Polyhedral cages (p-cages) describe the geometry of some families of artificial protein cages. We identify the p-cages made out of families of equivalent polygonal faces such that the faces of 1 family has 5 neighbours and P_1 edges, while those of the other family have 6 neighbours and P_2 edges. We restrict ourselves to polyhedral cages where the holes are adjacent to at most 4 faces. We characterise all the p-cages with a deformation of the faces, compared to regular polygons, not exceeding 10%.

Symmetry, as a key concept in geometry and the foundational principle of mathematical concepts, is a fundamental part of geometry and nature, creating patterns that help us conceptually organize our world. It is often linked with aesthetic beauty and can be observed in both natural phenomena and human-made objects, such as artworks and manufactured products, across the globe. The automorphism group of a graph is a mathematical structure that captures all the symmetries of the graph. symmetry refers to the property of a graph, indicating that it can be transformed without changing its structure, while the automorphism group is the collection of all permutations of the vertices that preserve the graph's adjacency relationships. In graph theory, the concepts of symmetry group and automorphism group play fundamental roles in understanding the symmetries and structural properties of graphs. This paper aims to explore the relationship between symmetry groups and automorphism groups, shedding light on their interconnected nature and their significance in graph theory. We delve into the definitions and properties of both symmetry groups and automorphism groups, highlighting their similarities and distinctions. Furthermore, we discuss their applications in various fields, including computer science, chemistry, and network analysis. Through this exploration, we aim to provide a comprehensive understanding of the relationship between symmetry groups and automorphism groups, deepening our comprehension of the symmetries and structures inherent in graphs.

Polyhedral Oligomeric Silsesquioxanes (POSS) are one of the smallest silica nano particles. The presence of ordered ladder-like or cage conformation in their structure leads to remarkable thermal stability far greater than that for a linear polysiloxane. Reinforcement with nano particles is one of the recently favored strategy used to enhance the performance of polymers materials. The cyclic silica structure present in POSS ensures excellent thermal and mechanical stability. As a reinforcing filler, POSS can enhance the resistance of polymer matrix against pyrolysis and corrosion. In this paper, amine-terminated POSS with ladder like structure was synthesized by condensation of hydrolyzed aminosilane coupling agent. The ladder-like structure of POSS severely limits the motion of the chain backbone. The absence of phase transition for the synthesized POSS was confirmed by differential scanning calorimetry (DSC). Thermogravimetric analysis (TGA) was utilized to evaluate the thermal stability of POSS. TGA data show that the ladder-like POSS reported here has high char yield of about 75 % in nitrogen atmosphere and 54% in air atmosphere at 800℃. The TGA derivative curves show that the degradation of ladder-like POSS involves multiple stages and that the degradation rate is directly proportional to the heating rate. With the aid of Fourier Transform Infrared Spectroscopy (FT-IR), the cyclic structure of Si-O-Si bonding was determined to be the predominant structure. By analyzing the FTIR spectra of the cured POSS residue obtained at the specified temperatures, the detailed degradation mechanism of POSS was inferred. From the FTIR spectra of the low temperature cured samples, it was shown that the terminal silanol group was eliminated below 400℃. Some intermediate polymeric siloxane structures were observed by means of FT-IR and they eventually transitioned into silica which is the final product of pyrolysis of POSS. The presence of the FTIR transmission peaks at 1000 and 1100 cm-1, due to asymmetrical vertical and horizontal stretching of Si-O-Si, were the key evidence for the inferred ladder-like structure of POSS.