A new polynomial on compositions of integers: on distinguishing caterpillars from their symmetric chromatic function

TL;DR

This paper introduces an algebraic framework using composition polynomials to distinguish non-isomorphic caterpillar trees based on their symmetric chromatic functions, advancing graph invariants.

Contribution

It proposes the composition-lattice polynomial and a unique factorization theorem, linking composition structure to graph invariants and conjecturing a method to distinguish caterpillars.

Findings

Defined the composition-lattice polynomial (L-polynomial) for compositions.

Proved a unique irreducible factorization theorem for compositions.

Established a condition under which caterpillars can be distinguished by their symmetric chromatic functions.

Abstract

In this paper, we propose an algebraic approach to determine whether two non-isomorphic caterpillar trees can have the same symmetric function generalization of the chromatic polynomial. On the set of all composition on integers, we introduce: An operation, which we call composition product; and a combinatorial polynomial, which we call the composition-lattice polynomial or L-polynomial, that mimics the weighted graph polynomial of Noble and Welsh. We prove a unique irreducible factorization theorem and establish a connection between the L-polynomial of a composition and its irreducible factorization, namely that reversing irreducible factors does not change L, and conjecture that is the only way of generating such compositions. Finally, we find a sufficient condition for two caterpillars have a different symmetric function generalization of the chromatic polynomial, and use this condition to show that if our conjecture were to hold, then the symmetric function generalization of the chromatic polynomial distinguishes among a large class of caterpillars.