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 . Let T n (CT n ) be the set of trees (chemical trees) with n vertices. Let T n,k (CT n,k ) be the set of trees (chemical trees) with n vertices and k pendent vertices.

Hexagonal chains are a special class of catacondensed benzenoid system and phenylene chains are a class of polycyclic aromatic compounds. Recently, A family of Sombor indices was introduced by Gutman in the chemical graph theory. It had been examined that these indices may be successfully applied on modeling thermodynamic properties of compounds. In this paper, we study the expected values of the Sombor indices in random hexagonal chains, phenylene chains, and consider the Sombor indices of some chemical graphs such as graphene, coronoid systems and carbon nanocones.

Let G = (V (G), E(G)) be a simple graph with vertex set V (G) = {v 1 , v 2 , • • • , v n } and edge set E(G). The p-Sombor matrix S p (G) of G is the square matrix of order n whose (i, j)-entry is equal to ((d i ) p + (d j ) p ) 1 p if v i ∼ v j , and 0 otherwise, where d i denotes the degree of vertex v i in G. In this paper, we study the relationship between p-Sombor index SO p (G) and p-Sombor matrix S p (G) by the k-th spectral moment N k and the spectral radius of S p (G). Then we obtain some bounds of p-Sombor Laplacian eigenvalues, p-Sombor spectral radius, p-Sombor spectral spread, p-Sombor energy and p-Sombor Estrada index. We also investigate the Nordhaus-Gaddum-type results for p-Sombor spectral radius and energy. At last, we give the regression model for boiling point and some other invariants.

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This article is devoted to obtain a general method of calculating the expected values and variances for degree-based topological indices in random hexagonal, phenylene and polyphenyl chains. Based on the general method, some important degree-based topological indices are discussed and the explicit analytical expressions of their expected values and variances are presented, in which some known results are included. Besides, the expected values and variances for degree-based topological indices in these random chains are compared. In the end, the extremal values and the average values for degree-based topological indices in these random chains are determined.