Chromatic polynomial of a picture fuzzy graph with application in traffic light control

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 (θ,ρ,σ)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(\theta , \rho , \sigma )$$\end{document}-level graph ((θ,ρ,σ)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(\theta , \rho , \sigma )$$\end{document}-LG) of a PFG, illustrating these concepts with an example. We then discuss the properties of the (θ,ρ,σ)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(\theta , \rho , \sigma )$$\end{document}-LGs of a PFG. Additionally, we introduce the concept of the chromatic polynomial (CP) of a PFG, providing a demonstration with an example using (θ,ρ,σ)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(\theta , \rho , \sigma )$$\end{document}-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?