  Providing an analytical solution for the problem of finding Fastest
Distributed Consensus (FDC) is one of the challenging problems in the field of
sensor networks. Most of the methods proposed so far deal with the FDC
averaging algorithm problem by numerical convex optimization methods and in
general no closed-form solution for finding FDC has been offered up to now
except in [3] where the conjectured answer for path has been proved. Here in
this work we present an analytical solution for the problem of Fastest
Distributed Consensus for the Path network using semidefinite programming
particularly solving the slackness conditions, where the optimal weights are
obtained by inductive comparing of the characteristic polynomials initiated by
slackness conditions.
