1,247 research outputs found
More on the skew-spectra of bipartite graphs and Cartesian products of graphs
Given a graph , let be an oriented graph of with the
orientation and skew-adjacency matrix . Then the spectrum
of is called the skew-spectrum of , denoted by
. It is known that a graph is bipartite if and only if
there is an orientation of such that . In
[D. Cui, Y. Hou, On the skew spectra of Cartesian products of graphs, Electron.
J. Combin. 20(2013), #P19], Cui and Hou conjectured that such orientation of a
bipartite graph is unique under switching-equivalence. In this paper, we prove
that the conjecture is true. Moreover, we give an orientation of the Cartesian
product of a bipartite graph and a graph, and then determine the skew-spectrum
of the resulting oriented product graph, which generalizes Cui and Hou's
result, and can be used to construct more oriented graphs with maximum skew
energy.Comment: 9 page
Skew-spectra and skew energy of various products of graphs
Given a graph , let be an oriented graph of with the
orientation and skew-adjacency matrix . Then the spectrum
of consisting of all the eigenvalues of is called
the skew-spectrum of , denoted by . The skew energy of
the oriented graph , denoted by , is defined
as the sum of the norms of all the eigenvalues of . In this paper,
we give orientations of the Kronecker product and the strong
product of and where is a bipartite graph and is an
arbitrary graph. Then we determine the skew-spectra of the resultant oriented
graphs. As applications, we construct new families of oriented graphs with
maximum skew energy. Moreover, we consider the skew energy of the orientation
of the lexicographic product of a bipartite graph and a graph .Comment: 11 page
Power domination and zero forcing
The power domination number arises from the monitoring of electrical networks
and its determination is an important problem. Upper bounds for power
domination numbers can be obtained by constructions. Lower bounds for the power
domination number of several families of graphs are known, but they usually
arise from specific properties of each family and the methods do not
generalize. In this paper we exploit the relationship between power domination
and zero forcing to obtain the first general lower bound for the power
domination number. We apply this bound to obtain results for both the power
domination of tensor products and the zero-forcing number of lexicographic
products of graphs. We also establish results for the zero forcing number of
tensor products and Cartesian products of graphs
Energy of signed digraphs
In this paper we extend the concept of energy to signed digraphs. We obtain
Coulson's integral formula for energy of signed digraphs. Formulae for energies
of signed directed cycles are computed and it is shown that energy of non cycle
balanced signed directed cycles increases monotonically with respect to number
of vertices. Characterization of signed digraphs having energy equal to zero is
given. We extend the concept of non complete extended sum (or briefly,
NEPS) to signed digraphs. An infinite family of equienergetic signed digraphs
is constructed. Moreover, we extend McClelland's inequality to signed digraphs
and also obtain sharp upper bound for energy of signed digraph in terms the
number of arcs. Some open problems are also given at the end.Comment: 19 page
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
On Cartesian product of matrices
Recently, Bapat and Kurata [\textit{Linear Algebra Appl.}, 562(2019),
135-153] defined the Cartesian product of two square matrices and as
A\oslash B=A\otimes \J+\J\otimes B, where \J is the all one matrix of
appropriate order and is the Kronecker product. In this article, we
find the expression for the trace of the Cartesian product of any finite number
of square matrices in terms of traces of the individual matrices. Also, we
establish some identities involving the Cartesian product of matrices. Finally,
we apply the Cartesian product to study some graph-theoretic properties.Comment: 14 page
Hermitian adjacency matrix of digraphs and mixed graphs
The paper gives a thorough introduction to spectra of digraphs via its
Hermitian adjacency matrix. This matrix is indexed by the vertices of the
digraph, and the entry corresponding to an arc from to is equal to the
complex unity (and its symmetric entry is ) if the reverse arc is
not present. We also allow arcs in both directions and unoriented edges, in
which case we use as the entry. This allows to use the definition also for
mixed graphs. This matrix has many nice properties; it has real eigenvalues and
the interlacing theorem holds for a digraph and its induced subdigraphs.
Besides covering the basic properties, we discuss many differences from the
properties of eigenvalues of undirected graphs and develop basic theory. The
main novel results include the following. Several surprising facts are
discovered about the spectral radius; some consequences of the interlacing
property are obtained; operations that preserve the spectrum are discussed --
they give rise to an incredible number of cospectral digraphs; for every
, all digraphs whose spectrum is contained in the
interval are determined.Comment: 35 pages, submitted to JG
Spectra and energy of bipartite signed digraphs
The set of distinct eigenvalues of a signed digraph together with their
multiplicities is called its spectrum. The energy of a signed digraph with
eigenvalues is defined as ,
where denotes real part of complex number . In this paper, we
show that the characteristic polynomial of a bipartite signed digraph of order
with each cycle of length negative and each cycle of
length positive is of the form \\
\\ where are nonnegative integers. We define a
quasi-order relation in this case and show energy is increasing. It is shown
that the characteristic polynomial of a bipartite signed digraph of order
with each cycle negative has the form
where are nonnegative integers. We study integral, real, Gaussian
signed digraphs and quasi-cospectral digraphs and show for each positive
integer there exists a family of cospectral, non symmetric,
strongly connected, integral, real, Gaussian signed digraphs (non cycle
balanced) and quasi-cospectral digraphs of order . We obtain a new family
of pairs of equienergetic strongly connected signed digraphs and answer to open
problem posed in Pirzada and Mushtaq, Energy of signed digraphs, Discrete
Applied Mathematics 169 (2014) 195-205.Comment: 23 page
The Hopf algebra of skew shapes, torsion sheaves on A^n/F_1, and ideals in Hall algebras of monoid representations
We study ideals in Hall algebras of monoid representations on pointed sets
corresponding to certain conditions on the representations. These conditions
include the property that the monoid act via partial permutations, that the
representation possess a compatible grading, and conditions on the support of
the module. Quotients by these ideals lead to combinatorial Hopf algebras which
can be interpreted as Hall algebras of certain sub-categories of modules. In
the case of the free commutative monoid on n generators, we obtain a
co-commutative Hopf algebra structure on -dimensional skew shapes, whose
underlying associative product amounts to a "stacking" operation on the skew
shapes. The primitive elements of this Hopf algebra correspond to connected
skew shapes, and form a graded Lie algebra by anti-symmetrizing the associative
product. We interpret this Hopf algebra as the Hall algebra of a certain
category of coherent torsion sheaves on
supported at the origin, where denotes the field of one element.
This Hopf algebra may be viewed as an -dimensional generalization of the
Hopf algebra of symmetric functions, which corresponds to the case
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
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