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Abstract
Let be a signed graph and let
be the adjacency matrix of
. The nullity of
is the multiplicity of eigenvalue zero in the spectrum of
. In this paper, we characterize the signed graphs of order
with nullity
or
, and introduce a graph transformation which preserves the nullity. As an application we determine the unbalanced bicyclic signed graphs of order
with nullity
or
, and signed bicyclic graphs (including simple bicyclic graphs) of order
with nullity
.
Keywords:
AMS Subject Classification:
Acknowledgments
The authors are very grateful to the referee for their valuable comments, especially for pointing out the inaccuracy of Theorem 2.4 in the original version. The authors would like to thank the following organizations for their support and funding: the National Natural Science Foundation of China (11071002, 11126178), the Program for New Century Excellent Talents in University, the Key Project of Chinese Ministry of Education (210091), the Specialized Research Fund for the Doctoral Program of Higher Education (20103401110002), the Science and Technological Fund of Anhui Province for Outstanding Youth (10040606Y33), the Project of Educational Department of Anhui Province (KJ2011A019), the Scientific Research Fund for Fostering Distinguished Young Scholars of Anhui University (KJJQ1001), and the Academic Innovation Team of Anhui University Project (KJTD001B).