5 research outputs found
Relation between the skew-rank of an oriented graph and the independence number of its underlying graph
An oriented graph is a digraph without loops or multiple arcs
whose underlying graph is . Let be the
skew-adjacency matrix of and be the independence number
of . The rank of is called the skew-rank of ,
denoted by . Wong et al. [European J. Combin. 54 (2016) 76-86]
studied the relationship between the skew-rank of an oriented graph and the
rank of its underlying graph. In this paper, the correlation involving the
skew-rank, the independence number, and some other parameters are considered.
First we show that , where
is the order of and is the dimension of cycle space of .
We also obtain sharp lower bounds for , and characterize all
corresponding extremal graphs.Comment: 16 Page; 1 figur
The skew-rank of oriented graphs
An oriented graph is a digraph without loops and multiple arcs,
where is called the underlying graph of . Let
denote the skew-adjacency matrix of . The rank of the skew-adjacency
matrix of is called the {\it skew-rank} of , denoted by
. The skew-adjacency matrix of an oriented graph is skew
symmetric and the skew-rank is even. In this paper we consider the skew-rank of
simple oriented graphs. Firstly we give some preliminary results about the
skew-rank. Secondly we characterize the oriented graphs with skew-rank 2 and
characterize the oriented graphs with pendant vertices which attain the
skew-rank 4. As a consequence, we list the oriented unicyclic graphs, the
oriented bicyclic graphs with pendant vertices which attain the skew-rank 4.
Moreover, we determine the skew-rank of oriented unicyclic graphs of order
with girth in terms of matching number. We investigate the minimum value of
the skew-rank among oriented unicyclic graphs of order with girth and
characterize oriented unicyclic graphs attaining the minimum value. In
addition, we consider oriented unicyclic graphs whose skew-adjacency matrices
are nonsingular.Comment: 17 pages, 4 figure
About the spectra of a real nonnegative matrix and its signings
For a real matrix , we denote by the spectrum of and by its absolute value, that is the matrix obtained from
by replacing each entry of by its absolute value. Let be a nonnegative
real matrix, we call a \emph{signing} of every real matrix such that
. In this paper, we study the set of all signings
of such that where is a complex unit number.
Our work generalizes some results obtained in [1, 5, 8]
On the spectral reconstruction problem for digraphs
The idiosyncratic polynomial of a graph with adjacency matrix is the
characteristic polynomial of the matrix , where is the
identity matrix and is the all-ones matrix. It follows from a theorem of
Hagos (2000) combined with an earlier result of Johnson and Newman (1980) that
the idiosyncratic polynomial of a graph is reconstructible from the multiset of
the idiosyncratic polynomial of its vertex-deleted subgraphs. For a digraph
with adjacency matrix , we define its idiosyncratic polynomial as the
characteristic polynomial of the matrix . By forbidding
two fixed digraphs on three vertices as induced subdigraphs, we prove that the
idiosyncratic polynomial of a digraph is reconstructible from the multiset of
the idiosyncratic polynomial of its induced subdigraphs on three vertices. As
an immediate consequence, the idiosyncratic polynomial of a tournament is
reconstructible from the collection of its -cycles. Another consequence is
that all the transitive orientations of a comparability graph have the same
idiosyncratic polynomial
A survey on the skew energy of oriented graphs
Let be a simple undirected graph with adjacency matrix . The energy
of is defined as the sum of absolute values of all eigenvalues of ,
which was introduced by Gutman in 1970s. Since graph energy has important
chemical applications, it causes great concern and has many generalizations.
The skew energy and skew energy-like are the generalizations in oriented
graphs. Let be an oriented graph of with skew adjacency matrix
. The skew energy of , denoted by
, is defined as the sum of the norms of all
eigenvalues of , which was introduced by Adiga, Balakrishnan and
So in 2010. In this paper, we summarize main results on the skew energy of
oriented graphs. Some open problems are proposed for further study. Besides,
results on the skew energy-like: the skew Laplacian energy and skew Randi\'{c}
energy are also surveyed at the end.Comment: This will appear as a chapter in Mathematical Chemistry Monograph
No.17: Energies of Graphs -- Theory and Applications, edited by I. Gutman and
X. L