The binding number of a graph G is defined as \(\begin{aligned} \operatorname {bind}(G)=\min \left\{ \frac{|N_G(S)|}{|S|}\,\bigg |\,\emptyset \ne S\subseteq V(G), N_G(S)\ne V(G)\right\} . \end{aligned}\) As a key structural parameter in graph factor theory, the binding number provides a useful tool for inferring the existence of various graph factors. We address the Brualdi–Solheid type problem for graphs with a given binding number and identify the extremal graph with the maximum spectral radius. For a real number \(r\ge 0\) , a graph G is called to be r-binding if \(\operatorname {bind}(G)\ge r\) . A natural question is to determine whether a graph is r-binding. For a positive integer r, Fan and Lin (2024) provided a spectral radius condition to guarantee a connected graph to be r-binding. In this paper, we extend their result by establishing two tight sufficient conditions based on the size and the spectral radius, respectively, for a connected graph to be \(\frac{1}{r}\) -binding.
