Kocay's Lemma, Whitney's Theorem, and some Polynomial Invariant Reconstruction Problems

Abstract

Given a graph $G$, an incidence matrix ${\cal N}(G)$ is defined on the set of distinct isomorphism types of induced subgraphs of $G$. It is proved that Ulam's conjecture is true if and only if the ${\cal N}$-matrix is a complete graph invariant. Several invariants of a graph are then shown to be reconstructible from its ${\cal N}$-matrix. The invariants include the characteristic polynomial, the rank polynomial, the number of spanning trees and the number of hamiltonian cycles in a graph. 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. 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.