THE COEFFICIENTS THEOREM OF SIGNED GRAPH AND ITS APPLICATION

The nullity of a graph is the multiplicity of the eigenvalue zero in its spectrum. The classic coefficients theorem of simple graph is generalized to the case of signed graph. With the help of this theorem, the nullity of signed graph is studied. The upper bound of the nullity of $n(n\geq 5)$-vertex unicyclic signed graphs is shown to be $n-4$. The unicyclic signed graphs with nullity equal to n-2(n-3 or n-4, respectively) are charaterized. Moreover, the nullity set is concerned.