Molecular Directed, Bidirected, and Multidirected Graphs
DOI:
https://doi.org/10.22105/kmisj.v2i4.99Keywords:
Directed graph, Bidirected graph, Multidirected graph, Molecular graphAbstract
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