On problems and conjectures on adjointly equivalent graphs

For a graph G, let h(G,x) denote its adjoint polynomial and β(G) denote the minimum real root of h(G,x). Two graphs H and G are said to be adjointly equivalent if h(H,x)=h(G,x). Let F1={G|β(G)>-4} and F2={G|β(G)⩾-4}. In this paper, we give a necessary and sufficient condition for two graphs H and G in Fi to be adjointly equivalent, where i=1,2. We also solve some problems and conjectures proposed by Dong et al. (Discrete Math. 258 (2002) 303-321).