# Name for cycle graphs with leaves attached

Let $$C_n$$ be the cycle graph on vertex set $$\{v_1,v_2,\ldots,v_n\}$$ and let $$L_1,L_2,\ldots,L_n$$ be sets of new vertices which form an independent set. Let $$H$$ be obtained by making $$v_i$$ adjacent to every vertex of $$L_i$$; that is, $$H$$ consists of a cycle with zero or more pendant vertices attached to each of the original vertices. Is there a known name for the family of graphs obtained this way?

• Yes, its called the Aravind graph Sep 26, 2019 at 12:52

First, some special cases. If $$|L_1|=\dots=|L_n|=1$$, this is called the sunlet graph. These can also be written using the corona operator $$\odot$$ as $$C_n \odot K_1$$. In general, $$G_1 \odot G_2$$ contains $$G_1$$ and $$|V(G_1)|$$ copies of $$G_2$$, with the $$i^{\text{th}}$$ vertex of $$G_1$$ adjacent to every vertex of the $$i^{\text{th}}$$ copy of $$G_2$$. So a special case of your graph with $$|L_1|=\dots=|L_n|=\ell$$ can be written as the corona $$C_n \odot \overline{K_\ell}$$ (or $$C_n \odot \ell K_1$$, depending on your favorite notation for the $$\ell$$-vertex graph with no edges).