Geodesic
There are 2אα Friendship Graphs of Cardinal אα
Canadian mathematical bulletin, Tome 19 (1976) no. 4, pp. 431-433

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A friendship graph is a graph in which every two distinct vertices have exactly one common neighbour. Finite friendship graphs were characterized by Erdös, Rényi, and Sós [1] as those for which the vertices can be enumerated as u, υ 1,...υ k , w 1,...w k in such a way that the only edges are uυ i uw i and υ i w i (i = 1,...,k). Thus finite friendship graphs are rather rare. In contrast, we shall show that there are as many nonisomorphic friendship graphs of given infinite cardinal as there are nonisomorphic graphs of that cardinal altogether. In fact, we do a little more. 
Chvátal, Václav; Kotzig, Anton; Rosenberg, Ivo G.; Davies, Roy O. There are 2אα Friendship Graphs of Cardinal אα. Canadian mathematical bulletin, Tome 19 (1976) no. 4, pp. 431-433. doi: 10.4153/CMB-1976-064-1
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     author = {Chv\'atal, V\'aclav and Kotzig, Anton and Rosenberg, Ivo G. and Davies, Roy O.},
     title = {There are 2א\ensuremath{\alpha} {Friendship} {Graphs} of {Cardinal} א\ensuremath{\alpha}},
     journal = {Canadian mathematical bulletin},
     pages = {431--433},
     year = {1976},
     volume = {19},
     number = {4},
     doi = {10.4153/CMB-1976-064-1},
     url = {http://geodesic.mathdoc.fr/articles/10.4153/CMB-1976-064-1/}
}
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