We perform a detailed (computational) scaling study of well-known general indices (the first and second variable Zagreb indices, $M_1^{\alpha }\left(G\right)$ and $M_2^{\alpha }\left(G\right)$, and the general sum-connectivity index, ${\chi }_{\alpha }\left(G\right)$) as well as of general versions of indices of interest: the general inverse sum indeg index $IS{I}_{\alpha }\left(G\right)$ and the general first geometric-arithmetic index $G{A}_{\alpha }\left(G\right)$ (with $\alpha \in \mathbb{R}$). We apply these indices on two models of random networks: Erdös–Rényi (ER) random networks $G_{\mathrm{ER}}(n_{\mathrm{ER}},p)$ and random geometric (RG) graphs $G_{\mathrm{RG}}(n_{\mathrm{RG}},r)$. The ER random networks are formed by $n_{\mathrm{ER}}$ vertices connected independently with probability $p\in [0,1]$; while the RG graphs consist of $n_{\mathrm{RG}}$ vertices uniformly and independently distributed on the unit square, where two vertices are connected by an edge if their Euclidean distance is less or equal than the connection radius $r\in [0,\sqrt{2}]$. Within a statistical random matrix theory approach, we show that the average values of the indices normalized to the network size scale with the average degree $⟨k⟩$ of the corresponding random network models, where $⟨k_{\mathrm{ER}}⟩=(n_{\mathrm{ER}}-1)p$ and $⟨k_{\mathrm{RG}}⟩=(n_{\mathrm{RG}}-1)(\pi r^2-8r^3/3+r^4/2)$. That is, $⟨X\left(G_{\mathrm{ER}}\right)⟩/n_{\mathrm{ER}}\approx ⟨X\left(G_{\mathrm{RG}}\right)⟩/n_{\mathrm{RG}}$ if $⟨k_{\mathrm{ER}}⟩=⟨k_{\mathrm{RG}}⟩$, with X representing any of the general indices listed above. With this work, we give a step forward in the scaling of topological indices since we have found a scaling law that covers different network models. Moreover, taking into account the symmetries of the topological indices we study here, we propose to establish their statistical analysis as a generic tool for studying average properties of random networks. In addition, we discuss the application of specific topological indices as complexity measures for random networks.

Copolymers of l-lactide and poly(propylene glycol) diglycidyl ether (PPGDGE₃₈₀) were synthesized by ring opening polymerization (ROP). Stannous octoate was used as the catalyst and 1-dodecanol as the initiator. The effect of the variables on the thermal properties of the copolymers was investigated by differential scanning calorimetry (DSC). Contact angle measurements were made in order to study the wettability of the synthesized copolymers. The copolymers differed widely in their physical characteristics, ranging from weak elastomers to tougher thermoplastics, according to the ratio of l-lactide and PPGDGE₃₈₀. The results showed that the copolymers were more hydrophilic than neat Poly(lactide) (PLA) and the monomer ratio had a strong influence on the hydrophilic properties.

The formation of a binary porphyrinic self-assembled system between meso-tetrakis(4-carboxyphenyl) porphyrin (TCPP) and meso-tetrakis(4-dimethyl amino) porphyrin (TDAP) was easily designed through non-covalent interactions in solution and adsorbed on a gold substrate. It was found that non-covalent interactions and geometrical conformations between porphyrins allow their self-assembly into a well-defined arrangement, which was confirmed by UV-Vis spectroscopy, electrochemistry, atomic force microscopy and density functional theory (DFT) studies.