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 is the same for every orientation σ 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 = ⌊3(n− 1)/2⌋. 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.