On problems and conjectures on adjointly equivalent graphs

For a graph G, let image denote its adjoint polynomial and image denote the minimum real root of image. Two graphs H and G are said to be adjointly equivalent if image. Let image and image. In this paper, we give a necessary and sufficient condition for two graphs H and G in image to be adjointly equivalent, where image. We also solve some problems and conjectures proposed by Dong et al. (Discrete Math. 258 (2002) 303–321).