Note on the Subgraph Component Polynomial

Tittmann, Averbouch and Makowsky [The enumeration of vertex induced subgraphs with respect to the number of components, European J. Combin. 32 (2011) 954–974] introduced the subgraph component polynomial Q(G;x, y) of a graph G, which counts the number of connected components in vertex induced subgraphs. This polynomial encodes a large amount of combinatorial information about the underlying graph, such as the order, the size, and the independence number. We show that several other graph invariants, such as the connectivity and the number of cycles of length four in a regular bipartite graph are also determined by the subgraph component polynomial. Then, we prove that several well-known families of graphs are determined by the polynomial Q(G;x, y). Moreover, we study the distinguishing power and find simple graphs which are not distinguished by the subgraph component polynomial but distinguished by the characteristic polynomial, the matching polynomial and the Tutte polynomial. These are partial answers to three open problems proposed by Tittmann et al.