• Aditya Ambarwati Departement of Mathematics, Faculty of Mathematics and Natural Sciences, Brawijaya University, Indonesia
  • Vira Hari Krisnawati Departement of Mathematics, Faculty of Mathematics and Natural Sciences, Brawijaya University, Indonesia
Keywords: Bicyclic Graph, Directed Graph, Directed Bicyclic Graph, Trans Jogja Routes


A bicyclic graph is a type of graph that consists of exactly two cycles. A cycle is a graph that is a closed path where no vertices are repeated except the first and last vertices which are the same. The cycles in bicyclic graph can be of different lengths and shapes, but they must have at least one common vertex. Bicyclic graphs can be divided into two categories based on the types of induced subgraphs they contain. One category consists of graphs that include an -graph as an induced subgraph, while the other category comprises graphs that contain a -graph as an induced subgraph. There are 3 types of bicyclic graph without pendant vertex. A directed graph, also referred to as a digraph, is a graph in which each edge is assigned a specific direction. A directed bicyclic graph is a special kind of directed graph that contains precisely two distinct directed cycles. This graph can be applied in transportation problem. In this article, we give some examples of directed bicyclic graph in Trans Jogja routes.


Download data is not yet available.


R. Diestel and R. Diestel, "The basics," Graph Theory, vol. 5, pp. 1-34, 2017.

G. Chartrand, L. Lesniak and P. Zhang, Graphs & digraphs, CRC press., 2010.

B. X. E. &. H. Y. Wu, "The spectral radius of trees on k pendant vertices," Linear Algebra and its Applications, vol. 395, pp. 343-350, 2005.

J. A. Bondy and U. S. R. Murty, Graph theory with applications (Vol. 290), London: Macmillan, 1976.

A. Ferber and M. Krivelevich, "Every graph contains a linearly sized induced subgraph with all degrees odd," Advances in Mathematics, vol. 406, p. 108534, 2022.

S. Lee and S. Nirjon, "SubFlow: A dynamic induced-subgraph strategy toward real-time DNN inference and training.," In 2020 IEEE Real-Time and Embedded Technology and Applications Symposium (RTAS) , pp. 15-29, 2020.

S. Hajebi, Y. Li and S. Spirkl, "Complexity dichotomy for list-5-coloring with a forbidden induced subgraph," SIAM Journal on Discrete Mathematics, vol. 36, no. 3, pp. 2004-2027, 2022.

S. K. Simić, "On the largest eigenvalue of bicyclic graphs," Publications de l'Institut Mathématique. Nouvelle Série, 46, 1-6, pp. 1-6, 1989.

S. X. T. &. L. B. Hu, "On the nullity of bicyclic graphs," Linear Algebra and its Applications, vol. 429, no. 7, pp. 1387-1391., 2008.

L. Y. J. Z. Y. &. Y. Z. You, "The maximal total irregularity of bicyclic graphs," Journal of Applied Mathematics, pp. 1-9, 2014.

J. S. Y. W. Z. &. Y. J. Ma, "On Wiener polarity index of bicyclic networks," Scientific reports, vol. 6, no. 1, pp. 1-7, 2016.

J. Bang-Jensen and G. Z. Gutin, Digraphs: theory, algorithms and applications., Springer Science & Business Media, 2008.

F. Harary, R. Z. Norman and D. Cartwright, Structural models: an introduction the theory of directed graphs., New York: Wiley, 1965.

W. H. Cunningham, "Decomposition of directed graphs," SIAM Journal on Algebraic Discrete Methods, vol. 3, no. 2, pp. 214-228, 1982.

V. Vo, T. Le, L. T. Vuong, H. Zhao, E. Bonilla and D. Phung, "Learning Directed Graphical Models with Optimal Transport," arXiv preprint arXiv., 2023.

W. Hong and L. You, " Spectral radius and signless Laplacian spectral radius of strongly connected digraphs," Linear Algebra and Its Applications, vol. 475, pp. 93-113, 2014.

J. Monsalve and J. Rada, "Bicyclic digraphs with maximal energy," Applied Mathematics and Computation, vol. 280, pp. 124-131, 2016.

K. Moon, A. M. Guerrero, V. M. Adams, D. Biggs, D. A. Blackman, L. Craven, ... and H. Ross, "Mental models for conservation research and practice," Conservation Letters, vol. 12, no. 3, p. e12642, 2019.

P. Mani, B. Vasudevan and M. Sivaraman, "Shortest path algorithm of a network via picture fuzzy digraphs and its application," Materials Today: Proceedings, vol. 45, pp. 3014-3018, 2021.

A. G. Marques, S. Segarra and G. Mateos, "Signal processing on directed graphs: The role of edge directionality when processing and learning from network data.," IEEE Signal Processing Magazine, vol. 37 , no. 6, pp. 99-116, 2020.

X. Yang and L. Wang, " The eccentricity matrix of a digraph," Discrete Applied Mathematics, vol. 322, pp. 61-73, 2022.

S. Jain and A. Sinha, "Social network sustainability for transport planning with complex interconnections," Sustainable Computing: Informatics and Systems, vol. 24, p. 100351, 2019.

P. J. Romadhona, "Fasilitas Keselamatan Pada Halte Transjogja," Warta Penelitian Perhubungan, vol. 23 , no. 4, pp. 388-401, 2011.

How to Cite
A. Ambarwati and V. Krisnawati, “CONSTRUCTION OF BICYCLIC GRAPH AND ITS APPLICATION IN TRANS JOGJA ROUTES”, BAREKENG: J. Math. & App., vol. 17, no. 4, pp. 2095-2106, Dec. 2023.