Is the Cheeger constant $\mathsf{NP}$-hard? I have read in uncountably many articles that determining the Cheeger constant of a graph is $\mathsf{NP}$-hard. It seems to be a folk theorem, but I have never found either a quote or a proof for this statement. Whom should I give credit for it? In an old paper (Isoperimetric Numbers of Graphs, J. Comb. Theory B, 1989) Mohar only proves this assertion "for graphs with multiple edges".
I too encountered this issue when I was writing a paper that required a citation to hardness of edge expansion (or Cheeger constant) defined as $\min_{S \subset V, |S| \leq |V|/2} |\delta(S)|/|S|$. The classic paper of Leighton and Rao on separators ( http://dl.acm.org/citation.cfm?id=331526 ) mentions that this is a hard problem and refers to the paper of Garey, Johnson and Stockmeyer (http://www.sciencedirect.com/science/article/pii/0304397576900591). I could not figure out for a while what they were referring to since there is no mention of edge expansion in the referred to paper. I communicated with Avi Wigderson about this. It finally transpired that one can use the hardness of Max-Cut as shown in the Garey et al paper to relatively easily show that edge-expansion is hard. I forget the details now but it should not be hard to recreate. The paper of Blum etal on hardness of checking whether a graph is a superconcentrator does not directly imply hardness of edge expansion. They are technically not the same problem.