The Exact Linear Turán Number of the Sail

A hypergraph is linear if any two of its edges intersect in at most one vertex. The sail (or $3$-fan) $F^3$ is the $3$-uniform linear hypergraph consisting of $3$ edges $f_1, f_2, f_3$ pairwise intersecting in the same vertex $v$ and an additional edge $g$ intersecting each $f_i$ in a vertex different from $v$. The linear Turán number $\mathrm{ex}_{\mathrm{lin}}(n, F^3)$ is the maximum number of edges in a $3$-uniform linear hypergraph on $n$ vertices that does not contain a copy of $F^3$. Füredi and Gyárfás proved that if $n = 3k$, then $\mathrm{ex}_{\mathrm{lin}}(n, F^3) = k^2$ and the only extremal hypergraphs in this case are transversal designs. They also showed that if $n = 3k+2$, then $\mathrm{ex}_{\mathrm{lin}}(n, F^3) = k^2+k$, and the only extremal hypergraphs are truncated designs (which are obtained from a transversal design on $3k+3$ vertices with $3$ groups by removing one vertex and all the hyperedges containing it) along with three other small hypergraphs. However, the case when $n =3k+1$ was left open. In this paper, we solve this remaining case by proving that $\mathrm{ex}_{\mathrm{lin}}(n, F^3) = k^2+1$ if $n = 3k+1$, answering a question of Füredi and Gyárfás. We also characterize all the extremal hypergraphs. The difficulty of this case is due to the fact that these extremal examples are rather non-standard. In particular, they are not derived from transversal designs like in the other cases.

Compressed graph representation for scalable molecular graph generation

Recently, deep learning has been successfully applied to molecular graph generation. Nevertheless, mitigating the computational complexity, which increases with the number of nodes in a graph, has been a major challenge. This has hindered the application of deep learning-based molecular graph generation to large molecules with many heavy atoms. In this study, we present a molecular graph compression method to alleviate the complexity while maintaining the capability of generating chemically valid and diverse molecular graphs. We designate six small substructural patterns that are prevalent between two atoms in real-world molecules. These relevant substructures in a molecular graph are then converted to edges by regarding them as additional edge features along with the bond types. This reduces the number of nodes significantly without any information loss. Consequently, a generative model can be constructed in a more efficient and scalable manner with large molecules on a compressed graph representation. We demonstrate the effectiveness of the proposed method for molecules with up to 88 heavy atoms using the GuacaMol benchmark.

Fast Algorithms for Diameter-Optimally Augmenting Paths and Trees

We consider the problem of augmenting an [Formula: see text]-vertex graph embedded in a metric space, by inserting one additional edge in order to minimize the diameter of the resulting graph. We present exact algorithms for the cases when (i) the input graph is a path, running in [Formula: see text] time, and (ii) the input graph is a tree, running in [Formula: see text] time. We also present an algorithm for paths that computes a [Formula: see text]-approximation in [Formula: see text] time.

Engineering additional edge sites on molybdenum dichalcogenides toward accelerated alkaline hydrogen evolution kinetics

Molybdenum dichalcogenidebased heterostructures deliver substantially improved catalytic activity over the individual nanosheets in alkaline media.

Extending Perfect Matchings to Gray Codes with Prescribed Ends

A binary (cyclic) Gray code is a (cyclic) ordering of all binary strings of the same length such that any two consecutive strings differ in a single bit. This corresponds to a Hamiltonian path (cycle) in the hypercube. Fink showed that every perfect matching in the $n$-dimensional hypercube $Q_n$ can be extended to a Hamiltonian cycle, confirming a conjecture of Kreweras. In this paper, we study the "path version" of this problem. Namely, we characterize when a perfect matching in $Q_n$ extends to a Hamiltonian path between two prescribed vertices of opposite parity. Furthermore, we characterize when a perfect matching in $Q_n$ with two faulty vertices extends to a Hamiltonian cycle. In both cases we show that for all dimensions $n\ge 5$ the only forbidden configurations are so-called half-layers, which are certain natural obstacles. These results thus extend Kreweras' conjecture with an additional edge, or with two faulty vertices. The proof for the case $n=5$ is computer-assisted.

Extremal graphs for the sum of the two largest signless Laplacian eigenvalues

Let G be a simple graph on n vertices and e(G) edges. Consider the signless Laplacian, Q(G) = D + A, where A is the adjacency matrix and D is the diagonal matrix of the vertices degree of G. Let q_1(G) and q_2(G) be the first and the second largest eigenvalues of Q(G), respectively, and denote by S_n^+ the star graph with an additional edge. It is proved that inequality q_1(G)+q_2(G) \leq e(G)+3 is tighter for the graph S_n^+ among all firefly graphs and also tighter to S_n^+ than to the graphs K_k \vee K_{n−k} recently presented by Ashraf, Omidi and Tayfeh-Rezaie. Also, it is conjectured that S_n^+ minimizes f(G) = e(G) − q_1(G) − q_2(G) among all graphs G on n vertices.