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Based on elementary geometry, a class of novel graph invariants was introduced by Gutman, of which the simplest is the Sombor index SO, defined as italicSO()G=∑italicuv∈EdG2()u+dG2()v, where G = (V, E) is a simple graph and dG(v) denotes the degree of v in G. In this paper, the chemical importance of the Sombor index is investigated and it is shown that the new index is useful in predicting physico‐chemical properties with high accuracy compared to some well‐established and often used indices. We obtain a sharp upper bound for the Sombor index among all molecular trees with fixed numbers of vertices, and characterize those molecular trees achieving the extremal value. Also, we obtain the extremal values of the reduced Sombor index for molecular trees.

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The Wiener index W (G) of a connected graph G is the sum of distances between all pairs of vertices in G. In this paper, we first give the recurrences or explicit formulae for computing the Wiener indices of spiro and polyphenyl hexagonal chains, which are graphs of a class of unbranched multispiro molecules and polycyclic aromatic hydrocarbons, then we establish a relation between the Wiener indices of a spiro hexagonal chain and its corresponding polyphenyl hexagonal chain, and determine the extremal values and characterize the extremal graphs with respect to the Wiener index among all spiro and polyphenyl hexagonal chains with n hexagons, respectively. An interesting result shows that the average value of the Wiener indices with respect to the set of all such hexagonal chains is exactly the average value of the Wiener indices of three special hexagonal chains, and is just the Wiener index of the meta-chain.