More geometric studies of vertex-degree-based graph indices--tangent Sombor index

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 Sombor, 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.