graph theory

[Edwin Clark]

I think what you are dealing with has also been called the corona of a 
graph. I think you will agree that if you "ciliate" (in your terms) a 
cycle you will get something that looks like a crown (or corona). 

Anyhow graph theorists have a somewhat more general concept 
involving two graphs:

"The corona $G_1\circ G_2$ of two graphs $G_1$ and $G_2$ was defined by 
Frucht and Harary as the graph $G$ obtained by taking one copy of $G_1$ 
which has $p_1$ vertices and $p_1$ copies of $G_2$ and then joining the 
$i$th vertex of $G_1$ to every vertex in the $i$th copy of $G_2$. 
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If you take G_2 to be the trivial one vertex graph you will get what some 
call the corona of G1. I have seen this construction several place without 
a name, but corona seems to have taken. If you search MathSciNet using 
Anywhere:  corona  and Anywhere: graph, you get a large number of hits.
Although I'm not sure they all lead to the same concept.

--Edwin Clark

PS. As for improving on your wording if I understand you correctly you 
need to mention the new edges as well as the new vertices. 

PPS The above definition was taken from the MathSciNet abstract:

MR1980055 (2004e:05166)
Bhat-Nayak, Vasanti N.(6-BOMB); Telang, Shanta(6-BOMB)
Cahit-$k$-equitability of $C\sb n\circ K\sb 1$, $k=n$ to $2n-1$, $n\geq 
3$. 
Proceedings of the Thirty-third Southeastern International Conference on 
Combinatorics, Graph Theory and Computing (Boca Raton, FL, 2002).
Congr. Numer. 155 (2002), 131--213.


On Fri, 7 Jan 2005, Emeric Deutsch wrote:

> Dear seqfans,
> In view of submitting new sequences to OEIS, I'd like to pick the 
> brain of graph theorists.
> 
> Let G be a graph with n vertices. By the ciliation of G we mean 
> the graph obtained from G by joining n new vertices to the n 
> vertices of G, respectively. 
> 
> For example, a path graph becomes a "comb". 
> 
> My first question: is "ciliation" a good term for this new graph? 
> Any suggestions?
> 
> My second question: can one improve on the wording of "the graph
> obtained from G by joining n new vertices to the n vertices of G,
> respectively" ?   
> 
> Many thanks and Happy 2005.
> Emeric
> 
> P.S. The second graph (the ciliation of G) is an "equible" graph 
> in the terminology of Farrell-Kennedy-Quintas-Wahid.
> 

-- 
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  W. Edwin Clark, Math Dept, University of South Florida
           http://www.math.usf.edu/~eclark/
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