A biclique in G is a subset of V(G) which induces a complete bipartite subgraph of G. In this paper, we applied the Binomial Theorem to establish the biclique polynomial of complete bipartite graphs. Moreover, we integrated the concepts of independent sets and independence polynomials in establishing the biclique polynomial of graphs resulting from the join of two connected graphs.

independent set, biclique, binomial theorem, biclique polynomial, independence polynomial

Received: December 12, 2023; Accepted: February 27, 2024; Published: March 9, 2024

How to cite this article: Shiena Mae B. Lumpayao, Regimar A. Rasid and Rosalio G. Artes Jr., On biclique polynomials, Advances and Applications in Discrete Mathematics 41(3) (2024), 231-237. http://dx.doi.org/10.17654/0974165824017

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References:

[1] R. G. Artes Jr. and R. A. Rasid, Balanced biclique polynomial of graphs, Global Journal of Pure and Applied Mathematics 12(5) (2016), 4427-4433.
[2] R. G. Artes Jr., N. H. R. Mohammad, A. A. Laja and N. H. M. Hassan, From graphs to polynomial rings: Star polynomial representation of graphs, Advances and Applications in Discrete Mathematics 37 (2023), 67-76.
https://doi.org/10.17654/0974165823012
[3] R. G. Artes Jr., N. H. R. Mohammad, Z. H. Dael and H. B. Copel, Star polynomial of the corona of graphs, Advances and Applications in Discrete Mathematics 39(1) (2023), 81-87. https://doi.org/10.17654/0974165823037
[4] R. G. Artes Jr. and R. A. Rasid, Combinatorial approach in counting the balanced bicliques in the join and corona of graphs, Journal of Ultra Scientist of Physical Sciences 29(5) (2017), 192-195.
[5] R. G. Artes Jr., J. I. C. Salim, R. A. Rasid, J. I. Edubos and B. J. Amiruddin-Rajik, Geodetic closure polynomial of graphs, International Journal of Mathematics and Computer Science 19(2) (2024), 439-443.
[6] J. A. Bondy and U. S. R. Murty, Graph Theory and Related Topics, Academic Press, New York, 1979.
[7] J. I. Brown and R. J. Nowakowski, The neighbourhood polynomial of a graph, Australian Journal of Combinatorics 42 (2008), 55-68.
[8] J. Ellis-Monaghan and J. Merino, Graph Polynomials and their Applications II: Interrelations and Interpretations, Birkhauser, Boston, 2011.
[9] J. L. Gross and J. Yellen, Graph Theory and its Applications, Chapman & Hall, New York, 2006.
[10] I. Gutman and F. Harary, Generalizations of the matching polynomial, Utilitas Mathematica 24 (1983), 97-106.
[11] F. Harary, Graph Theory, CRC Press, Boca Raton, 2018.
[12] C. Hoede and X. Li, Clique polynomials and independent set polynomials of graphs, Discrete Mathematics 125 (1994), 219-228.
[13] L. S. Laja and R. G. Artes Jr., Zeros of convex subgraph polynomials, Applied Mathematical Sciences 8(59) (2014), 2917-2923.
http://dx.doi.org/10.12988/ams.2014.44285
[14] L. S. Laja and R. G. Artes Jr., Convex subgraph polynomials of the join and the composition of graphs, International Journal of Mathematical Analysis 10(11) (2016), 515-529. http://dx.doi.org/10.12988/ijma.2016.512296
[15] R. E. Madalim, R. G. Eballe, A. H. Arajaini and R. G. Artes Jr., Induced cycle polynomial of a graph, Advances and Applications in Discrete Mathematics 38(1) (2023), 83-94. https://doi.org/10.17654/0974165823020
[16] C. A. Villarta, R. G. Eballe and R. G. Artes Jr., Induced path polynomial of graphs, Advances and Applications in Discrete Mathematics 39(2) (2023), 183-190. https://doi.org/10.17654/0974165823045