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

Let Gn,& gamma; be the set of all connected graphs on n vertices with domination number & gamma;. A graph is called a minimizer graph if it attains the minimum spectral radius among Gn,& gamma;. Very recently, Liu, Li and Xie (2023) [17] proved that the minimizer graph over all graphs in Gn,& gamma; must be a tree. Moreover, they determined the minimizer graph among Gn,Ln21 for even n, and posed the conjecture on the minimizer graph among Gn,L n2 1 for odd n. In this paper, we disprove the conjecture and completely determine the unique minimizer graph among Gn,L n2 1 for odd n. & COPY; 2023 Published by Elsevier Inc.