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A fundamental problem in mathematics and network analysis is to find conditions under which a graph can be partitioned into smaller pieces. The most important tool for this partitioning is the Fiedler vector or discrete Cheeger inequality. These results relate the graph spectrum (eigenvalues of the normalized adjacency matrix) to the ability to break a graph into two pieces, with few edge deletions. An entire subfield of mathematics, called spectral graph theory, has emerged from these results. Yet these results do not say anything about the rich community structure exhibited by real-world networks, which typically have a significant fraction of edges contained in numerous densely clustered blocks. Inspired by the properties of real-world networks, we discover a new spectral condition that relates eigenvalue powers to a network decomposition into densely clustered blocks. We call this the \emph{spectral triadic decomposition}. Our relationship exactly predicts the existence of community structure, as commonly seen in real networked data. Our proof provides an efficient algorithm to produce the spectral triadic decomposition. We observe on numerous social, coauthorship, and citation network datasets that these decompositions have significant correlation with semantically meaningful communities.