On the Graovac-Ghorbani index for bicyclic graphs with no pendant vertices
Abstract
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 $$ABC_{GG}(G)= \sum_{uv \in E(G)} \sqrt{\frac{n_{u}+n_{v}-2} {n_{u} n_{v}}},$$ where $n_u$ is the number of vertices closer to vertex $u$ than vertex $v$ of the edge $uv \in E(G)$ and $n_{v}$ 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 $\mathcal{B}_{n} = B_1(n) \cup B_2(n) \cup B_3(n).$ In this paper, we give an lower bound to the $ABC_{GG}$ index for all graphs in $B_1(n)$ 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 $\mathcal{B}_{n}.$
- Publication:
-
arXiv e-prints
- Pub Date:
- April 2020
- DOI:
- 10.48550/arXiv.2005.02141
- arXiv:
- arXiv:2005.02141
- Bibcode:
- 2020arXiv200502141P
- Keywords:
-
- Mathematics - General Mathematics