  We introduce a family of multi-way Cheeger-type constants $\{h_k^{\sigma},
k=1,2,\ldots, n\}$ on a signed graph $\Gamma=(G,\sigma)$ such that
$h_k^{\sigma}=0$ if and only if $\Gamma$ has $k$ balanced connected components.
These constants are switching invariant and bring together in a unified
viewpoint a number of important graph-theoretical concepts, including the
classical Cheeger constant, those measures of bipartiteness introduced by
Desai-Rao, Trevisan, Bauer-Jost, respectively, on unsigned graphs,, and the
frustration index (originally called the line index of balance by Harary) on
signed graphs. We further unify the (higher-order or improved) Cheeger and dual
Cheeger inequalities for unsigned graphs as well as the underlying algorithmic
proof techniques by establishing their corresponding versions on signed graphs.
In particular, we develop a spectral clustering method for finding $k$
almost-balanced subgraphs, each defining a sparse cut. The proper metric for
such a clustering is the metric on a real projective space. We also prove
estimates of the extremal eigenvalues of signed Laplace matrix in terms of
number of signed triangles ($3$-cycles).
