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In this paper, we first introduce the explicit analytical formulas for the expected values of the Gutman and Schultz indices for a random cyclooctane chain COCn. Meanwhile, the explicit formulas of the variances of the Gutman and Schultz indices for a random cyclooctane chain are determined and we prove these two indices are asymptotically subject to normal distribution. Furthermore, we are surprised to find the expected values of the Sombor index SOCOCn, the reduced Sombor index RSOCOCn, the modified Sombor index mSOCOCn and the reduced and modified Sombor index mRSOCOCn for a random cyclooctane chain. [ABSTRACT FROM AUTHOR]

The study of topological indices of some special chemical graphs has become a hot topic in chemical theory. Let Ln be the dicyclobutadieno derivative of linear phenylene which consists of n hexagons and (n + 1) quadrangles and let Ln2 be the strong prism of the dicyclobutadieno derivative of linear phenylenes. Firstly, we obtain that the normalized Laplacian spectrum of Ln2 consists of the eigenvalues of symmetric matrices LA and LS according to the decomposition theorem of normalized Laplacian characteristic polynomial. Then, by taking full advantage of the relationship between the roots and coefficients of the characteristic polynomials of the above two matrices, we determine the explicit expressions of the multiplicative degree‐Kirchhoff index and the number of spanning trees of the graph Ln2. [ABSTRACT FROM AUTHOR]

The re‐establishment of submerged macrophytes is crucial in shallow lake restoration transforming a turbid into a clear water system, but it might be inhibited by juvenile omni‐benthivorous fish. These fishes disturb the sediment, and even uproot new sprouts, creating turbid water through sediment resuspension and release of nutrients stored in the sediment. Lanthanum modified bentonite (LMB), an in‐situ capping material, can immobilise phosphate and consolidate sediments, potentially weakening the negative effects of juvenile omni‐benthivorous fish.We hypothesised that applying LMB would improve water quality and promote the growth of submerged macrophytes by eliminating or weakening the increase of suspended solids, nutrient concentration, and algal reproduction due to juvenile omni‐benthivorous fish disturbance. To test our hypothesis, a two‐by‐two factorial mesocosm experiment with two densities of juvenile crucian carp (absence and 10 g/m3) and two doses of LMB (absence and 800 g/m2) was conducted.Juvenile crucian carp significantly inhibited the growth and reproduction of the macrophyte Vallisneria natans by reducing the underwater light and direct effects such as uprooting. No significant inhibition or promotion by LMB on the growth of V. natans in the absence of juvenile crucian carp was observed. However, LMB significantly inhibited the fish‐induced sediment resuspension and nutrients increases. Consequently, algal production was reduced and underwater light conditions were improved, which promoted the growth and reproduction of V. natans, e.g., more ramets, biomass, and leaves of V. natans.Our results indicated that submerged macrophytes could benefit from LMB application under high intensity disturbance by juvenile omni‐benthivorous fish, thereby conducive to the maintenance of a clear water state in shallow lakes. [ABSTRACT FROM AUTHOR]