Chromatic Signed-Symmetric Functions of Signed Graphs

TL;DR

This paper introduces the chromatic signed-symmetric function for signed graphs, generalizes Stanley's reciprocity theorem, and verifies the conjecture that these functions distinguish certain classes of signed trees.

Contribution

It extends the concept of chromatic symmetric functions to signed graphs and proves a generalized reciprocity theorem, advancing the understanding of graph invariants.

Findings

Generalized Stanley's reciprocity theorem for signed graphs

Verified the conjecture for certain classes of signed paths

Introduced a new graph invariant for signed graphs

Abstract

Stanley introduced the chromatic symmetric function of a simple graph, which is a generalization of a chromatic polynomial. This is expressed in terms of the integer points of the complements of the corresponding graphic arrangement. Stanley proved a combinatorial reciprocity theorem for chromatic functions. This is considered as an Ehrhart-type reciprocity theorem for the graphic arrangement. We introduce the chromatic signed-symmetric function of a signed graph, an analogue of the chromatic symmetric function, by the integer points of the complements of the corresponding signed-graphic arrangement and prove a generalization of Stanley's reciprocity theorem. Stanley has conjectured that the chromatic symmetric function distinguishes trees. This conjecture is also generalized for signed trees. We verify the conjecture for certain classes of signed paths.