Disproof of a conjecture on the minimum spectral radius and the domination number

Abstract

Let Gn,γ be the set of all connected graphs on n vertices with domination number γ. A graph is called a minimizer graph if it attains the minimum spectral radius among Gn,γ. Very recently, Liu, Li and Xie (2023) [17] proved that the minimizer graph over all graphs in Gn,γ must be a tree. Moreover, they determined the minimizer graph among Gn,⌊n2⌋ for even n, and posed the conjecture on the minimizer graph among Gn,⌊n2⌋ for odd n. In this paper, we disprove the conjecture and completely determine the unique minimizer graph among Gn,⌊n2⌋ for odd n.