The study of uncertainty advanced significantly with the introduction of fuzzy sets (FS) by Zadeh.¹ In a FS, each element is characterized by a single membership value in the interval [0, 1] whereas the intuitionistic fuzzy sets (IFS) generalize this concept. It represents both information as a membership degree and an non-membership degree to each element in the given interval [0, 1].² This extension is particularly important when the non-membership degree cannot be obtained simply as the complement of the membership degree. However, in uncertain situations it is difficult to determine the membership or the non-membership degree for elements of a set.³ It requires computing hesitant part like draw matches is beyond win and loss of the match. To address such cases, neutrosophic set (NS) theory extends IFS by introducing an additional degree-the indeterminacy degree.³ In this framework, each element is described in terms of its membership, indeterminacy, and non-membership values.

However, challenges arise when defining the complement of the indeterminacy degree in NSs.⁴ The complement of the indeterminate part is either partially true or partially false, or unknown. To address the unknown case or human quantum consciousness is involved the Turiyam set theory was developed.⁵ A graph-theoretic framework based on this set has been developed, and its fundamental properties, as well as its applications in real-world domains such as social networks and Wi-Fi communication systems, have been thoroughly investigated^6–8 Recently, domination concepts for Turiyam graphs⁹ have been developed by extending the corresponding domination concepts in neutrosophic graphs.

Ambiguous set theory¹⁰ was introduced as a generalization of NSs to handle situations involving partial truth and partial falsity as similar to Quadripartitioned set. This theory model aspect of unconscious human cognition where distinguishing the boundaries between true and false becomes difficult. The Ambiguous sets explore the indeterminacy component of NSs with partially true and partially false values within the given interval. It helps in detailed representation of uncertainty.¹⁰ Hence one can say that it is totally works like Aristotle logic. It can be useful for medical data sets where partial truth and partial false values exists. Due to which, Singh et al.¹¹ demonstrated the effectiveness of this framework through real-life applications, particularly in medical image segmentation, where ambiguous sets more accurately capture inherent ambiguity than existing methods. The authors further proposed a new image-clustering algorithm based on ambiguous set theory, which proved especially useful for CT scan images.¹² These applications illustrate that ambiguous sets can successfully model uncertainty that is neither entirely true nor entirely false—an area where neutrosophic sets often fall short.

Since its introduction, ambiguous set theory has gained attention from researchers who have examined its theoretical foundations and its relevance to unconscious human perception.^11,13 These studies highlight how ambiguous sets better reflect the way humans interpret unclear or ambiguous information, offering a richer analytical framework than models relying solely on indeterminacy.^3,14 Despite these promising findings, the literature on ambiguous set theory remains limited, and its potential—particularly in graph theory—has yet to be fully explored.

In this paper, we extend the concept of neutrosophic graphs (NGs)¹⁴ to ambiguous graphs (AGs). We derive several essential properties of AGs and investigate their theoretical foundations. Furthermore, we illustrate the practical applicability of AGs by modeling social networks that contain inherent ambiguity. Social networks often involve relationships that are neither fully true nor fully false but lie within a spectrum of partial truth and partial falsity. Ambiguous graphs offer a more expressive tool for modeling such relationships, addressing cases where neutrosophic graphs are insufficient.

The next section of the paper is ordered as the basic mathematical background about NSS and ambiguous set, ambiguous graph and its application and discussion followed by conclusion.

In this section basic of NSS and Ambiguous set is explained. In this regard, U is considered as a universe set:

In this case, N is a vertex set of G and R is an edge set of G.

Where ∨ and ∧ are maximum and minimum operator respectively.

In this section, we define ambiguous graphs and derive some of their properties. Let’s consider G = (U, E) as a graph.

The following theorem provides how one-half multiplier impacts the consistency of the ambiguous graph.

Proof: From the definition ambiguous set, min{tA∘(ai),tA∘(bj)}≤1 implies that ptR∘({aibj})≤12min{ptA∘(ai),ptA∘(bj)} ≤ 12 and similarly pfR∘(aibj)≤ 12 . Adding, ptR∘({aibj}) + pfR∘(aibj)≤12 + 12 = 1. Thus, the total uncertainty of any edge never exceeds 1. This bounded uncertainty contributes to maintaining the condition 0 ≤tR∘({aibj})+ptR∘({aibj})+pfR∘({aibj})+fR∘({aibj})≤2 there by improving the consistency of the ambiguous graph.

In this ambiguous graph, u4 is an isolated vertex.

In this ambiguous graph ÂG , the vertex u4 is a pendent vertex while the vertex u3 is a non-pendant vertex. Also, the edge (u1u4) is a pendent edge while the edge (u2u3) is a non-pendent edge. The vertex u1 is the support of u1u4 while u4 is not its support.

