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Let $G$ be a simple graph with no even cycle, called an odd-cycle graph.Cavers et al. [Cavers et al. Skew-adjacency matrices of graphs, Linear AlgebraAppl. 436(2012), 4512--1829] showed that the spectral radius of $G^\sigma$ isthe same for every orientation $\sigma$ of $G$, and equals the maximum matchingroot of $G$. They proposed a conjecture that the graphs which attain themaximum skew spectral radius among the odd-cycle graphs $G$ of order $n$ areisomorphic to the odd-cycle graph with one vertex degree $n-1$ and size$m=\lfloor 3(n-1)/2\rfloor$. This paper, by using the Kelmans transformation,gives a proof of the conjecture. Moreover, sharp upper bounds of the maximummatching roots of the odd-cycle graphs with given order $n$ and size $m$ aregiven and extremal graphs are characterized.

Solution to a conjecture on the maximum skew-spectral radius of odd-cycle graphs