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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...
Let β(G) denote the minimum real root of the σ-polynomial of the complement of a graph G and δ(G) the minimum degree of G. In this paper, we give a characterization of all connected graphs G with β(G)>=-4. Using these results, we establish a sufficient and necessary condition for a graph G with p vertices and δ(G)>=p-3, to be chromatically unique. Many previously known results are generalized...
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