In 2022, the idea of a bivariate polynomial which represents the number of complete subgraphs of a graph with corresponding common neighborhood systems has been introduced in [3]. In the present work, we extend this notion to a more restricted case by considering maximal connected common neighborhood systems of cliques in a given graph. Besides characterizing the cliques in the corona of two connected graphs, we establish the clique connected common neighborhood polynomial of the graph resulting from the corona of two connected graphs.

clique, clique polynomial, common neighborhood system, clique connected common neighborhood polynomial.

Received: June 2, 2023; Accepted: July 5, 2023; Published: July 18, 2023

How to cite this article: Amelia L. Arriesgado, Jeffrey Imer C. Salim and Rosalio G. Artes Jr., Polynomial representation of the neighborhood systems of cliques in the corona of graphs, Advances and Applications in Discrete Mathematics 40(1) (2023), 11-18. http://dx.doi.org/10.17654/0974165823054

This Open Access Article is Licensed under Creative Commons Attribution 4.0 International License

References:

[1] A. L. Arriesgado, S. C. Abdurasid and R. G. Artes, Jr., Connected common neighborhood systems of cliques in a graph: A polynomial representation, Advances and Applications in Discrete Mathematics 38(1) (2023), 69-81.
DOI: 10.17654/0974165823019
[2] A. L. Arriesgado, J. I. C. Salim and R. G. Artes, Jr., Clique connected common neighborhood polynomial of the join of graphs, International Journal of Mathematics and Computer Science 18(4) (2023), 655-659.
[3] R. G. Artes, Jr., M. A. Langamin and A. B. Calib-og, Clique common neighborhood polynomial of graphs, Advances and Applications in Discrete Mathematics 35 (2022), 77-85. DOI: 10.17654/0974165822053
[4] J. I. Brown and R. J. Nowakowski, The neighborhood polynomial of a graph, Australian Journal of Combinatorics 42 (2008), 55-68.
[5] J. Ellis-Monaghan and J. Merino, Graph Polynomials and their Applications II: Interrelations and Interpretations, Birkhauser, Boston, 2011.
[6] F. S. Gella and R. G. Artes, Jr., Clique cover of graphs, Applied Mathematical Sciences 8(87) (2014), 4301-4307. DOI: 10.12988/ams.2014.45343
[7] F. S. Gella and R. G. Artes, Jr., Clique cover of the join and the corona of graphs, International Journal of Mathematical Analysis 9(32) (2015), 1579-1583. DOI: 10.12988/ijma.2015.53111
[8] F. Harary, Graph Theory, CRC Press, Boca Raton, 2018.
[9] C. Hoede and X. Li, Clique polynomials and independent set polynomials of graphs, Discrete Mathematics 125 (1994), 219-228.