nontrivial cyclic subgroup of order from a finite group and there exists a cyclic subgroup of order that properly includes where is an odd prime other than 3, then cannot be an extreme vertex in the -divisible graph of the finite group .

Proof. Take and such that and . It is given that there exists a cyclic subgroup of order that properly includes where is the odd prime other than 3, then and . So, there exist two cyclic subgroups of order and that properly includes . Using the above theorem, cannot be an extreme vertex in the -divisible graph of the finite group .

Corollary 3.6. If is a nontrivial cyclic subgroup of order from a finite group and , then cannot be an extreme vertex in the -divisible graph of the finite group .

Proof. We know that there exists a unique subgroup of order where from the group and each subgroup of order where that properly includes . So, there exists a cyclic subgroup of order that properly includes . Taking in above corollary 3.5, then cannot be an extreme vertex in the -divisible graph of the finite group .

According to theorems 3.3 ,3.4 and corollary 3.6, if be a nontrivial subgroup of is extreme vertices of order in the -divisible graph of the finite group , then is a prime power and { \frac { n - 2 } { 5 } } \end{document} ]]>.

Theorem 3.7.If is a nontrivial subgroup of is an extreme vertex of order in the -divisible graph of the finite group if and only if is an odd prime power and { \frac { n - 2 } { 5 } } \end{document} ]]>.

Proof. It is given that

is an odd prime power, so there exists an odd prime such that . The list of divisors of greater than 1 is only.

Take and where , then there exist two cyclic subgroups and of order and such that .

So, and .

Also, .

Hence, by Lemmas 2.2 and 3.2, we get and . Therefore, and are adjacent to each other. If we take , then . Hence, we conclude that are subgroups of that are adjacent to and also adjacent to each other.

Now, we determine all subgroups of order where from that properly include and , so . If where is an odd prime, then possible value for is otherwise \frac { n - 2 } { 5 } \end{document} ]]> fails.

Now two cases arise on the basis of value exists or not.

Case 1: If no such exists.

In this case is adjacent to and they are also adjacent to each other. Hence, is an extreme vertex in the -divisible graph of the finite group .

Case 2: If

So, we have a subgroup of order which includes subgroups and . Therefore includes subgroups and because proper subgroup of is adjacent to in the -divisible graph of the finite group where . Hence, is an extreme vertex in the -divisible graph of the finite group .

In this article, we obtain the necessary and suficient conditions for a vertex of the finite cyclic group to be an extreme vertex in the -divisible graph under:

be a non-trivial subgroup of is an extreme vertex of order in the -divisible graph of the finite group if and only if is an odd prime power and { \frac { n - 2 } { 5 } } \end{document} ]]>.

The result of the extreme vertices of -divisible graph of group reported in this article may be extended to the group .