Molecular trees with extremal values of Sombor indices

Abstract

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.

More on Sombor indices of chemical graphs and their applications to the boiling point of benzenoid hydrocarbons

“…If n ≥ 4, then . Therefore, we have the following conclusion.Corollary [2, 8] Let ( n ≥ 4). Then and SO red ( T ) ≤ ( n − 1)( n − 2) , with equality if and only if T ≅ S n .…”

“…The Sombor index of G is defined as [2] whereas the reduced Sombor index is [2] Recent results of Sombor index can be found in References [2–19].…”

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Let G be a connected graph with vertex set V(G) and edge set E(G). The Sombor index of G is defined asRecent results of Sombor index can be found in References [2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19].A chemical tree is a tree with d v ≤ 4 for all v ∈ V(G). For details on chemical trees, see .

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“…Recent results of Sombor index can be found in References [2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19].…”

The expected values of Sombor indices in random hexagonal chains, phenylene chains and Sombor indices of some chemical graphs

“…Das, Cevik, Cangul and Shang presented lower and upper bounds on the Sombor index of graphs by using some graph parameters and obtain several relations on Sombor index with the first and second Zagreb indices of graphs [7]. More results of Sombor indices can be found in [2,3,[8][9][10][11][12][13][14][15][16][17]37]. In Section 2, we study the expected values of the Sombor indices in the random hexagonal chains and random phenylene chains, and make a comparison between the expected values.…”