  In this paper we define cycle-star graph CSk,n-k to be a graph on n vertices
consisting of the cycle of length k and n-k leafs appended to the same vertex
of the cycle. Also, we define cycle-path graph CPk,n-k to be a graph on n
vertices consisting of the cycle of length k and of path on n-k vertices whose
one end is linked to a vertex on a cycle. We establish that cycle-star graph
CS3,n-3 is the only maximal graph with respect to additively weighted Harary
index among all unicyclic graphs on n vertices, while cycle-path graph CP3,n-3
is the only minimal unicyclic graph (here n must be at least 5). The values of
additively weighted Harary index for extremal unicyclic graphs are established,
so these values are the upper and the lower bound for the value of additively
weighted Harary index on the class of unicyclic graphs on n vertices.
