Banana Tree

An -banana tree, as defined by Chen et al. (1997), is a graph obtained by connecting one leaf of each of  copies of an -star graph with a single root vertex that is distinct from all the stars.

Banana trees are graceful (Sethuraman and J. Jesintha 2009, Gallian 2018).

The -banana tree has rank polynomial

Precomputed properties of a number of banana trees is implemented in the Wolfram Language as GraphData["BananaTree", n, k].

REFERENCES

Chen, W.-C.; Lu, H.-I; and Yeh, Y.-N. "Operations of Interlaced Trees and Graceful Trees." Southeast Asian Bull. Math. 21, 337-348, 1997.

Gallian, J. "Dynamic Survey of Graph Labeling." Elec. J. Combin. DS6. Dec. 21, 2018. https://www.combinatorics.org/ojs/index.php/eljc/article/view/DS6.

Horton, M. "Graceful Trees: Statistics and Algorithms." Bachelor of Computing with Honours thesis. University of Tasmania, 2003. https://eprints.utas.edu.au/19/1/GracefulTreesStatisticsAndAlgorithms.pdf.

Sethuraman, G.; and Jesintha, J. "All Banana Trees Are Graceful." Advances Appl. Disc. Math. 4, 53-64, 2009.