Molecular Directed, Bidirected, and Multidirected Graphs

Authors

DOI:

https://doi.org/10.22105/kmisj.v2i4.99

Keywords:

Directed graph, Bidirected graph, Multidirected graph, Molecular graph

Abstract

A directed graph (or digraph) consists of a finite vertex set V and a set of ordered edges E ⊆ V × V , each edge (u, v) indicating a one-way connection from u (source) to v (target). A bidirected graph is a generalization of an undirected graph where each edge is assigned a direction at each of its endpoints independently, allowing more expressive edge orientation. A multidirected graph is a structure with vertices and edges, where edges may repeat, sources and targets are assigned, and multiplicities recorded. A molecular graph models a molecule with atoms as vertices and bonds as edges, representing its structural connectivity. In this paper, we examine definitions such as molecular bidirected graphs and multidirected graphs. These are concepts that extend molecular graphs by incorporating directional information.

References

Diestel, R. (2024). Graph theory. Springer. https://doi.org/10.1007/978-3-662-70107-2

Zhang, P., & Chartrand, G. (2006). Introduction to graph theory. New York: Tata McGraw-Hill. https://openli-

brary.org/books/OL37078322M/Introduction_to_graph_theory

Gurski, F., Rehs, C., & Rethmann, J. (2018). Directed path-width of sequence digraphs. International conference on combinatorial optimization and applications (pp. 79-93). Cham: Springer International Publishing. https://doi.org/10.1007/978-3-030-04651-4_6

Xu, R., & Zhang, C. Q. (2005). On flows in bidirected graphs. Discrete mathematics, 299(1-3), 335-343.

https://doi.org/10.1016/j.disc.2004.06.023

Kita, N. (2017). Bidirected graphs I: Signed general Kotzig-Lov'asz decomposition.

https://doi.org/10.48550/arXiv.1709.07414

Weith Jr, A. J., Hobbs, M. E., & Gross, P. M. (1948). The electric moments of hydrogen fluoride, hydrogen chloride and

hydrogen bromide in several non-polar solvents1. Journal of the American chemical society, 70(2), 805-811. https://doi.org/10.1021/ja01182a110

Plumley, J. A., & Evanseck, J. D. (2007). Covalent and ionic nature of the dative bond and account of accurate ammonia

borane binding enthalpies. The journal of physical chemistry a, 111(51), 13472-13483. https://doi.org/10.1021/jp074937z

Pardo-Guerra, S., George, V. K., Morar, V., Roldan, J., & Silva, G. A. (2024). Extending undirected graph techniques to

directed graphs via category theory. Mathematics, 12(9), 1357. https://doi.org/10.3390/math12091357

Fujita, T. (2025). Extensions of multidirected graphs: Fuzzy, Neutrosophic, Plithogenic, Rough, soft, hypergraph, and su-

perhypergraph variants. International journal of topology, 2(3), 11. https://doi.org/10.3390/ijt2030011

Pardo-Guerra, S., George, V. K., & Silva, G. A. (2025). On the graph isomorphism completeness of directed and multidi-

rected graphs. Mathematics, 13(2), 228. https://doi.org/10.3390/math13020228

Braunschweig, H., Dellermann, T., Dewhurst, R. D., Ewing, W. C., Hammond, K., Jimenez-Halla, J. O. C., ... & Vargas,

A. (2013). Metal-free binding and coupling of carbon monoxide at a boron–boron triple bond. Nature chemistry, 5(12), 1025-1028. https://doi.org/10.1038/nchem.1778

Kearnes, S., McCloskey, K., Berndl, M., Pande, V., & Riley, P. (2016). Molecular graph convolutions: Moving beyond fin-

gerprints. Journal of computer-aided molecular design, 30(8), 595-608. https://doi.org/10.1007/s10822-016-9938-8

