Papers › On the Graovac-Ghorbani index for bicyclic graphs with no pendant vertices
On the Graovac-Ghorbani index for bicyclic graphs with no pendant vertices
Diego Pacheco, Leonardo de Lima, Carla Silva Oliveira
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Let G=(V,E) be a simple undirected and connected graph on n vertices. The Graovac--Ghorbani index of a graph G is defined as ABC_(GG)(G)= ∑_(uv ∈E(G)) √((nᵤ+nᵥ-2)/(nᵤ nᵥ)), where nᵤ is the number of vertices closer to vertex u than vertex v of the edge uv ∈E(G) and nᵥ 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) ∪B₂(n) ∪B₃(n). In this paper, we give an lower bound to the ABC_(GG) index for all graphs in B₁(n) and prove it is sharp by presenting its extremal graphs. Additionally, we conjecture a sharp lower bound to the ABC_(GG) index for all graphs in ℬₙ.
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