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

Given a graph G, an incidence matrix 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 N -matrix is a complete graph invariant. Several invariants of a graph are then shown to be reconstructible from its N -matrix. The invariants include the characteristic polynomial, the rank polynomial, the numbe...

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A ug 2 00 5 Kocay ’ s lemma , Whitney ’ s theorem , and some polynomial invariant reconstruction problems

Given a graph G, an incidence matrix 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 N -matrix is a complete graph invariant. Several invariants of a graph are then shown to be reconstructible from its N -matrix. The invariants include the characteristic polynomial, the rank polynomial, the numbe...

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0 50 73 75 v 2 1 1 A ug 2 00 5 Kocay ’ s lemma , Whitney ’ s theorem , and some polynomial invariant reconstruction problems

Given a graph G, an incidence matrix 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 N -matrix is a complete graph invariant. Several invariants of a graph are then shown to be reconstructible from its N -matrix. The invariants include the characteristic polynomial, the rank polynomial, the numbe...

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Reconstruction Lemma and Fluctuation-Dissipation Theorem

We discuss a new approach to nonequilibrium statistical thermodynamics based on mappings of the microscopic dynamics into the macroscopic dynamics. Near stationary solutions, this mapping results in a compact formula for the macroscopic vector field without a hypothesis of a separation of time scales. Relations of this formula to short-memory approximation, the Green-Kubo formula, and expressio...

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Some Problems in Invariant Theory

We present summaries of a number of research problems in invariant theory. These problems were posed during the organized problem sessions held during the April 2002, Kingston Ontario, Invariant Theory Workshop and Conference.

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

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