Solution to a Conjecture on the Maximum Skew-Spectral Radius of Odd-Cycle Graphs

Let $G$ be a simple graph with no even cycle, called an odd-cycle graph. Cavers et al. [Linear Algebra Appl. 436(12):4512-1829, 2012] showed that the spectral radius of $G^\sigma$ is the same for every orientation $\sigma$ of $G$, and equals the maximum matching root of $G$. They proposed a conjecture that the graphs which attain the maximum skew spectral radius among the odd-cycle graphs $G$ of order $n$ are isomorphic to the odd-cycle graph with one vertex degree $n-1$ and size $m=\lfloor 3(n-1)/2\rfloor$. By using the Kelmans transformation, we give a proof to the conjecture. Moreover, sharp upper bounds of the maximum matching roots of the odd-cycle graphs with given order $n$ and size $m$ are given and extremal graphs are characterized.