New families of adjointly unique graphs (Q1366788)

The adjoint polynomial \(h(G,x)\) of a graph \(G\) is derived from the chromatic polynomial of its complement. The graph \(G\) is called adjointly unique if \(h(H,x)=h(G,x)\) implies that \(H\) is isomorphic with \(G\). It is clear that a graph is adjointly unique if and only if its complement is chromatically unique. In the paper is presented a class of adjointly unique graphs, which is formed from paths, cycles and \(D_n\) graphs. \(D_n\) is a graph obtained from the union of \(P_{n-2}\) and \(K_3\) by identifying a vertex from \(K_3\) with one endvertex of \(P_{n-2}\).