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    On the Graovac-Ghorbani index for bicyclic graphs with no pendant vertices

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    Let G=(V,E)G=(V,E) be a simple undirected and connected graph on nn vertices. The Graovac--Ghorbani index of a graph GG is defined as ABCGG(G)=βˆ‘uv∈E(G)nu+nvβˆ’2nunv,ABC_{GG}(G)= \sum_{uv \in E(G)} \sqrt{\frac{n_{u}+n_{v}-2} {n_{u} n_{v}}}, where nun_u is the number of vertices closer to vertex uu than vertex vv of the edge uv∈E(G)uv \in E(G) and nvn_{v} 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 Bn=B1(n)βˆͺB2(n)βˆͺB3(n).\mathcal{B}_{n} = B_1(n) \cup B_2(n) \cup B_3(n). In this paper, we give an lower bound to the ABCGGABC_{GG} index for all graphs in B1(n)B_1(n) and prove it is sharp by presenting its extremal graphs. Additionally, we conjecture a sharp lower bound to the ABCGGABC_{GG} index for all graphs in $\mathcal{B}_{n}.
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