论文标题
在没有吊坠顶点的双环图形的Graovac-Ghorbani指数上
On the Graovac-Ghorbani index for bicyclic graphs with no pendant vertices
论文作者
论文摘要
令$ g =（v，e）$为$ n $顶点上的简单无向且连接的图形。图$ g $的graovac-- ghorbani索引定义为$$ abc_ {gg}（g）= \ sum_ {uv \ in E（g）} \ sqrt {\ sqrt {\ frac {n_ {n_ {n_ {u}与顶点$ u $接近的顶点相比，e（g）$和$ n_ {v} $的edge $ uv \的顶点$ v $类似地定义。 It is well-known that all bicyclic graphs with no pendant vertices are composed by three families of graphs, which we denote by $\mathcal{B}_{n} = B_1(n) \cup B_2(n) \cup B_3(n).$ In this paper, we give an lower bound to the $ABC_{GG}$ index for all graphs in $B_1(n)$ and通过呈现其极端图，证明它是锋利的。此外，我们猜想了$ \ Mathcal {b} _ {n}中所有图的$ abc_ {gg} $索引的尖锐下限。
Let $G=(V,E)$ be a simple undirected and connected graph on $n$ vertices. The Graovac--Ghorbani index of a graph $G$ is defined as $$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 $\mathcal{B}_{n}.$
