The fuzzy graph theory has many applications in solving various problems in multiple fields, including networking, communications, clustering, planning, and scheduling. In this article, we introduce the fundamental set of a picture fuzzy graph (PFG) and the \((\theta , \rho , \sigma )\) -level graph ( \((\theta , \rho , \sigma )\) -LG) of a PFG, illustrating these concepts with an example. We then discuss the properties of the \((\theta , \rho , \sigma )\) -LGs of a PFG. Additionally, we introduce the concept of the chromatic polynomial (CP) of a PFG, providing a demonstration with an example using \((\theta , \rho , \sigma )\) -LGs of a PFG and exploring various properties of CP of a PFG. Furthermore, we construct an algorithm, which is executed using Matlab. This algorithm enables us to determine the number of distinct ways to colour a PFG with a specified number of colours. The application of the work presented is to answer the question: How does the picture fuzzy model propose to rectify the inefficiency in traffic signal timing where equal time is allocated to each vehicle in a flow based on vehicle count, leading to potential delays for vehicles with different characteristics, such as motorcycles and containers?
