A New Two-variable Chromatic Polynomial for Signed Graphs

It is natural to study vertex colorings in graph theory. The function that counts the number of colorings of a graph is the chromatic polynomial. One way to compute this polynomial is through the deletion contraction method involving the recursive combination of its subgraphs. Such colorings can also be done with signed graphs.  A signed graph is a graph consisting of an unsigned graph along with a sign function that labels each edge and loop positive or negative. The sign function is defined for all edges except halfedges. Coloring a signed graph requires signed colors and it has a chromatic polynomial with the same enumerative and algebraic properties as for unsigned graphs. Signed graphs were first developed in the social sciences in the 1950's to model social situations.  They have applications in many fields because they give a relationship, positive or negative, between two nodes.  In addition to social networks, signed graphs show up in knot theory to show positive or negative crossings, biology to model different classes of biological networks, physics to compute the ground state energy in Ising model, computer science to cluster data by similarities, and qualitative economics to model changes. In 2003, Dohmen, Ponitz and Tittmann introduced a two-variable chromatic polynomial by adding improper colors. These colors can be referred to as "wildcard" colors. We extend this polynomial to signed graphs and explore some interesting properties that realize other graph concepts such as independence and antibalance as special evaluation.