  Finding optimal weights for the problem of Fastest Distributed Consensus on
networks with different topologies has been an active area of research for a
number of years. Here in this work we present an analytical solution for the
problem of Fastest Distributed Consensus for a network formed from fusion of
two different symmetric star networks or in other words a network consists of
two different symmetric star networks which share the same central node. The
solution procedure consists of stratification of associated connectivity graph
of network and Semidefinite Programming (SDP), particularly solving the
slackness conditions, where the optimal weights are obtained by inductive
comparing of the characteristic polynomials initiated by slackness conditions.
Some numerical simulations are carried out to investigate the trade-off between
the parameters of two fused star networks, namely the length and number of
branches.
