WIENER TYPE TOPOLOGICAL INDICES OF SOME WELL-KNOWN UNICYCLIC GRAPHS

Topological indices are mathematical tools that numerically express the topological properties of molecular structures that can be represented by graphs. These indices are widely used in various disciplines such as biology, computer science, and network theory, as well as chemistry. In graph theory, many topological indices have been defined to measure different topological properties. However, explicit formulations of certain Wiener-type distance-based indices that take into account the odd–even structure of the number of vertices and their behavior on some special unicyclic graph families remain limited in the literature. In this work, Wiener-type distance-based topological indices (Wiener, Wiener polarity, hyper-Wiener, Harary, reciprocal complementary Wiener, and terminal Wiener) are discussed. First, these indices for the well-known cycle graph C_n and the path graph P_n are recalculated by considering the odd and even cases of n. Then, these Wiener-type indices for the turnip graph, lollipop graph, and sun graph, which are well-known unicyclic graphs, are calculated and presented depending on the graph parameters and the odd–even status of n. The results obtained in this study extend the existing results in the literature by providing explicit expressions for these indices under parity conditions and for specific unicyclic graph structures. Therefore, this work contributes to a better understanding and comparison of the topological properties of these graph families and provides a useful basis for future studies on topological index calculations for graphs with similar structures.

Topological indices are mathematical tools that numerically express the topological properties of molecular structures that can be represented by graphs. These indices are widely used in various disciplines such as biology, computer science, and network theory, as well as chemistry. In graph theory, many topological indices have been defined to measure different topological properties. However, explicit formulations of certain Wiener-type distance-based indices that take into account the odd–even structure of the number of vertices and their behavior on some special unicyclic graph families remain limited in the literature. In this work, Wiener-type distance-based topological indices (Wiener, Wiener polarity, hyper-Wiener, Harary, reciprocal complementary Wiener, and terminal Wiener) are discussed. First, these indices for the well-known cycle graph C_n and the path graph P_n are recalculated by considering the odd and even cases of n. Then, these Wiener-type indices for the turnip graph, lollipop graph, and sun graph, which are well-known unicyclic graphs, are calculated and presented depending on the graph parameters and the odd–even status of n. The results obtained in this study extend the existing results in the literature by providing explicit expressions for these indices under parity conditions and for specific unicyclic graph structures. Therefore, this work contributes to a better understanding and comparison of the topological properties of these graph families and provides a useful basis for future studies on topological index calculations for graphs with similar structures.