The limiting behaviours for the Gutman index, Schultz index, multiplicative degree-Kirchhoff index and additive degree-Kirchhoff index of a random polyphenylene chain

The expected values and limiting behaviours for the Gutman index, Schultz index, multiplicative degree-Kirchhoff index and additive degree-kirchhoff index of a random cyclooctane chain

“…Motivated by [23], we use four random variables Z 1 n , Z 2 n , Z 3 n and Z 4 n to represent our choice. If our choice is COC i n , we put Z i n = 1, otherwise Z i n = 0 (i = 1, 2, 3, 4).…”

“…Motivated by [22,23,28,29], the rest of the paper is organized as follows. In section 2, we determine the explicit formulas of E Gut(COC n ) and E S(COC n ) for random cyclooctane chains.…”

The expected values, variances and limiting distributions of Gutman index, Schultz index, multiplicative degree-Kirchhoff index and additive degree-Kirchhoff index for a class of random chain networks

“…Refer to ref. [45], by the properties of variance, Eq. (3.23) and exchanging the order of l and m, we can directly find out that…”

“…Motivated by [39], we use two random variables Z 1 n and Z 2 n to represent our choices. If our choice is P G i n , we put Z i n = 1, otherwise Z i n = 0 (i = 1, 2).…”

“…At the same way, Zhang, Li, Li and Zhang [44] obtained the simple formulas of E Gut(G n ) , E S(G n ) , E Kf * (G n ) and E Kf + (G n ) for a class of random chain networks. Motivated by [44][45][46][47][48], for this article, in section 2, we determine the explicit formulas of E Gut(P G n ) , E S(P G n ) , E Kf * (P G n ) and E Kf + (P G n ) for the random chain networks. In section 3, we obtain explicit formulas of V ar Gut(P G n ) , V ar S(P G n ) , V ar Kf * (P G n ) and V ar Kf + (P G n ) based on the known results of others' paper, and prove that these four indices of the random chain networks asymptotically obey normal distributions.…”