We consider a graph polynomial ξ(G;x, y, z) introduced by Ilia Averbouch, Benny Godlin, and Johann A. Makowsky (2008). This graph polynomial simultaneously generalizes the Tutte polynomial as well as a bivariate chromatic polynomial defined by Klaus Dohmen, André Pönitz, and Peter Tittmann (2003). We derive an identity which relates the graph polynomial ξ of a thickened graph (i.e. a graph with...

GraphsG andH are T -equivalent if they have the same Tutte polynomial. G is T -unique if any arbitrary graph H being T -equivalent to G implies that H is isomorphic to G. We show that the generalized Petersen graph P (m, 2) and the line graph of P (m, 2) are T -unique. 1 The Tutte Polynomial Let G be a graph with vertex set V and edge set E, and let G[A] denote the subgraph ofG induced by the e...

An independent dominating set of the simple graph G=(V,E) is a vertex subset that both and in G. The domination polynomial G Di(G,x)=∑Ax|A|, summed over all subsets A⊆V. A root Di(G,x) called an independence root. We investigate polynomials some generalized compound graphs. As consequence, we construct graphs whose roots are real. Also, consider related to paths study number their sets.