Uncertainty-aware prediction of chemical reaction yields with graph neural networks

In this paper, we present a data-driven method for the uncertainty-aware prediction of chemical reaction yields. The reactants and products in a chemical reaction are represented as a set of molecular graphs. The predictive distribution of the yield is modeled as a graph neural network that directly processes a set of graphs with permutation invariance. Uncertainty-aware learning and inference are applied to the model to make accurate predictions and to evaluate their uncertainty. We demonstrate the effectiveness of the proposed method on benchmark datasets with various settings. Compared to the existing methods, the proposed method improves the prediction and uncertainty quantification performance in most settings.

Computing Correlation among the Graphs under Lexicographic Product via Zagreb Indices

A topological index (TI) is a numerical descriptor of a molecule structure or graph that predicts its different physical, biological, and chemical properties in a theoretical way avoiding the difficult and costly procedures of chemical labs. In this paper, for two connected (molecular) graphs G 1 and G 2 , we define the generalized total-sum graph consisting of various (molecular) polygonal chains by the lexicographic product of the graphs T k G 1 and G 2 , where T k G 1 is obtained by applying the generalized total operation T k on G 1 with k ≥ 1 as some integral value. Moreover, we compute the different degree-based TIs such as first Zagreb, second Zagreb, forgotten Zagreb, and hyper-Zagreb. In the end, a comparison among all the aforesaid TIs is also conducted with the help of certain statistical tools for some particular families of generalized total-sum graphs under lexicographic product.

Some Bond Incident Degree Indices of (Molecular) Graphs with Fixed Order and Number of Cut Vertices

A bond incident degree (BID) index of a graph G is defined as ∑ u v ∈ E G f d G u , d G v , where d G w denotes the degree of a vertex w of G , E G is the edge set of G , and f is a real-valued symmetric function. The choice f d G u , d G v = a d G u + a d G v in the aforementioned formula gives the variable sum exdeg index SEI a , where a ≠ 1 is any positive real number. A cut vertex of a graph G is a vertex whose removal results in a graph with more components than G has. A graph of maximum degree at most 4 is known as a molecular graph. Denote by V n , k the class of all n -vertex graphs with k ≥ 1 cut vertices and containing at least one cycle. Recently, Du and Sun [AIMS Mathematics, vol. 6, pp. 607–622, 2021] characterized the graphs having the maximum value of SEI a from the set V n k for a > 1 . In the present paper, we not only characterize the graphs with the minimum value of SEI a from the set V n k for a > 1 , but we also solve a more general problem concerning a special type of BID indices. As the obtained extremal graphs are molecular graphs, they remain extremal if one considers the class of all n -vertex molecular graphs with k ≥ 1 cut vertices and containing at least one cycle.

Entropy Measures of Distance Matrix

Bonchev and Trinajstic defined two distance based entropy measures to measure the molecular branching of molecular graphs in 1977 [Information theory, distance matrix, and molecular branching, J. Chem. Phys., 38 (1977), 4517–4533]. In this paper we use these entropy measures which are based on distance matrices of graphs. The first one is based on distribution of distances in distance matrix and the second one is based on distribution of distances in upper triangular submatrix. We obtain the two entropy measures of paths, stars, complete graphs, cycles and complete bipartite graphs. Finally we obtain the minimal trees with respect to these entropy measures with fixed diameter.

Modified Reduced Neighborhood Topological Indices of Cycloalkanes, Clycloalkenes, and Cycloalkynes

In this paper, some newly introduced topological indices, namely, modified reduced neighborhood first Zagreb index (〖RNM〗_1^* (G)) and modified forgotten reduced neighborhood index (〖RNM〗_3^* (G)) of a graph, G are used to establish the general formula for the molecular graphs of cycloalkanes, cycloalkenes, and cycloalkynes. We also calculate the exact values of some special types of these molecular graphs by attaching the alkyls instead of hydrogen atoms. Also, the QSPR analysis for these newly introduced indices is studied, and we exhibit a good correlation with characteristics on these molecular graphs.

ON THE WIENER INDEX OF F_H SUMS OF GRAPHS

Wiener index is the first among the long list of topological indices which was used to correlate structural and chemical properties of molecular graphs. In \cite{Eli} M. Eliasi, B. Taeri defined four new sums of graphs based on the subdivision of edges with regard to the cartesian product and computed their Wiener index. In this paper, we define a new class of sums called $F_H$ sums and compute the Wiener index of the resulting graph in terms of the Wiener indices of the component graphs so that the results in \cite{Eli} becomes a particular case of the Wiener index of $F_H$ sums for $H = K_1$, the complete graph on a single vertex.

Characterization of (Molecular) Graphs with Fractional Metric Dimension as Unity

Distance-based dimensions provide the foreground for the identification of chemical compounds that are chemically and structurally different but show similarity in different reactions. The reason behind this similarity is the occurrence of a set S of atoms and their same relative distances to some ordered set T of atoms in both compounds. In this article, the aforementioned problem is considered as a test case for characterising the (molecular) graphs bearing the fractional metric dimension (FMD) as 1. For the illustration of the theoretical development, it is shown that the FMD of path graph is unity. Moreover, we evaluated the extremal values of fractional metric dimension of a tetrahedral diamond lattice.

On Computation and Analysis of Entropy Measures for Crystal Structures

In recent years, the study of topological indices associated to different molecular tubes and structures gained a lot of attention of the researchers—working in Chemistry and Mathematics. These descriptors play an important role in describing different properties associated to the objects of study. Moreover, Shannon’s entropy concept—a slightly different but more effective approach—provides structural information related to the molecular graphs. In this article, we have computed and analyzed different entropy measures associated to different crystallographic structures. In particular, we have worked on the Zagreb entropies, hyper and augmented Zagreb entropies, and forgotten and Balaban entropies for the crystallographic structures of the cuprite Cu 2 O and titanium difluoride TiF 2 .