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    Geometric approach to vertex-degree-based topological indices-Elliptic Sombor index, theory and application
    (Wiley, 2024) Gutman, Ivan; Furtula, Boris; Oz, Mert Sinan
    A novel geometric method is proposed for constructing vertex-degree-based molecular structure descriptors (topological indices). The model is based on an ellipse whose focal points represent the degrees of a pair of adjacent vertices. This approach enables a geometric interpretation of several previously known topological indices, and lead to design of a few new. The area of the ellipse induces a vertex-degree-based topological index of remarkable simplicity, which we call elliptic Sombor index(ESO$$ ESO $$). The main mathematical properties of ESO$$ ESO $$ are established, especially its relations to other, earlier known, indices. Then the applicative potential of ESO$$ ESO $$ is analyzed. The elliptic Sombor index (ESO) is a topological molecular descriptor derived from the geometric properties of an ellipse. It is defined by using the equation of the area of an ellipse. image
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    More geometric studies of vertex-degree-based graph indices-tangent Somb or index
    (Hacettepe Univ, Fac Sci, 2025) Oz, Mert Sinan; Gutman, Ivan
    The representation of an edge of a graph in a 2-dimensional coordinate system (shown in Fig. 1) made it possible to get a geometric interpretation of several earlier proposed vertex-degree-based graph indices. In particular, the sum of sine, cosine, and secant of the angle alpha (shown in Fig. 1) over all edges of the underlying graph yields, respectively, the second Somb or, inverse symmetric division deg, and symmetric division deg indices. Analogous trigonometric relations for the cosecant and cotangent of alpha are not possible. Therefore, the only remaining such relation is for the tangent of alpha, resulting in a new vertex-degree-based topological index, the tangent Sombor index, Tan. In this paper, the basic properties of Tan are established. Connected graphs and trees reaching extremal Tan-values are characterized. Inequalities between Tan and other graph indices are established. The chemical usefulness of Tan in terms of structure sensitivity, abruptness, degeneracy, and correlation with some physicochemical properties of octane isomers and other indices is investigated.