A Graph Polynomial Arising from Community Structure

Inspired by the study of community structure in connection networks; we introduce the graph polynomial Q (G;x,y), as a bivariate generating function which counts the number of connected components in induced subgraphs. We analyze the features of the new polynomial. First, we re-define it as a subset expansion formula. Second; we give a recursive definition of Q (G; x, y) using vertex deletion; vertex contraction and deletion of a vertex together with its neighborhood, and prove a universality property. We relate Q (C; y) to the universal edge elimination polynomial introduced by I. Averbouch, B. Godlin and J.A. Makowsky (2008), which subsumes other known graph invariants and graph polynomials, among them the Tutte polynomial; the independence and matching polynomials; and the bivariate extension of the chromatic polynomial introduced by K. Dohmen, A. Ponitz, and P. Tittmann (2003). Finally we show that the computation of Q (C; X y) is #P-hard, but Fixed Parameter Tractable for graphs of bounded tree-width and clique-width.