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A note on the lacking polynomial of the complete bipartite graph

Amal Alofi, Mark Dukes

Abstract

The lacking polynomial is a graph polynomial introduced by Chan, Marckert, and Selig in 2013 that is closely related to the Tutte polynomial of a graph. It arose by way of a generalization of the Abelian sandpile model and is essentially the generating function of the level statistic on the set of recurrent configurations, called stochastically recurrent states, for that model. In this note we consider the lacking polynomial of the complete bipartite graph. We classify the stochastically recurrent states of the stochastic sandpile model on the complete bipartite graphs $K_{2,n}$ and $K_{m,2}$ where the sink is always an element of the set counted by the first index. We use these characterizations to give explicit formulae for the lacking polynomials of these graphs. Log-concavity of the sequence of coefficients of these two lacking polynomials is proven, and we conjecture log-concavity holds for this general class of graphs.

A note on the lacking polynomial of the complete bipartite graph

Abstract

The lacking polynomial is a graph polynomial introduced by Chan, Marckert, and Selig in 2013 that is closely related to the Tutte polynomial of a graph. It arose by way of a generalization of the Abelian sandpile model and is essentially the generating function of the level statistic on the set of recurrent configurations, called stochastically recurrent states, for that model. In this note we consider the lacking polynomial of the complete bipartite graph. We classify the stochastically recurrent states of the stochastic sandpile model on the complete bipartite graphs and where the sink is always an element of the set counted by the first index. We use these characterizations to give explicit formulae for the lacking polynomials of these graphs. Log-concavity of the sequence of coefficients of these two lacking polynomials is proven, and we conjecture log-concavity holds for this general class of graphs.

Paper Structure

This paper contains 1 section, 8 theorems, 43 equations, 2 tables.

Table of Contents

  1. Acknowledgments

Key Result

Theorem 2

A stable configuration $c$ on $G$ is stochastically recurrent if and only if there exists an orientation $\mathcal{O}$ on $G$ such that $c\in \mathrm{comp}(\mathcal{O})$. It implies that $\mathsf{Sto}(G)$, the set of all stochastically recurrent states on $G$, may be written where the union is taken over all orientations on $G$.

Theorems & Definitions (19)

  • Definition 1: Chan et al. ssm
  • Theorem 2: Chan et al. ssm
  • Definition 3: Chan et al. ssm
  • Proposition 4
  • proof
  • Example 5
  • Proposition 7
  • proof
  • Theorem 8
  • proof
  • ...and 9 more