  Exploiting the power of the expectation operator and indicator (or Bernoulli)
random variables, we present the exact governing equations for both the SIR and
SIS epidemic models on \emph{networks}. Although SIR and SIS are basic epidemic
models, deductions from their exact stochastic equations \textbf{without}
making approximations (such as the common mean-field approximation) are scarce.
An exact analytic solution of the governing equations is highly unlikely to be
found (for any network) due to the appearing pair (and higher order)
correlations. Nevertheless, the maximum average fraction $y_{I}$ of infected
nodes in both SIS and SIR can be written as a quadratic form of the graph's
Laplacian. Only for regular graphs, the expression for the maximum of $y_{I}$
can be simplied to exhibit the explicit dependence on the spectral radius. From
our new Laplacian expression, we deduce a general \textbf{upper} bound for the
epidemic SIS threshold in any graph.