Graph theory have an important application in the analysis of social networks.^16–18 But the problem exists when uncertainty arising from different human perceptions, interpretations, opinions, or assumptions about information occurs in social networks. Human interactions, on social media like WhatsApp, Facebook, and Instagram partially influence customer’s purchasing decisions. Companies can analyze the partial acceptance and rejection of products via social media advertisements with relatively minimum effort and cost. Advertising products on these platforms facilitates effective human interaction, but often the information provided about the brand is general rather than specific, leading to ambiguity which may arise because social media interactions do not always reflect genuine personal relationships. In such situations, social media relationships may develop through online interactions without direct personal acquaintance. It means partial friendship (or not-friendship) concept arises based on their interaction on social media rather than actual friendship. To represent these types of partial truth or partial false friendship can be represented via ambiguous set.

In such cases, an ambiguous digraph (ADG) offers a more suitable approach to assessing the partial influence exerted by individuals within a group, where traditional methods like neutrosophic digraphs (NDG) may fall short. The ADG allows for the evaluation of a person’s positive, negative, and ambiguous influence on others’ perceptions. By using ADGs, we can identify individuals who hold dominant and influential positions within a given social group.

Consider the set H={Abdi,Birraa,Ahmed,Chala,Gadise,Tolassa} consisting of six individuals in a social group involved in Instagram advertisements. In advertising, concise and compelling messages that vividly describe a brand and its products are preferred over factual information applied to any brand or product.

Assume that C={(x,t(x),pt(x),pf(x),f(x))} describe an individual ambiguity set on platform. Then the ambiguous data sets for Abdi, Chala, Gadise and Tolassa given as follows: C={(Abdi,0.12,0.09,0.51,0.22),(Birra,0.25,0.80,0.13,0.32),(Ahmed,0.40,0.15,0.23,0.08),(Chala,0.13,0.71,0.21,0.42),(Gadise,0.19,0.52,0.21,0.33),(Tolassa,0.32,0.27,0.06,0.53)} represent the ambiguous sets on the set H, where he truth value of each individual denotes their positive influence on others, the partial truth value denotes their partially positive influence on others, the partial falsity value denotes their partially negative influence on others and the falsity value represents their negative influence on others.

Let R={(Chala,Birra),(Ahmed,Abdi),(Ahmed,Chala),(Ahmed,Gadise),(Ahmed,Tolassa),(Gadise,Birra),(Gadise,Chala),(Tolassa,Abdi),(Tolassa,Birra) be the set of relations on H. Define E as the ambiguous set on the relation R, as shown in Table 1.

The truth, partial truth,partial falsity and falsity values of each edge are computed as tE(ab)≤min{tA∘(a),tA∘(b)},ptE(ab)≤12min{ptA∘(a),ptA∘(b)},pfE(ab)≤12max{pfA∘(a),pfA∘(b)},andfE(ab)≤max{fA∘(a),fA∘(b)} respectively.

Consider this situation as diambiguous graph from the first person to second person in the coordinate. From this graph, the most partially influenced we focus on the incoming edges and their partial truth ( pt ) and partial falsity ( pf ) values because influenced is received with incoming relationships. The incoming edges for each person is given in Table 2.

Based on the aggregated partial truth and partial falsity values, we identify the individuals who are most positively and most negatively influenced in the network. As shown in Table 2, Birraa is the most partially positively influenced individual, having the highest total incoming partial truth value (0.30). Similarly, Birraa is the most partially negatively influenced individual, as he also has the highest total incoming partial falsity value (0.30). These results indicate that Birraa’s opinions or decisions are the most strongly affected by uncertain, incomplete, or ambiguous information received from other members of the social network. This finding is further supported by the network structure, since Birraa has the highest in-degree (3), meaning that he receives influence from more individuals than any other node in the network.

The following is the general procedure algorithm for the given applications:

Graph theories have several versions, like FG, IFG and NG, for modeling uncertainty environment. In this study, an ambiguous graph was established. Then, the ambiguous (di) graph applied to model uncertainty in social media influence. This ambiguous graph analysis offers a richer picture than FG, IFG and NG, revealing not only who is influenced person but how (i.e., whether he or she influenced through positive influence or through negative, ambiguous effects. Also, the novelty of this graph in a social network is clearly observed when compared to the other graphs.

The existing graphs including neutrosophic graph (NG) framework fail to adequately describe situations where uncertainty in the vertices or edges, or both, of the graph exhibit partially true and partially false characteristics. Therefore, in this paper, we have developed the theory of ambiguous graphs (AGs) to address such scenarios. We have introduced various properties of AGs, including complete ambiguous graphs, degrees, sizes, and orders. Additionally, we have provided illustrative examples to clarify the introduced concepts.

Furthermore, we have applied the developed AG framework to social networks, demonstrating its novel contribution in this area by comparing it with other graph theory aspects such as NG, Intuitionistic Fuzzy Graphs (IFG), and Fuzzy Graphs (FG).

Looking ahead, our future research will focus on exploring additional types of ambiguous graphs, including interval-valued ambiguous graphs, regular and irregular ambiguous graphs, as well as bipolar and interval-valued ambiguous graphs. We aim to extend the applications of ambiguous graphs to areas such as image processing, decision making, and organizational networks.

Not applicable.