scholarly journals On oriented graphs with minimal skew energy

2014 ◽  
Vol 27 ◽  
Author(s):  
Shi-Cai Gong ◽  
Xueliang Li ◽  
Guanghui Xu
Keyword(s):  
Adjacency Matrix ◽  
Variable Neighborhood Search ◽  
Integral Formula ◽  
Oriented Graph ◽  
Singular Values ◽  
Neighborhood Search ◽  
External Energy ◽  
Oriented Graphs ◽  
Skew Energy ◽  
Skew Adjacency Matrix

Let S(G^σ) be the skew-adjacency matrix of an oriented graph Gσ. The skew energy of G^σ is the sum of all singular values of its skew-adjacency matrix S(G^σ). This paper first establishes an integral formula for the skew energy of an oriented graph. Then, it determines all oriented graphs with minimal skew energy among all connected oriented graphs on n vertices with m (n ≤ m < 2(n − 2)) arcs, which is analogous to the conjecture for the energy of undirected graphs proposed by Caporossi et al. [G. Caporossi, D. Cvetkovic, I. Gutman, and P. Hansen. Variable neighborhood search for extremal graphs. 2. Finding graphs with external energy. J. Chem. Inf. Comput. Sci., 39:984–996, 1999].

scholarly journals Skew Spectra of Oriented Graphs

10.37236/270 ◽  
2009 ◽  
Vol 16 (1) ◽  
Author(s):  
Bryan Shader ◽  
Wasin So
Keyword(s):  
Directed Graph ◽  
Adjacency Matrix ◽  
Undirected Graph ◽  
Oriented Graph ◽  
Oriented Graphs ◽  
Underlying Graph ◽  
Adjacency Spectrum ◽  
The Relationship ◽  
Skew Adjacency Matrix

An oriented graph $G^{\sigma}$ is a simple undirected graph $G$ with an orientation $\sigma$, which assigns to each edge a direction so that $G^{\sigma}$ becomes a directed graph. $G$ is called the underlying graph of $G^{\sigma}$, and we denote by $Sp(G)$ the adjacency spectrum of $G$. Skew-adjacency matrix $S( G^{\sigma} )$ of $G^{\sigma}$ is introduced, and its spectrum $Sp_S( G^{\sigma} )$ is called the skew-spectrum of $G^{\sigma}$. The relationship between $Sp_S( G^{\sigma} )$ and $Sp(G)$ is studied. In particular, we prove that (i) $Sp_S( G^{\sigma} ) = {\bf i} Sp(G)$ for some orientation $\sigma$ if and only if $G$ is bipartite, (ii) $Sp_S(G^{\sigma}) = {\bf i} Sp(G)$ for any orientation $\sigma$ if and only if $G$ is a forest, where ${\bf i}=\sqrt{-1}$.

scholarly journals Characteristic Polynomials of Skew-Adjacency Matrices of Oriented Graphs

10.37236/643 ◽  
2011 ◽  
Vol 18 (1) ◽  
Author(s):  
Yaoping Hou ◽  
Tiangang Lei
Keyword(s):  
Characteristic Polynomial ◽  
Directed Graph ◽  
Adjacency Matrix ◽  
Undirected Graph ◽  
Oriented Graph ◽  
Bipartite Graphs ◽  
Characteristic Polynomials ◽  
Oriented Graphs ◽  
Underlying Graph ◽  
Skew Adjacency Matrix

An oriented graph $\overleftarrow{G}$ is a simple undirected graph $G$ with an orientation, which assigns to each edge a direction so that $\overleftarrow{G}$ becomes a directed graph. $G$ is called the underlying graph of $\overleftarrow{G}$ and we denote by $S(\overleftarrow{G})$ the skew-adjacency matrix of $\overleftarrow{G}$ and its spectrum $Sp(\overleftarrow{G})$ is called the skew-spectrum of $\overleftarrow{G}$. In this paper, the coefficients of the characteristic polynomial of the skew-adjacency matrix $S(\overleftarrow{G}) $ are given in terms of $\overleftarrow{G}$ and as its applications, new combinatorial proofs of known results are obtained and new families of oriented bipartite graphs $\overleftarrow{G}$ with $Sp(\overleftarrow{G})={\bf i} Sp(G) $ are given.

