Definitions of Turiyam Wheels, Star, Trees, and Forest

<ns5:p>Background Turiyam graphs widen neutrosophic graphs by incorporating a fourth component that represents the liberal state. This featres enables more effective modeling of complex and uncertain relationships. Although basic definitions of Turiyam graphs exist, specific families such as wheels, stars, trees, and forests have not yet been formally defined. The absence of these definitions limits the applicability of Turiyam graph. Methods In this study, we present the concepts of Turiyam wheels, stars, trees, and forests through the four-part Turiyam set framework. For each type, we expand standard graph definitions to incorporate truth, indeterminacy, falsity, and liberal values for both vertices and edges. The definitions are supported with illustrative examples for consistency and validity, and are compared to neutrosophic and fuzzy graphs. Results We introduced the new Turiyam concepts: the Turiyam wheel, Turiyam star, Turiyam tree, and Turiyam forest. We also provide their formal definitions. Examples show that these structures maintain the key properties of their classical versions while allowing for the representation of four-dimensional uncertainty. Conclusions The proposed definitions and concepts provide a strong foundation for further exploration of Turiyam graph theory. This includes ideas such as spanning trees, cycles, connectivity, and algorithmic uses. These concepts also allow for modeling real-world systems where truth, uncertainty, falsehood, and neutrality coexist. This approach extends the application of Turiyam graph in network science and intelligent systems.</ns5:p>