  In 1978 Gutman introduced the energy of a graph as the sum of the absolute
values of graph eigenvalues, and ever since then graph energy has been
intensively studied.
  Since graph energy is the trace norm of the adjacency matrix, matrix norms
provide a natural background for its study. Thus, this paper surveys research
on matrix norms that aims to expand and advance the study of graph energy.
  The focus is exclusively on the Ky Fan and the Schatten norms, both
generalizing and enriching the trace norm. As it turns out, the study of
extremal properties of these norms leads to numerous analytic problems with
deep roots in combinatorics.
  The survey brings to the fore the exceptional role of Hadamard matrices,
conference matrices, and conference graphs in matrix norms. In addition, a vast
new matrix class is studied, a relaxation of symmetric Hadamard matrices.
  The survey presents solutions to just a fraction of a larger body of similar
problems bonding analysis to combinatorics. Thus, open problems and questions
are raised to outline topics for further investigation.
