We propose a new paradigm for describing complex networks in terms of the spectrum of the adjacency matrix and its submatrices. We show that a variety of basic node information, such as degree, clique, and subgraph centrality, can be calculated analytically. Moreover, we find that energy of spectrum series can uncover randomness and complexity of network structure. Interestingly, it presents an universal linear growth pattern with the growth of networks. Furthermore, the spectrum series of synthetic and real networks present clearly self-similarity characteristics for which the associated scaling exponents remain constant. Our work reveals that spectrum series representation will provide an alternative perspective for studying and understanding structure and function of complex networks rather than connectivity.