In recent years, extensive research has been conducted on graphs related to groups. In real-life applications, graphs are essential in various fields, especially in timetable scheduling and route optimization, such as in Google Maps. In timetable scheduling, nodes represent time slots, classrooms, or teachers, and edges show possible assignments. Algorithms help to create schedules without conflicts, ensuring resources like rooms and teachers are not double-booked. Similarly, Google Maps can represent as graph where locations as vertices and roads as edges, with distances or travel times as weights. The application uses algorithms to find the fastest or shortest route, adjusting for trafic in real-time. These applications show how graphs help solve complex problems eficiently [1].

This study extends the idea of subgroup graphs to the graph of non-normal cyclic subgroups. If a subgroup graph H of a group G has the same left and right cosets, it is categorized as a normal subgroup. Most researchers focus on an undirected graphs, including the study on undirected graph on the data complexity of regular trails [2]. However, there are some research in the directed graph which are related to group theory. For instance, [3] defined an order graph which is a new representation of finite groups. Next, the subgroup graph was introduced by Anderson et al. [4]. According to Anderson et al. [4], the subgroup graph, denoted as is a graph which consists of a vertex set of group G such that x is the starting point while y is the endpoint of the directed graph if and only if , where H is a subgroup of G. The subgroup graph is then extended to a non-normal subgroup graph, which has been defined by Rahin et al. [5]. Then, Rahin et al. introduced the non-normal subgroup graphs of some dihedral groups in [5] and generalized quaternion groups in [6]. There are some research on graphs that apply into groups such as [7] that study about coprime graph of generalized quaternion group. In 2024, Rahin [8] found the general form of the non-normal cyclic subgroup graph for dihedral and generalized quaternion groups. Hence, this paper will explore on the non-normal cyclic subgroup graph of quasidihedral groups of order 16, with the goal of preparing for future work, where a subsequent paper will focus in determining the general form of the non-normal cyclic subgroup graph for the quasidihedral groups.

A potential application of the non-normal cyclic subgroup graphs in cryptography is cryptographic hash functions which lies in their ability to generate complex, collision-resistant hash values. The structure of the graph can be used to define a hashing algorithm where inputs are mapped to elements of the graph, and the hash value is derived from specific subgroup relations. Due to the unpredictability and intricate subgroup interactions within the graph, even small changes in the input would produce vastly diferent hash outputs, a key property of secure hash functions. This makes the graph-based hashing a promising approach for applications such as digital signatures, blockchain security, and password storage.

This paper will discuss on the non-normal subgroup graphs of quasidihedral groups of order 16 with the help of Groups, Algorithms and Programming software (GAP). GAP software is very helpful in defining the normal and non-normal subgroups of the groups. Futhermore, Maple 2016 software has been used to visualize the graphs based on the subgroups obtained from GAP software.

This section contains several definitions related to subgroup graphs.

Definition 2.1. Subgroup graph [9] Let G be a group and H is a subgroup of G. denotes as the subgroup graph which is a directed graph with the vertex set and two distinct elements x and y are adjacent if and only if

Definition 2.2. Non-normal cyclic subgroup graph [8] Let be the non-normal cyclic subgroup graph of the non-normal cyclic subgroup of a group denotes as H. Then, is a directed graph with vertex set G such that x is the initial vertex and y is the terminal vertex of an edge if any only and denoted by

In 2024, Rahin [8] introduced the non-normal cyclic subgroup graph and studied on the non-normal cyclic subgroup graph for dihedral and generalized quaternion groups. Apart from that, the general form of the graph of those two groups is determined together with the general form of the incidence matrix for the nonnormal cyclic subgroup graphs.

The quasidihedral group, which is denoted as , is a non-abelian group of order where and . The group presentation of quasidihedral groups is given as:

Recently, many researchers have conducted studies on quasidihedral groups, particularly focusing on the graphs of groups. For example, Pahil Muhidin [10] constructed the prime order and composite order Cayley graphs of quasidihedral groups. In addition, the concept of commuting graphs has been extended by defining the generalized commuting graph of dihedral, quasidihedral and semi-dihedral groups [11]. Other researchers have also investigated the conjugacy class graph of quasidihedral and generalized quaternion groups using some topological indices [12].

Next, Proposition 2.3 and 2.4, as determined in [8], are stated as follows.

Proposition 2.3.[8] Let be the dihedral groups of order

with 3n-2 edges, where

Proposition 2.4.[8] be the generalized quaternion groups of order 4n, . Then,

These propositions which include the general forms of the non-normal cyclic subgroup graph for the dihedral and generalized quaternion groups will be used to determine some graph properties, which are graph coloring and chromatic number of the non-normal cyclic subgroup graph for dihedral group of order 16, and generalized quaternion group of order 16, to compare with the graph properties for the graph of quasidihedral group of order 16, in the next section.

