ABSTRACT
A signed graph is a graph with a sign attached to each of its edges, where G is the underlying graph of
. Let
denote the adjacency matrix of
, and
be the adjacency matrix of G. The rank of G (resp.
), written as
(resp.
), refers to the rank of
(resp.
)). A natural and interesting problem is: What is the relation between the rank of a signed graph and the rank of its underlying graph? In this paper, we focus our attention on this problem. Sharp lower bounds and upper bounds on
are determined. All the corresponding extremal signed graphs
are characterized, respectively.
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Acknowledgements
The author is very grateful to Professor Shuchao Li for some of his ideas, discussions, and many useful suggestions. I am indebted to the referees for carefully reading the manuscript and providing numerous comments.
Disclosure statement
No potential conflict of interest was reported by the author.