A note on the lacking polynomial of the complete bipartite graph

TL;DR

This paper analyzes the lacking polynomial of complete bipartite graphs, providing explicit formulas, classifying recurrent states in the sandpile model, and proving log-concavity of the polynomial coefficients.

Contribution

It classifies stochastically recurrent states for certain bipartite graphs and derives explicit lacking polynomial formulas, advancing understanding of these graph invariants.

Findings

Explicit lacking polynomial formulas for K_{2,n} and K_{m,2}

Classification of recurrent states in the sandpile model on these graphs

Proof of log-concavity of polynomial coefficients

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.