A new two-variable generalization of the chromatic polynomial for signed graphs

The function that counts the number of proper colorings of a graph is the chromatic\ud polynomial. Such colorings can also be done with signed graphs, graphs consisting\ud of an unsigned graph and a sign function labeling each edge and loop positive or negative. A signed graph has a chromatic polynomial with the same enumerative and algebraic properties as unsigned graphs. In 2003, Dohmen, Ponitz and Tittmann introduced a two-variable chromatic polynomial by adding improper colors. We extend this polynomial to signed graphs and explore some interesting properties that realize other graph concepts as special evaluation. Furthermore, we extend Stanley???s reciprocity theorem for the chromatic polynomial and its connections to acyclic orientations to the two-variable chromatic polynomial for unsigned and signed graphs