Distribution of Coefficients of Rank Polynomials for Random Sparse Graphs

Dmitry Jakobson; Calum MacRury; Sergey Norin; Lise Turner

doi:10.37236/7133

Distribution of Coefficients of Rank Polynomials for Random Sparse Graphs

The Electronic Journal of Combinatorics ◽

10.37236/7133 ◽

2018 ◽

Vol 25 (4) ◽

Author(s):

Dmitry Jakobson ◽

Calum MacRury ◽

Sergey Norin ◽

Lise Turner

Keyword(s):

Limiting Distribution ◽

Regular Graphs ◽

Newton Polygons ◽

Sparse Graphs ◽

General Graph ◽

Rank Polynomial

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.

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(1, 2)-Factorizations of General Eulerian Nearly Regular Graphs

Combinatorics Probability Computing ◽

10.1017/s0963548300001000 ◽

1994 ◽

Vol 3 (1) ◽

pp. 87-95 ◽

Cited By ~ 2

Author(s):

Roland Häggkvist ◽

Anders Johansson

Keyword(s):

Regular Graphs ◽

General Graph ◽

Edge Colouring

Every general graph with degrees 2k and 2k − 2, k ≥ 3, with zero or at least two vertices of degree 2k − 2 in each component, has a k-edge-colouring such that each monochromatic subgraph has degree 1 or 2 at every vertex.In particular, if T is a triangle in a 6-regular general graph, there exists a 2-factorization of G such that each factor uses an edge in T if and only if T is non-separating.

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Rate of Convergence of the Short Cycle Distribution in Random Regular Graphs Generated by Pegging

The Electronic Journal of Combinatorics ◽

10.37236/133 ◽

2009 ◽

Vol 16 (1) ◽

Cited By ~ 1

Author(s):

Pu Gao ◽

Nicholas Wormald

Keyword(s):

Rate Of Convergence ◽

Total Variation ◽

Upper Bound ◽

Mixing Time ◽

Limiting Distribution ◽

Total Variation Distance ◽

Regular Graphs ◽

Short Cycle ◽

Variation Distance

The pegging algorithm is a method of generating large random regular graphs beginning with small ones. The $\epsilon$-mixing time of the distribution of short cycle counts of these random regular graphs is the time at which the distribution reaches and maintains total variation distance at most $\epsilon$ from its limiting distribution. We show that this $\epsilon$-mixing time is not $o(\epsilon^{-1})$. This demonstrates that the upper bound $O(\epsilon^{-1})$ proved recently by the authors is essentially tight.

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