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abstract: 'Given a graph $G$, an incidence matrix $\mathcal{N}(G)$ 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 $\mathcal{N}$-matrix is a complete graph invariant. Several invariants of a graph are then shown to be reconstructible from its $\mathcal{N}$-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.'
author:
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    Bhalchandra D. Thatte\
    Allan Wilson Centre for Molecular Ecology and Evolution,\
    and Institute of Fundamental Sciences,\
    Massey University, Palmerston North, New Zealand\
    `b.thatte@massey.ac.nz`
date: |
     