Gutman, I., & Estrada, E. (1996). Topological indices based on the line graph of the molecular graph. Journal of chemical

information and computer sciences, 36(3), 541-543. https://doi.org/10.1021/ci950143i

You, J., Liu, B., Ying, R., Pande, V., & Leskovec, J. (2018). Graph convolutional policy network for goal-directed molecular

graph generation. Advances in neural information processing systems, 31, 6410–6421. https://proceedings.neurips.cc/paper/2018/file/d60678e8f2ba9c540798ebbde31177e8-Paper.pdf

Gasteiger, J., Groß, J., & Günnemann, S. (2020). Directional message passing for molecular graphs.

https://doi.org/10.48550/arXiv.2003.03123

Jin, W., Barzilay, R., & Jaakkola, T. (2020). Hierarchical generation of molecular graphs using structural motifs. Interna-

tional conference on machine learning (pp. 4839-4848). PMLR. https://proceedings.mlr.press/v119/jin20a/jin20a.pdf

Jin, W., Barzilay, R., & Jaakkola, T. (2018). Junction tree variational autoencoder for molecular graph generation. Inter-

national conference on machine learning (pp. 2323-2332). PMLR. https://proceedings.mlr.press/v80/jin18a/jin18a.pdf

Fujita, T. (2025). An introduction and reexamination of molecular hypergraph and molecular n-superhypergraph. Asian

journal of physical and chemical sciences, 13(3), 1-38. https://doi.org/10.9734/ajopacs/2025/v13i3248

Rahman, A., Poirel, C. L., Badger, D. J., & Murali, T. M. (2012). Reverse engineering molecular hypergraphs. Proceedings

of the ACM conference on bioinformatics, computational biology and biomedicine (pp. 68-75). Association for Computing Machinery (ACM). https://doi.org/10.1145/2382936.2382945

Chen, J., & Schwaller, P. (2024). Molecular hypergraph neural networks. The journal of chemical physics, 160(14), 144307.

https://doi.org/10.1063/5.0193557

Kajino, H. (2019). Molecular hypergraph grammar with its application to molecular optimization. International confer-

ence on machine learning (pp. 3183-3191). PMLR. https://proceedings.mlr.press/v97/kajino19a/kajino19a.pdf

Crabtree, R. H. (1995). Aspects of methane chemistry. Chemical reviews, 95(4), 987-1007. https://pire-

ecci.ucsb.edu/pire-ecci-old/summerschool/papers/ChemRev.pdf

Chai, W. S., Bao, Y., Jin, P., Tang, G., & Zhou, L. (2021). A review on ammonia, ammonia-hydrogen and ammonia-

methane fuels. Renewable and sustainable energy reviews, 147, 111254. https://doi.org/10.1016/j.rser.2021.111254

Shen, D., Song, H., Zou, T., Raza, A., Li, P., Li, K., & Xiong, J. (2022). Reduction of sodium chloride: A review. Journal of

the science of food and agriculture, 102(10), 3931-3939. https://doi.org/10.1002/jsfa.11859

Scatena, L. F., & Richmond, G. L. (2001). Orientation, hydrogen bonding, and penetration of water at the organic/water

interface. The journal of physical chemistry b, 105(45), 11240-11250. https://doi.org/10.1021/jp0132174

Yoon, Y. K., & Carpenter, G. B. (1959). The crystal structure of hydrogen chloride monohydrate. Acta crystallographica,

(1), 17-20. https://doi.org/10.1107/S0365110X59000056

Mutikainen, I. L. P. O., & Lumme, P. A. A. V. O. (1980). The structure of Diammine (Orotato) copper (II). Structural sci-

ence, 36(10), 2233-2237. https://doi.org/10.1107/S0567740880008485

Herrebout, W. A., Lundell, J., & Van der Veken, B. J. (1999). Carbon−carbon triple bonds as Nucleophiles: Adducts of

