  Elastic Optical Network (EON) has been considered as a promising optical
networking technology to architect the next-generation backbone networks.
Routing and Spectrum Assignment (RSA) is the fundamental problem in EONs to
realize service provisioning. Generally, the RSA is solved by routing the
requests with lightpaths first and then assigning spectrum resources to the
lightpaths to optimize the spectrum usage. Thus, the spectrum assignment
explicitly decide the final spectrum usage of EONs. However, besides the
spectrum assignment, there are three other factors, the network topology,
traffic distribution and routing scheme, implicitly impact on the spectrum
usage. Few related work involves in the implicit impact mechanism. In this
paper, we aim to provide a thoroughly theoretical analysis on the impact of the
three key factors on the spectrum usage. To this end, two theoretical chains
are proposed: (1) The optimal spectrum usage can be measured by the chromatic
number of the conflict graph, which is positively correlated to the
intersecting probability, \emph{i.e.}, the smaller the intersecting
probability, the smaller the optimal spectrum usage; (2) The intersecting
probability is decided by the network topology, traffic distribution and
routing scheme via a quadratic programming parameterized with a matrix of
conflict coefficients. The effectiveness of our theoretical analysis has been
validated by extensive numerical results. Meanwhile, our theoretical deductions
also permit to give several constant approximation ratios for RSA algorithms.
