We develop a new method that uses walk counts for comparing spectral radii of graphs similar in a precisely defined fashion. The method is applied to the cases where a path-like or a star-like structure is coalesced to a graph, in order to prove weak inequality in the conjectured inequality of Belardo, Li Marzi and Simić, and to resolve the Brualdi-Solheid problem for the classes of graphs consisting of rooted products with the same rooted graph.