Ethyne and Propyne with Boron Trifluoride. The journal of physical chemistry a, 103(38), 7639-7645. https://doi.org/10.1021/jp992010w

Sheline, R. K. (1951). The spectra and structure of iron Carbonyls. II. iron Tetracarbonyl. Journal of the American chemical

society, 73(4), 1615-1618. https://doi.org/10.1021/ja01148a060

Guais, A., Brand, G., Jacquot, L., Karrer, M., Dukan, S., Grévillot, G., ... & Schwartz, L. (2011). Toxicity of carbon dioxide:

A review. Chemical research in toxicology, 24(12), 2061-2070. https://doi.org/10.1021/tx200220r

Sakakura, T., Choi, J. C., & Yasuda, H. (2007). Transformation of carbon dioxide. Chemical reviews, 107(6), 2365-2387.

https://doi.org/10.1021/cr068357u

Hughes, A. K., & Wade, K. (2000). Metal–metal and metal–Ligand bond strengths in metal Carbonyl clusters. Coordination

chemistry reviews, 197(1), 191-229. https://doi.org/10.1016/S0010-8545(99)00208-8

Rosenfeld, A. (1975). Fuzzy graphs. In Fuzzy sets and their applications to cognitive and decision processes (pp. 77-95). Aca-

demic Press. https://doi.org/10.1016/B978-0-12-775260-0.50008-6

Parvathi, R., Karunambigai, M. G., & Atanassov, K. T. (2009). Operations on intuitionistic fuzzy graphs. 2009 IEEE inter-

national conference on fuzzy systems (pp. 1396-1401). IEEE. https://doi.org/10.1109/FUZZY.2009.5277067

Akram, M., Davvaz, B., & Feng, F. (2013). Intuitionistic fuzzy soft k-algebras. Mathematics in computer science, 7(3), 353-

https://doi.org/10.1007/s11786-013-0158-5

Ghosh, J., & Samanta, T. K. (2012). Hyperfuzzy set and hyperfuzzy group. International journal of advanced science and

technology, 41, 27-38. https://article.nadiapub.com/IJAST/vol41/3.pdf

Berge, C. (1984). Hypergraphs: Combinatorics of finite sets. Elsevier. https://www.semanticscholar.org/paper/Hypergraphs

%3A-Combinatorics-of-Finite-Sets-Berge/cd39f475d805c5eb0e0e52d51ce72fce041493f8

Bretto, A. (2013). Hypergraph theory. An introduction. Mathematical engineering. Cham: Springer.

https://doi.org/10.1007/978-3-319-00080-0

Florentin Smarandache. (2020). Extension of hypergraph to n-superhypergraph and to Plithogenic n-superhypergraph, and

extension of hyperalgebra to n-ary (classical-/Neutro-/anti-) hyperalgebra. Neutrosophic sets and systems, 33, 290–296. https://doi.org/10.5281/zenodo.3783103

Hamidi, M., Smarandache, F., & Davneshvar, E. (2022). Spectrum of superhypergraphs via flows. Journal of mathematics,

(1), 9158912. https://doi.org/10.1155/2022/9158912

Akram, M., Malik, H. M., Shahzadi, S., & Smarandache, F. (2018). Neutrosophic soft rough graphs with application. Axioms,

(1), 14. https://doi.org/10.3390/axioms7010014

Sultana, F., Gulistan, M., Ali, M., Yaqoob, N., Khan, M., Rashid, T., & Ahmed, T. (2022). A study of Plithogenic graphs:

Applications in spreading coronavirus disease (Covid-19) globally. Journal of ambient intelligence and humanized computing, 14, 13139–13159. https://doi.org/10.1007/s12652-022-03772-6

Published

2025-12-04

How to Cite

Fujita, T. (2025). Molecular Directed, Bidirected, and Multidirected Graphs. Karshi Multidisciplinary International Scientific Journal, 2(4), 212-222. https://doi.org/10.22105/kmisj.v2i4.99