scholarly journals On the Skew Spectra of Cartesian Products of Graphs

10.37236/2864 ◽  
2013 ◽  
Vol 20 (2) ◽  
Author(s):  
Cui Denglan ◽  
Hou Yaoping
Keyword(s):  
Directed Graph ◽  
Adjacency Matrix ◽  
Undirected Graph ◽  
Oriented Graph ◽  
Bipartite Graphs ◽  
Cartesian Products ◽  
Product Graph ◽  
Underlying Graph ◽  
Skew Energy ◽  
Skew Adjacency Matrix

An oriented graph ${G^{\sigma}}$ is a simple undirected graph $G$ with an orientation, which assigns to each edge of $G$ a direction so that ${G^{\sigma}}$ becomes a directed graph. $G$ is called the underlying graph of ${G^{\sigma}}$ and we denote by $S({G^{\sigma}})$ the skew-adjacency matrix of ${G^{\sigma}}$ and its spectrum $Sp({G^{\sigma}})$ is called the skew-spectrum of ${G^{\sigma}}$. In this paper, the skew spectra of two orientations of the Cartesian products are discussed, as applications, new families of oriented bipartite graphs ${G^{\sigma}}$ with $Sp({G^{\sigma}})={\bf i} Sp(G)$ are given and the orientation of a product graph with maximum skew energy is obtained.

scholarly journals A Note on Skew Energy of Hadamard Graph

2019 ◽  
Vol 9 (1S4) ◽  
pp. 1010-1013
Keyword(s):  
Adjacency Matrix ◽  
Oriented Graph ◽  
Eigen Values ◽  
Skew Energy ◽  
The Absolute

The skew spectrum and skew energy of an oriented graph are respectively the set of eigenvalues of the adjacency matrix of and the sum of the absolute values of the eigen values of the adjacency matrix of . In this work, we find and study the skew spectrum and the skew energy of Hadamard graph for a particular orientation.

SINGULARITY OF ORIENTED GRAPHS FROM SEVERAL CLASSES

2019 ◽  
Vol 102 (1) ◽  
pp. 7-14
Author(s):  
XIAOXUAN CHEN ◽  
JING YANG ◽  
XIANYA GENG ◽  
LONG WANG
Keyword(s):  
Adjacency Matrix ◽  
Oriented Graph ◽  
Common Vertex ◽  
Oriented Graphs ◽  
Common Path ◽  
Bicyclic Graphs ◽  
Vertex Disjoint

A digraph is called oriented if there is at most one arc between two distinct vertices. An oriented graph $D$ is nonsingular if its adjacency matrix $A(D)$ is nonsingular. We characterise all nonsingular oriented graphs from three classes: graphs in which cycles are vertex disjoint, graphs in which all cycles share exactly one common vertex and graphs formed by cycles sharing a common path. As a straightforward corollary, the singularity of oriented bicyclic graphs is determined.

scholarly journals Skew-rank of an oriented graph in terms of the rank and dimension of cycle space of its underlying graph

Filomat ◽  
2018 ◽  
Vol 32 (4) ◽  
pp. 1303-1312 ◽  
Author(s):  
Yong Lu ◽  
Ligong Wang ◽  
Qiannan Zhou
Keyword(s):  
Lower Bound ◽  
Upper Bound ◽  
Adjacency Matrix ◽  
Oriented Graph ◽  
Connected Components ◽  
Cycle Space ◽  
Vertex Number ◽  
Underlying Graph ◽  
Skew Adjacency Matrix

Let G? be an oriented graph and S(G?) be its skew-adjacency matrix, where G is called the underlying graph of G?. The skew-rank of G?, denoted by sr(G?), is the rank of S(G?). Denote by d(G) = |E(G)|-|V(G)| + ?(G) the dimension of cycle spaces of G, where |E(G)|, |V(G)| and ?(G) are the edge number, vertex number and the number of connected components of G, respectively. Recently, Wong, Ma and Tian [European J. Combin. 54 (2016) 76-86] proved that sr(G?) ? r(G) + 2d(G) for an oriented graph G?, where r(G) is the rank of the adjacency matrix of G, and characterized the graphs whose skew-rank attain the upper bound. However, the problem of the lower bound of sr(G?) of an oriented graph G? in terms of r(G) and d(G) of its underlying graph G is left open till now. In this paper, we prove that sr(G?) ? r(G)-2d(G) for an oriented graph G? and characterize the graphs whose skew-rank attain the lower bound.

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