论文标题
边界莫斯塔尔指数
Bounding the Mostar index
论文作者
论文摘要
Došlić等。 defined the Mostar index of a graph $G$ as $Mo(G)=\sum\limits_{uv\in E(G)}|n_G(u,v)-n_G(v,u)|$, where, for an edge $uv$ of $G$, the term $n_G(u,v)$ denotes the number of vertices of $G$ that have a smaller distance in $G$ to $u$ than to $ V $。他们推测$ mo（g）\ leq 0. \ + overline {148} n^3 $对于订单$ n $的每个图$ g $。作为Mostar索引上的自然上限，Geneson和Tsai隐含地考虑参数$ MO^\ star（g）= \ sum \ limits_ {uv \ in E（g）} \ big big（n- \ min \ {d_g（d_g（d_g（u））对于订单$ n $的图$ g $，他们表明$ mo^\ star（g）\ leq \ frac {5} {24}（1+o（1））n^3 $。 我们将此绑定到$ mo^\ star（g）\ leq \ left（\ frac {2} {\ sqrt {3}}}} - 1 \ right）n^3 $，这是最好的较低顺序。此外，我们表明$ mo^\ star（g）\ leq \ left（2 \ left（\fracδ{n} \ right）^2+\左（\fracδ{n} \ right）-2 \ left（\fracΔ{n} \ right）\ sqrt {\ left（\fracδ{n} \ right）^2+\ left（\fracδ{n} \ right）} \ right）n^3 $前提是$ g $具有最高度$δ$。
Došlić et al. defined the Mostar index of a graph $G$ as $Mo(G)=\sum\limits_{uv\in E(G)}|n_G(u,v)-n_G(v,u)|$, where, for an edge $uv$ of $G$, the term $n_G(u,v)$ denotes the number of vertices of $G$ that have a smaller distance in $G$ to $u$ than to $v$. They conjectured that $Mo(G)\leq 0.\overline{148}n^3$ for every graph $G$ of order $n$. As a natural upper bound on the Mostar index, Geneson and Tsai implicitly consider the parameter $Mo^\star(G)=\sum\limits_{uv\in E(G)}\big(n-\min\{ d_G(u),d_G(v)\}\big)$. For a graph $G$ of order $n$, they show that $Mo^\star(G)\leq \frac{5}{24}(1+o(1))n^3$. We improve this bound to $Mo^\star(G)\leq \left(\frac{2}{\sqrt{3}}-1\right)n^3$, which is best possible up to terms of lower order. Furthermore, we show that $Mo^\star(G)\leq \left(2\left(\fracΔ{n}\right)^2+\left(\fracΔ{n}\right)-2\left(\fracΔ{n}\right)\sqrt{\left(\fracΔ{n}\right)^2+\left(\fracΔ{n}\right)}\right)n^3$ provided that $G$ has maximum degree $Δ$.
