A sharp Randi\'c bound for K\"onig--Egerv\'ary graphs and a conjecture of Aouchiche, Hansen, and Zheng

Abstract

Let $\alpha'(G)$ be the matching number of a graph $G$, and let its Randi\'c 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 $R(G)-\alpha'(G)$ 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 K\"onig--Egerv\'ary graph, and in particular every bipartite graph, satisfies \[ R(G)\le\sqrt{\alpha'(G)\left(n-\alpha'(G)\right)}, \] and we characterize the graphs attaining equality as the bipartite graphs all of whose components are semiregular with a common degree ratio. The K\"onig--Egerv\'ary hypothesis cannot be dropped, but the Berge--Tutte formula reduces the general case to it, and in this way we determine the maximum of $R(G)-\alpha'(G)$ 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.