The name of a cycle with chords added to one of its vertices Consider a cycle $C$ and its vertex $v$. If $C$ has $n$ vertices, then we can add at most $n-3$ chords to $v$. In the case of $n-3$ chords, the obtained graph is called shell and denoted as $C(n,n-3)$. What is the name of the graph if we add $k\leq n-3$ chords, i.e. $C(n,k)$? I want to publish some results on these graphs and going to call them pre-shells, but don't think it's a good idea.
 A: This graph is the fan graph $F_{1,n-1}$, where a vertex $v$ is joined to all vertices of an $(n-1)$-vertex path. (The fan graph notation is more general; $F_{k,n}$ involves taking an independent set of order $k$ and a path of order $n$, and adding all edges between them.)
There are other names for special cases: the graph you call $C(5,2)$ is also called the gem graph, for example.
There is no notation (that I am aware of) for the graphs you are considering calling "pre-shells". Part of that is that your definition of these graphs is not well-defined: there are multiple non-isomorphic graphs that are obtained from a cycle $C_n$ by adding $k$ chords out of the same vertex. So referring to a graph as $C(n,k)$ for $0<k<n-3$ would be ambiguous.
You can, of course, refer to the class of all graphs of this type. Conventionally, I would expect notation like $\mathcal C(n,k)$ or $\mathcal C_{n,k}$ (with the letter C as fancy as you can make it) to refer to a class of graphs, and an ordinary $C$ to refer to a single graph.
