Distribution of Coefficients of Rank Polynomials for Random Sparse Graphs

We study the distribution of coefficients of rank polynomials of random sparse graphs. We first discuss the limiting distribution for general graph sequences that converge in the sense of Benjamini-Schramm. Then we compute the limiting distribution and Newton polygons of the coefficients of the rank polynomial of random $d$-regular graphs.

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

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.