AI 中文总结
研究科尼格 - 埃格瓦里图的兰迪奇指数，证明其满足特定不等式并刻画等式成立的图，结合贝尔热 - 图特公式确定\(R(G)-\alpha'(G)\)的最大值及极值图，否定了奥奇切等人2006年的猜想。
AI 中文摘要
图\(G\)的兰迪奇指数\(R(G)=\sum_{uv\in E(G)}1/\!\sqrt{d(u)d(v)}\)，其中\(d(v)\)是\(v\)的度，匹配数\(\alpha'(G)\)是\(G\)中最大匹配的大小。我们证明每个\(n\)顶点的科尼格 - 埃格瓦里图，特别是每个二分图，满足\(R(G)\le\sqrt{\alpha'(G)(n - \alpha'(G))}\)，并刻画了达到等式的图。结合贝尔热 - 图特公式，我们确定了\(n\ge4\)的所有\(n\)顶点图中\(R(G)-\alpha'(G)\)的最大值及每个极值图。这否定了奥奇切、汉森和郑在2006年的一个猜想：最小反例是\(K_{10,55}\)，最优部分大小由比例\(\frac{2 - \sqrt2}{4}\)而非\(\frac{1}{7}\)确定，极值图不仅是完全二分图，对于\(n = 10\)等式就不成立了。两个比例给出的渐近斜率相差\(3.7\cdot10^{-5}\)，这解释了为什么该猜想在小阶图的搜索中未被发现，允许两个最优部分大小的阶数由佩尔方程\(x^2 - 2y^2 = 1\)产生。
英文摘要
Let $α'(G)$ be the matching number of a graph $G$, and let its Randić index be $R(G)=\sum_{uv\in E(G)}(d(u)d(v))^{-1/2}$. In 2006, Aouchiche, Hansen, and Zheng conjectured that the maximum of $R(G)-α'(G)$ over all $n$-vertex graphs is attained by the complete bipartite graph whose smaller part has $\lfloor\frac{n+4}{7}\rfloor$ vertices; the conjecture has remained open since then. In this paper, we prove that every $n$-vertex König--Egerváry graph, and in particular every bipartite graph, satisfies \[ R(G)\le\sqrt{α'(G)\left(n-α'(G)\right)}, \] and we characterize the graphs attaining equality as the bipartite graphs all of whose components are semiregular with a common degree ratio. The König--Egerváry hypothesis cannot be dropped, but the Berge--Tutte formula reduces the general case to it, and in this way we determine the maximum of $R(G)-α'(G)$ for every $n\ge4$, together with all extremal graphs. The conjecture is therefore false, and it fails for infinitely many orders: the optimal part size is governed by the proportion $\frac{2-\sqrt2}{4}$ rather than by $\frac17$. The two proportions give asymptotic slopes differing by less than $3.7\cdot10^{-5}$, which is why a search over graphs of small order does not distinguish them. The equality statement fails as well, since the extremal graphs are not only the complete bipartite ones.
Comments15 pages
