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Abstract
Let be a weighted graph and
be the adjacency matrix of
. The rank of
is the rank of
. If the weight of each edge of
is 1 or
,
is also called a signed graph. In this paper, we characterize weighted graphs with rank 2, weighted
-free graphs with rank 3 and weighted graphs containing pendent vertices with rank 4. We also characterize signed graphs with rank 4.
Keywords:
AMS SUBJECT CLASSIFICATION:
Acknowledgements
The authors would like to thank the anonymous referees for their valuable comments, which have considerably improved the presentation of this paper.
Notes
No potential conflict of interest was reported by the authors.
Additional information
Funding
This work was supported by the National Natural Science Foundation of China [grant number 11101027], [grant number 11371193]; the Fundamental Research Funds for the Central Universities of China [grant number 2015JBM107]; the Beijing Higher Education Young Elite Teacher Project [grant number YETP0573].