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On the Graovac-Ghorbani index for bicyclic graphs with no pendant vertices
Let be a simple undirected and connected graph on vertices. The
Graovac--Ghorbani index of a graph is defined as where is the
number of vertices closer to vertex than vertex of the edge and is defined analogously. It is well-known that all bicyclic
graphs with no pendant vertices are composed by three families of graphs, which
we denote by In this paper,
we give an lower bound to the index for all graphs in and
prove it is sharp by presenting its extremal graphs. Additionally, we
conjecture a sharp lower bound to the index for all graphs in
$\mathcal{B}_{n}.