The non-normal cyclic subgroup graph corresponding to the quasidihedral groups of order 16 is identified in this section. The methodology is illustrated in Figure 1.

The quasidihedral groups of order 16, with its elements is with . The non-normal cyclic subgroups of are , ⟨b, aba⟩, ， and . The code to obtain the non-normal cyclic subgroups of are as shown in Figure 2:

Based on Figure 2, the non-normal cyclic subgroups that have been found using GAP software can be simplified as shown in Table 1. In addition, their elements and their orders are shown in Table 1 too.

After all the non-normal cyclic subgroup of are identified, determination on the direction of elements in subgroups is shown in following propositions.

Proposition 3.1. Let be the non-normal cyclic subgroup graph of , where . Then, is as shown in Figure 3.

Proof. Let . The product of the elements of and the directed edges are determined by using the Cayley table shown as follows.

To determine the directed edges for the graph, each group element is represented as a vertex. By examining Table 2, when two elements from a row and a column are multiplied, the result shaded in the table indicates a directed edge from vertex of row to vertex of column. This process is repeated for all possible pairs of group elements, and the corresponding directed edges are included in the graph.

By Definition 2.2, x → y if and only if and . Thus

and since

Therefore, is the graph shown in Figure 3.

Proposition 3.2.Let be the non-normal cyclic subgroup graph of , where . Then, is as shown in Figure 4.

Proof. Let . The product of the elements of and the directed edges are determined by using the Cayley table shown in Table 2.

By the Definition 2.2,

and since ,

and since

Therefore, is the graph shown in Figure 4.

Proposition 3.3.Let be the non-normal cyclic subgroup graph of , where . Then, is as shown in Figure 5.

Proof. Let . The product of the elements of and the directed edges are determined by using the Cayley table shown in Table 2.

By the Definition 2.2,

and since 2

and since

Therefore, is the graph shown in Figure 5.

Proposition 3.4.Let be the non-normal cyclic subgroup graph of , where . Then, is as shown in Figure 6.

Proof. Let . The product of the elements of and the directed edges are determined by using the Cayley table shown in Table 2.

By the Definition 2.2,

and since

and since

Therefore, is the graph shown in Figure 6.

Proposition 3.5. Let be the non-normal cyclic subgroup graph of , where . Then, is as shown in Figure 7

By the Definition 2.2,

Therefore, is the graph shown in Figure 7.

Proposition 3.6. Let be the non-normal cyclic subgroup graph of , where . Then, is as shown in Figure 8.

By the Definition 2.2,

Therefore, is the graph shown in Figure 8.

Proposition 3.7. Let be the non-normal cyclic subgroup graph of , where . Then, is as shown in Figure 9.

Proof. Let . The product of the elements of and the directed edges are determined by using the Cayley table shown in Table 2.

By the Definition 2.2,

Therefore, is the graph shown in Figure 9.

Proposition 3.8. Let be the non-normal cyclic subgroup graph of , where . Then, is as shown in Figure 10.

Proof. Let . The product of the elements of and the directed edges are determined by using the Cayley table shown in Table 2.

By the Definition 2.2,

Therefore, is the graph shown in Figure 10.

Based on Propositions 3.1 to 3.8, the non-normal cyclic subgroup graph of the quasidihedral groups of order 16 is observed to be the union of some connected graphs. Since the results are limited to , hence its general form has not yet been found.

Next, based on Propositions 2.3 and 2.4, as found by Rahin [8], are used to study some graph properties of the dihedral and generalized quaternion groups of order 16. The graph properties that will be investigated are the maximal and minimal cyclic subgroups, graph coloring, and chromatic number, which are presented in Table 3. This comparison highlights key diferences between the non-normal cyclic subgroup graph of dihedral and generalized quaternion group of order 16.

Once all the non-normal cyclic subgroup graph associated with quasidihedral groups of order 16 have been identified, in Proposition 3.1 - 3.8 an analysis of their graph properties can be conducted. Table 4 and Table 5 present some graph properties which are the graph coloring, chromatic number and identification of whether the subgroup is minimal or maximal.

This research has focused on determining the non-normal cyclic subgroup graphs for a quasidihedral group, specifically , where . The non-normal cyclic subgroup graph of the quasidihedral group of order 16 can be viewed as a union of several connected graphs. Its chromatic number is either 3 or 4, which is consistent with the chromatic numbers of the corresponding graphs in the dihedral and generalized quaternion groups. For future studies, the non-normal cyclic subgroup graphs for quasidihedral groups of other orders can be explored using a similar approach. In subsequent research, the general form of these graphs could also be established. Additionally, future work could investigate the topological indices of the non-normal cyclic subgroup graph of quasidihedral groups.