Kocay's lemma, Whitney's theorem, and some polynomial invariant reconstruction problems

Abstract

Given a graph G, an incidence matrix N(G) is defined for the set of distinct isomorphism types of induced subgraphs of G. If Ulam's conjecture is true, then every graph invariant must be reconstructible from this matrix, even when the graphs indexing the rows and the columns of N(G) are unspecified. It is proved that the characteristic polynomial, the rank polynomial, and the number of spanning trees of a graph are reconstructible from its N-matrix. These results are stronger than the original results of Tutte in the sense that actual subgraphs are not used. It is also proved that the characteristic polynomial of a graph with minimum degree 1 can be computed from the characteristic polynomials of all its induced proper subgraphs. The ideas in Kocay's lemma play a crucial role in most proofs. Here Kocay's lemma is used to prove Whitney's subgraph expansion theorem in a simple manner. The reconstructibility of the characteristic polynomial is then demonstrated as a direct consequence of Whitney's theorem as formulated here.