Knots are some of the most remarkable topological features in nature. Self-assembly of knotted polymers without breaking or forming covalent bonds is challenging, as the chain needs to be threaded through previously formed loops in an exactly defined order. Here we describe principles to guide the folding of highly knotted single-chain DNA nanostructures as demonstrated on a nano-sized square pyramid. Folding of knots is encoded by the arrangement of modules of different stability based on derived topological and kinetic rules. Among DNA designs composed of the same modules and encoding the same topology, only the one with the folding pathway designed according to the ‘free-end’ rule folds efficiently into the target structure. Besides high folding yield on slow annealing, this design also folds rapidly on temperature quenching and dilution from chemical denaturant. This strategy could be used to design folding of other knotted programmable polymers such as RNA or proteins.

Natural proteins are characterised by a complex folding pathway defined uniquely for each fold. Designed coiled-coil protein origami (CCPO) cages are distinct from natural compact proteins, since their fold is prescribed by discrete long-range interactions between orthogonal pairwise-interacting coiled-coil (CC) modules within a single polypeptide chain. Here, we demonstrate that CCPO proteins fold in a stepwise sequential pathway. Molecular dynamics simulations and stopped-flow Förster resonance energy transfer (FRET) measurements reveal that CCPO folding is dominated by the effective intra-chain distance between CC modules in the primary sequence and subsequent folding intermediates, allowing identical CC modules to be employed for multiple cage edges and thus relaxing CCPO cage design requirements. The number of orthogonal modules required for constructing a CCPO tetrahedron can be reduced from six to as little as three different CC modules. The stepwise modular nature of the folding pathway offers insights into the folding of tandem repeat proteins and can be exploited for the design of modular protein structures based on a given set of orthogonal modules.

In this paper the problem of the existence of regular nut graphs is addressed. A generalization of Fowler?s Construction which is a local enlargement applied to a vertex in a graph is introduced to generate nut graphs of higher order. Let N (?) denote the set of integers n such that there exists a regular nut graph of degree ? and order n. It is proven that N (3) = {12} ? {2k : k ? 9} and that N (4) = {8, 10, 12} ? {n : n ? 14}. The problem of determining N (?) for ? > 4 remains completely open.

For a finite group $G$ and subset $S$ of $G,$ the Haar graph $H(G,S)$ is a bipartite regular graph, defined as a regular $G$-cover of a dipole with $|S|$ parallel arcs labelled by elements of $S$. If $G$ is an abelian group, then $H(G,S)$ is well-known to be a Cayley graph; however, there are examples of non-abelian groups $G$ and subsets $S$ when this is not the case. In this paper we address the problem of classifying finite non-abelian groups $G$ with the property that every Haar graph $H(G,S)$ is a Cayley graph. An equivalent condition for $H(G,S)$ to be a Cayley graph of a group containing $G$ is derived in terms of $G, S$ and $\mathrm{Aut } G$. It is also shown that the dihedral groups, which are solutions to the above problem, are $\mathbb{Z}_2^2,D_3,D_4$ and $D_{5}$.

A nut graph is a simple graph whose adjacency matrix has the eigenvalue zero with multiplicity one such that its corresponding eigenvector has no zero entries. It is known that there exist no cubic circulant nut graphs. A bicirculant (resp. tricirculant) graph is defined as a graph that admits a cyclic group of automorphisms having two (resp. three) orbits of vertices of equal size. We show that there exist no cubic bicirculant nut graphs and we provide a full classification of cubic tricirculant nut graphs.

A nut graph is a simple graph for which the adjacency matrix has a single zero eigenvalue such that all non-zero kernel eigenvectors have no zero entry. If the isolated vertex is excluded as trivial, nut graphs have seven or more vertices; they are connected, non-bipartite, and have no leaves. It is shown that a nut graph $G$ always has at least one more edge orbit than it has vertex orbits: $o_e(G) \geq o_v(G) + 1$, with the obvious corollary that edge-transitive nut graphs do not exist. We give infinite familes of vertex-transitive nut graphs with two orbits of edges, and infinite families of nut graphs with two orbits of vertices and three of edges. Several constructions for nut graphs from smaller starting graphs are known: double subdivision of a bridge, four-fold subdivision of an edge, a construction for extrusion of a vertex with preservation of the degree sequence. To these we add multiplier constructions that yield nut graphs from regular (not necessarily nut graph) parents. In general, constructions can have different effects on automorphism group and counts of vertex and edge orbits, but in the case where the automorphism group is ‘preserved’, they can be used in a predictable way to control vertex and edge orbit numbers.