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On the hierarchical product of graphs | ||
| Journal of Discrete Mathematics and Its Applications | ||
| مقاله 1، دوره 9، شماره 3، آذر 2024، صفحه 163-171 اصل مقاله (298.83 K) | ||
| نوع مقاله: Full Length Article | ||
| شناسه دیجیتال (DOI): 10.22061/jdma.2024.11171.1085 | ||
| نویسندگان | ||
| Wilfried Imrich* 1؛ Gabriela Makar2؛ Juliana Palmen2؛ Piotr Zajac2 | ||
| 1Montanuniversität Leoben, 8700 Leoben, Austria | ||
| 2AGH University of Krakow, 30-059 Krakow, Poland | ||
| تاریخ دریافت: 12 مرداد 1403، تاریخ بازنگری: 25 مرداد 1403، تاریخ پذیرش: 27 مرداد 1403 | ||
| چکیده | ||
| The hierarchical product of graphs is a variant of the Cartesian product. It is associative, not commutative, and finite connected graphs have unique first prime factors with respect to it. We present examples of infinite graphs with different first prime factors, and show that homogeneous trees of finite degree have unique prime factoriza-tions with respect to the hierarchical product. On the way, we pose two problems. | ||
| کلیدواژهها | ||
| hierarchical products of finite and infinite graphs؛ prime factor-izations؛ trees | ||
| مراجع | ||
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[1] S. E. Anderson, Y. Guo, A. Tenney, K. A. Wash, Prime factorization and domination in the hierar- chical product of graphs, Discuss. Math. Graph Theory 37 (2017) 873–890, doi:10.7151/dmgt.1952. [2] L. Barrière, F. Comellas, C. Dalfó, and M. A. Fiol, On the hierarchical product of graphs and the generalized binomial tree, Linear Multilinear Algebra 57 (2009) 695–712, doi:10.1080/03081080802305381. [3] L. Barriére, F. Comellas, C. Dalfó, M. A. Fiol, The hierarchical product of graphs, Discrete Appl. Math. 157 (2009) 36–48, doi:10.1016/j.dam.2008.04.018. [4] L. Barriére, C. Dalfó, M. A. Fiol, M. Mitjana, The generalized hierarchical product of graphs, Dis- crete Math. 309 (2009) 3871–3881, doi:10.1016/j.disc.2008.10.028. [5] C. D. Godsil, B. D. McKay, A new graph product and its spectrum, Bull. Aust. Math. Soc. 18 (1978) 21–28, doi:10.1017/S0004972700007760. [6] W. Imrich, R. Kalinowski, M. Pil´ sniak, Hierarchical product graphs and their prime factorization, Art Discrete Appl. Math. accepted for publication 2024. | ||
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