  Learning properties of large graphs from samples has been an important
problem in statistical network analysis since the early work of Goodman
\cite{Goodman1949} and Frank \cite{Frank1978}. We revisit a problem formulated
by Frank \cite{Frank1978} of estimating the number of connected components in a
large graph based on the subgraph sampling model, in which we randomly sample a
subset of the vertices and observe the induced subgraph. The key question is
whether accurate estimation is achievable in the \emph{sublinear} regime where
only a vanishing fraction of the vertices are sampled. We show that it is
impossible if the parent graph is allowed to contain high-degree vertices or
long induced cycles. For the class of chordal graphs, where induced cycles of
length four or above are forbidden, we characterize the optimal sample
complexity within constant factors and construct linear-time estimators that
provably achieve these bounds. This significantly expands the scope of previous
results which have focused on unbiased estimators and special classes of graphs
such as forests or cliques.
  Both the construction and the analysis of the proposed methodology rely on
combinatorial properties of chordal graphs and identities of induced subgraph
counts. They, in turn, also play a key role in proving minimax lower bounds
based on construction of random instances of graphs with matching structures of
small subgraphs.
