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.

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 New Bounds for the α-Indices of Graphs

Mathematics ◽  
2020 ◽  
Vol 8 (10) ◽  
pp. 1668
Author(s):  
Eber Lenes ◽  
Exequiel Mallea-Zepeda ◽  
Jonnathan Rodríguez
Keyword(s):  
Lower Bound ◽  
Spectral Radius ◽  
Upper Bound ◽  
Adjacency Matrix ◽  
Independence Number ◽  
Minimal Degree ◽  
Diagonal Matrix ◽  
The Matrix

Let G be a graph, for any real 0≤α≤1, Nikiforov defines the matrix Aα(G) as Aα(G)=αD(G)+(1−α)A(G), where A(G) and D(G) are the adjacency matrix and diagonal matrix of degrees of the vertices of G. This paper presents some extremal results about the spectral radius ρα(G) of the matrix Aα(G). In particular, we give a lower bound on the spectral radius ρα(G) in terms of order and independence number. In addition, we obtain an upper bound for the spectral radius ρα(G) in terms of order and minimal degree. Furthermore, for n>l>0 and 1≤p≤⌊n−l2⌋, let Gp≅Kl∨(Kp∪Kn−p−l) be the graph obtained from the graphs Kl and Kp∪Kn−p−l and edges connecting each vertex of Kl with every vertex of Kp∪Kn−p−l. We prove that ρα(Gp+1)<ρα(Gp) for 1≤p≤⌊n−l2⌋−1.

scholarly journals On Graphs whose Orientations are Determined by their Hermitian Spectra

10.37236/9640 ◽  
2020 ◽  
Vol 27 (3) ◽  
Author(s):  
Yi Wang ◽  
Bo-Jun Yuan
Keyword(s):  
Upper Bound ◽  
Connected Graph ◽  
Simple Graph ◽  
Equivalence Classes ◽  
Cycle Space ◽  
Mixed Graph ◽  
Cycle Basis ◽  
Underlying Graph ◽  
Mixed Graphs ◽  
Special Cycle

A mixed graph $D$ is obtained from a simple graph $G$, the underlying graph of $D$, by orienting some edges of $G$. A simple graph $G$ is said to be ODHS (all orientations of $G$ are determined by their $H$-spectra) if every two $H$-cospectral graphs in $\mathcal{D}(G)$ are switching equivalent to each other, where $\mathcal{D}(G)$ is the set of all mixed graphs with $G$ as their underlying graph. In this paper, we characterize all bicyclic ODHS graphs and construct infinitely many ODHS graphs whose cycle spaces are of dimension $k$. For a  connected graph $G$ whose cycle space is of dimension $k$, we also obtain an achievable upper bound $2^{2k-1} + 2^{k-1}$ for the number of switching equivalence classes in $\mathcal{D}(G)$, which naturally is an upper bound of the number of  cospectral classes in $\mathcal{D}(G)$. To achieve these, we propose a valid method to estimate the number of switching equivalence classes in $\mathcal{D}(G)$ based on the strong cycle basis, a special cycle basis  introduced in this paper.

Energy of m-Polar Fuzzy Digraphs

2020 ◽  
pp. 469-491 ◽  
Author(s):  
Muhammad Akram ◽  
Danish Saleem ◽  
Ganesh Ghorai
Keyword(s):  
Lower Bound ◽  
Upper Bound ◽  
Adjacency Matrix ◽  
Fuzzy Graph ◽  
Laplacian Energy ◽  
Signless Laplacian ◽  
Different Types ◽  
Matrix Eigenvalues

In this chapter, firstly some basic definitions like fuzzy graph, its adjacency matrix, eigenvalues, and its different types of energies are presented. Some upper bound and lower bound for the energy of this graph are also obtained. Then certain notions, including energy of m-polar fuzzy digraphs, Laplacian energy of m-polar fuzzy digraphs and signless Laplacian energy of m-polar fuzzy digraphs are presented. These concepts are illustrated with several example, and some of their properties are investigated.

scholarly journals The Random Graph Intuition for the Tournament Game

2015 ◽  
Vol 25 (1) ◽  
pp. 76-88 ◽  
Author(s):  
DENNIS CLEMENS ◽  
HEIDI GEBAUER ◽  
ANITA LIEBENAU
Keyword(s):  
Lower Bound ◽  
Random Graph ◽  
Complete Graph ◽  
Upper Bound ◽  
Winning Strategy ◽  
Constant Factor ◽  
Additive Constant ◽  
Underlying Graph ◽  
Straightforward Application

In the tournament game two players, called Maker and Breaker, alternately take turns in claiming an unclaimed edge of the complete graph Kn and selecting one of the two possible orientations. Before the game starts, Breaker fixes an arbitrary tournament Tk on k vertices. Maker wins if, at the end of the game, her digraph contains a copy of Tk; otherwise Breaker wins. In our main result, we show that Maker has a winning strategy for k = (2 − o(1))log2n, improving the constant factor in previous results of Beck and the second author. This is asymptotically tight since it is known that for k = (2 − o(1))log2n Breaker can prevent the underlying graph of Maker's digraph from containing a k-clique. Moreover, the precise value of our lower bound differs from the upper bound only by an additive constant of 12.We also discuss the question of whether the random graph intuition, which suggests that the threshold for k is asymptotically the same for the game played by two ‘clever’ players and the game played by two ‘random’ players, is supported by the tournament game. It will turn out that, while a straightforward application of this intuition fails, a more subtle version of it is still valid.Finally, we consider the orientation game version of the tournament game, where Maker wins the game if the final digraph – also containing the edges directed by Breaker – possesses a copy of Tk. We prove that in that game Breaker has a winning strategy for k = (4 + o(1))log2n.

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 Multiple closed orbits for N-body-type problems

1998 ◽  
Vol 58 (1) ◽  
pp. 1-13 ◽  
Author(s):  
Shiqing Zhang
Keyword(s):  
Lower Bound ◽  
Periodic Solutions ◽  
Upper Bound ◽  
Critical Value ◽  
Body Type ◽  
Fixed Energy ◽  
Closed Orbits

Using the equivariant Ljusternik-Schnirelmann theory and the estimate of the upper bound of the critical value and lower bound for the collision solutions, we obtain some new results in the large concerning multiple geometrically distinct periodic solutions of fixed energy for a class of planar N-body type problems.

Limit Cycle Bifurcations for Piecewise Smooth Hamiltonian Systems with a Generalized Eye-Figure Loop

2016 ◽  
Vol 26 (12) ◽  
pp. 1650204 ◽  
Author(s):  
Jihua Yang ◽  
Liqin Zhao
Keyword(s):  
Lower Bound ◽  
Limit Cycle ◽  
Hamiltonian Systems ◽  
Limit Cycles ◽  
Upper Bound ◽  
Melnikov Function ◽  
Piecewise Smooth ◽  
First Order ◽  
Period Annulus

This paper deals with the limit cycle bifurcations for piecewise smooth Hamiltonian systems. By using the first order Melnikov function of piecewise near-Hamiltonian systems given in [Liu & Han, 2010], we give a lower bound and an upper bound of the number of limit cycles that bifurcate from the period annulus between the center and the generalized eye-figure loop up to the first order of Melnikov function.

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