Enumerate the Number of Vertices Labeled Connected Graph of Order Seven Containing No Parallel Edges

Muslim Ansori, Wamiliana, Fitriani, Yudi Antoni, Desiana Putri


A graph that is connected G(V,E) is a graph in which there is at least one path connecting every two vertices in G; otherwise, it is called a disconnected graph. Labels or values can be assigned to the vertices or edges of a graph. A vertex-labeled graph is one in which only the vertices are labeled, and an edges-labeled graph is one in which only edges are assigned values or labels. If both vertices and edges are labeled, the graph is referred to as total labeling. If given n vertices and m edges, numerous graphs can be made, either connected or disconnected. This study will be discussed the number of disconnected vertices labeled graphs of order seven containing no parallel edges and may contain loops. The results show that number of vertices labeled connected graph of order seven with no parallel edges is N(G7,m, g)l= 6,727×Cm6; while for 7≤g≤ 21, N(G7,m, g)l= kg C(m−(g−6))g−1, where k7 =30,160, k8 = 30,765, k9=21,000, k10 =28,364, k11= 26,880, k12=26,460 , k13 = 20,790, k14 =10,290, k15 = 8,022, k16 = 2,940, k17 =4,417, k18 = 2,835, k19 =210, k20 = 21, k21= 1.


Agnarsson, G. and R. Greenlaw (2006). Graph Theory: Modeling, Applications, and Algorithms. Prentice-Hall, Inc.

Al Etaiwi, W. M. (2014). Encryption Algorithm Using Graph Theory. Journal of Scientific Research and Reports, 3(19); 2519–2527

Álvarez, M. C. and D. Ehnts (2015). The Roads Not Taken: Graph Theory and Macroeconomic Regimes in Stock-Flow Consistent Modeling. Working Paper

Amanto, A., N. Notiragayu, L. Zakaria, and W. Wamiliana (2021). The Relationship of the Formulas for the Number of Connected Vertices Labeled Graphs with Order Five and Order Six without Loops. Desimal: Jurnal Matematika, 4(3); 357–364

Amanto, A., W. Wamiliana, M. Usman, and R. Permatasari (2017). Counting the Number of Disconnected Vertex Labelled Graphs with Order Maximal Four. Science International Lahore, 29(6); 1181–1186

Ansori, M., W. Wamiliana, and F. Puri (2021). Determining the Number of Connected Vertex Labeled Graphs of Order Seven without Loops by Observing the Patterns of Formula for Lower Order Graphs with Similar Property. Science and Technology Indonesia, 6(4); 328–336

Bogart, K. P. (2004). Combinatorics Through Guided Discovery. LibreText

Bona, M. (2007). Introduction to Enumerative Combinatorics. McGraw-Hill Science/Engineering/Math

Brandes, U. and S. Cornelsen (2009). Phylogenetic Graph Models Beyond Trees. Discrete Applied Mathematics, 157(10); 2361–2369

Burch, K. J. (2019). Chemical Applications of Graph Theory. In Mathematical Physics in Theoretical Chemistry. Elsevier, pages 261–294

Cayley, P. (1874). LVII. On the Mathematical Theory of Isomers. The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, 47(314); 444–447

Gramatica, R., T. Di Matteo, S. Giorgetti, M. Barbiani, D. Bevec, and T. Aste (2014). Graph Theory Enables Drug Repurposing–How a Mathematical Model Can Drive the Discovery of Hidden Mechanisms of Action. PloS One, 9(1); e84912

Holmes, T. D., R. H. Rothman, and W. B. Zimmerman (2021). Graph Theory Applied to Plasma Chemical Reaction Engineering. Plasma Chemistry and Plasma Processing, 41(2); 531–557

Hsu, L. H. and C. K. Lin (2008). Graph Theory and Interconnection Networks. CRC Press

Huson, D. H. and D. Bryant (2006). Application of Phylogenetic Networks in Evolutionary Studies. Molecular Biology and Evolution, 23(2); 254–267

Kannimuthu, S., D. Bhanu, and K. Bhuvaneshwari (2020). A Novel Approach for Agricultural Decision Making Using Graph Coloring. SN Applied Sciences, 2(1); 1–6

Kawakura, S. and R. Shibasaki (2018). Grouping Method Using Graph Theory for Agricultural Workers Engaging in Manual Tasks. Journal of Advanced Agricultural Technologies, 5(3); 173–181

Mathur, R. and N. Adlakha (2016). A Graph Theoretic Model for Prediction of Reticulation Events and Phylogenetic Networks for DNA Sequences. Egyptian journal of Basic and Applied Sciences, 3(3); 263–271

Ni, B., R. Qazi, S. U. Rehman, and G. Farid (2021). Some Graph-Based Encryption Schemes. Journal of Mathematics, 2021; 1–8

Priyadarsini, P. (2015). A Survey on Some Applications of Graph Theory in Cryptography. Journal of Discrete Mathematical Sciences and Cryptography, 18(3); 209–217

Puri, F., M. Usman, M. Ansori, and Y. Antoni (2021). The Formula to Count the Number of Vertices Labeled Order Six Connected Graphs with Maximum Thirty Edges without Loops. Journal of Physics: Conference Series, 1751(1); 012023

Singh, R. P. (2014). Application of Graph Theory in Computer Science and Engineering. International Journal of Computer Applications, 104(1)

Vasudev, C. (2006). Graph Theory with Applications. New Age International

Wamiliana, W., A. Amanto, M. Usman, M. Ansori, and F. C. Puri (2020). Enumerating the Number of Connected Vertices Labeled Graph of Order Six with Maximum Ten Loops and Containing No Parallel Edges. Science and Technology Indonesia, 5(4); 131–135

Wamiliana, W., A. Nuryaman, A. Sutrisno, and N. Prayoga (2019). Determining the Number of Connected Vertices Labelled Graph of Order Five with Maximum Number of Parallel Edges is Five and Containing No Loops. Journal of Physics: Conference Series, 1338; 012043


Muslim Ansori
wamiliana.1963@fmipa.unila.ac.id (Primary Contact)
Yudi Antoni
Desiana Putri
Ansori, M., Wamiliana, Fitriani, Antoni, Y., & Putri, D. (2022). Enumerate the Number of Vertices Labeled Connected Graph of Order Seven Containing No Parallel Edges. Science and Technology Indonesia, 7(3), 392–399. https://doi.org/10.26554/sti.2022.7.3.392-399

Article Details

Most read articles by the same author(s)

Similar Articles

You may also start an advanced similarity search for this article.