Abstract
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.
Highlights
Dichromatic polynomial is the most general graph invariant satisfying deletion-contraction properties
We focus on the asymptotic properties of the rank polynomial, focusing primarily on the sparse graphs
That we have reviewed Benjamini-Schramm convergence, we define a probability measure that describes the relative size of the coefficients of the rank polynomial of a graph G
Summary
Dichromatic polynomial is the most general graph invariant satisfying deletion-contraction properties. It contains important information about G, in particular about its connectivity properties, and about nowhere-zero flows on G. We focus on the asymptotic properties of the rank polynomial, focusing primarily on the sparse graphs. Several asymptotic properties of the rank and Tutte polynomials have not been considered before, to our knowledge. We first study Newton polygons for the rank polynomials for random regular graphs in 2.1. We define probability measures describing the concentration of the (normalized) coefficients of those polynomials. The questions considered in this paper were suggested by numerical experiments in the paper [JLMRT], where some a priori results on the coefficient measures for the Tutte polynomial were established
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