Note on the subgraph component polynomial

Abstract

Tittmann, Averbouch and Makowsky [P. Tittmann, I. Averbouch, J.A. Makowsky, The enumeration of vertex induced subgraphs with respect to the number of components, European Journal of Combinatorics, 32 (2011) 954-974], introduced the subgraph component polynomial $Q(G;x,y)$ which counts the number of connected components in vertex induced subgraphs. It contains much of the underlying graph's structural information, e.g. the order, the size, the independence number. We show that there are several other graph invariants, like the connectivity, the number of cycles of length four in a regular bipartite graph, are determined by the subgraph component polynomial. Using the obtained results, we find several well-known families of graphs are determined by $Q(G;x,y)$: paths, cycles, tadpole graphs, complete bipartite graphs, friendship graphs, book graphs and hypercubes. Moreover, we study the distinguish power and find some simple graphs which are not distinguished by the subgraph component polynomial but distinguished by the Tutte polynomial and the character polynomial. These are answers to three questions concerning the subgraph component polynomial proposed by Tittmann et al.