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### Bipartite graphs obtained from adjacency matrices of orientations of graphs

K.B. Reid
1988 Discrete Mathematics
G" can be obtained from 6' by successively reversing the orientation of single arcs, and the number of components in the successive resulting bipartite graphs starts with m(G) and ends with M(G).  ...  Combining this If G denotes a graph of order n, then the adjacency matf;ix of an orientation G of G can be thought of as the adjacency matrix of a bipartite graph B(G) of order 2n, where the rows and columns  ...

### Bipartite Graphs Obtained from Adjacency Matrices of Orientations of Graphs [chapter]

K.B. Reid
1988 Annals of Discrete Mathematics
G" can be obtained from 6' by successively reversing the orientation of single arcs, and the number of components in the successive resulting bipartite graphs starts with m(G) and ends with M(G).  ...  Combining this If G denotes a graph of order n, then the adjacency matf;ix of an orientation G of G can be thought of as the adjacency matrix of a bipartite graph B(G) of order 2n, where the rows and columns  ...

### Skew-spectra and skew energy of various products of graphs [article]

Xueliang Li, Huishu Lian
2013 arXiv   pre-print
Given a graph G, let G^σ be an oriented graph of G with the orientation σ and skew-adjacency matrix S(G^σ).  ...  Moreover, we consider the skew energy of the orientation of the lexicographic product H[G] of a bipartite graph H and a graph G.  ...  Let H τ and G σ be oriented graphs of H and G with the skew-adjacency matrices S 1 and S 2 , respectively.  ...

### A Note on Skew Energy of Hadamard Graph

2019 International Journal of Engineering and Advanced Technology
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  ...  In this work, we find and study the skew spectrum and the skew energy of Hadamard graph for a particular orientation.  ...  If and are two orientations of a graph , then and are said to be switching-equivalent if one can be obtained from other by a switch with respect to some subset of Consider a bipartite graph with bipartition  ...

### A note on skew spectrum of graphs [article]

Yanna Wang, Bo Zhou
2013 arXiv   pre-print
., Skew-adjacency matrices of graphs, Linear Algebra Appl. 436 (2012) 4512--4529].  ...  We give some properties of skew spectrum of a graph, especially, we answer negatively a problem concerning the skew characteristic polynomial and matching polynomial in [M.  ...  This work was supported by the Guangdong Provincial Natural Science Foundation of China (no. S2011010005539). The authors thank the referee for constructive comments.  ...

M. Cavers, S.M. Cioabă, S. Fallat, D.A. Gregory, W.H. Haemers, S.J. Kirkland, J.J. McDonald, M. Tsatsomeros
2012 Linear Algebra and its Applications
The spectra of the skew-adjacency matrices of a graph are considered as a possible way to distinguish adjacency cospectral graphs.  ...  skew-adjacency matrices; skew-spectral radii and an analogue of the Perron-Frobenius theorem; and the number of skew-adjacency matrices of a graph with distinct spectra.  ...  Acknowledgements This paper contains research begun by the authors at the workshop on Theory and Applications of Matrices Described by Patterns, held at the Banff International Research Station (BIRS),  ...

### Permanental polynomials of skew adjacency matrices of oriented graphs [article]

Shunyi Liu, Heping Zhang
2014 arXiv   pre-print
Let G^σ be an orientation of a simple graph G. In this paper, the permanental polynomial of an oriented graph G^σ is introduced.  ...  The coefficients of the permanental polynomial of G^σ are interpreted in terms of the graph structure of G^σ, and it is proved that all orientations G^σ of G have the same permanental polynomial if and  ...  Consider the new orientation of G obtained from G σ by only reversing the orientation of e.  ...

### Permanents, Pfaffian orientations, and even directed circuits (extended abstract)

William McCuaig, Neil Robertson, P. D. Seymour, Robin Thomas
1997 Proceedings of the twenty-ninth annual ACM symposium on Theory of computing - STOC '97
This is equivalent to finding Pfaffian orientations of bipartite graphs and to the even circuit problem for directed graphs.  ...  The algorithm is based on a structural characterization of bipartite graphs that admit a Pfaffian orientation.  ...  The proof outlined here was discovered by the last three authors, who were not aware at the time of the prior work of the first author.  ...

### On the Permanental Polynomials of Matrices

Wei Li, Heping Zhang
2014 Bulletin of the Malaysian Mathematical Sciences Society
matrix of its orientation graph.  ...  Moreover, the characterization of a totally convertible matrix provides an equivalent condition to compute the permanental polynomial of a bipartite graph by the characteristic polynomial of the skew adjacency  ...  For a graph G, the skew adjacency matrix A s ( ⃗ G) of an orientation graph ⃗ G is a signing of the adjacency matrix A(G).  ...

### Eigenvalues of oriented-graph matrices

Jiong-Sheng Li
1995 Linear Algebra and its Applications
We also consider spectral properties of the bipartite oriented-graph matrices and the multiequipartite oriented-graph matrices.  ...  We give bounds on the real and imaginary parts of the eigenvalues of an oriented-graph matrix, prove that each of the irreducible oriented-graph matrices of order n 2 3 has at least three distinct eigenvalues  ...  Clearly, A is the adjacency matrix of some simple undirected graph G of order n, and M is the adjacency matrix of an oriented graph D obtained by assigning an orientation to each edge of G.  ...

### Upper bounds on the numbers of 1-factors and 1-factorizations of hypergraphs

Anna Taranenko
2015 Electronic Notes in Discrete Mathematics
We estimate the number of 1-factors of uniform hypergraphs and the number of 1-factorizations of complete uniform hypergraphs by means of permanents of their adjacency matrices.  ...  A 1-factor of a hypergraph G is a set of hyperedges such that every vertex of the hypergraph is incident to exactly one hyperedge from the set.  ...  The upper bound follows from Bregman's theorem for the permanent of (0,1)-matrices  and from the result of  . There exist several bounds on the number of 1-factorizations of other graphs.  ...

### Some Inequalities on The Skew Spectral Radii of Oriented Graphs

Guang-Hui Xu
2012 Journal of Inequalities and Applications
Let G be a simple graph and G σ be an oriented graph obtained from G by assigning a direction to each edge of G. The adjacency matrix of G is A(G) and the skew-adjacency matrix of G σ is S(G σ ).  ...  As an application of these results, we obtain a sharp upper bound of the skew-spectral radius of an oriented unicyclic graph. MSC: 05C50; 15A18  ...  Acknowledgements This work was supported by the National Natural Science Foundation of China (No. 11171373).  ...

### Computing the permanental polynomials of bipartite graphs by Pfaffian orientation [article]

Heping Zhang, Wei Li
2010 arXiv   pre-print
From the result that a bipartite graph G admits an orientation G^e such that every cycle is oddly oriented if and only if it contains no even subdivision of K_2,3, Yan and Zhang showed that the permanental  ...  polynomial of such a bipartite graph G can be expressed as the characteristic polynomial of the skew adjacency matrix A(G^e).  ...  An orientation G e obtained by Algorithm 4.1 is given in Figure 5(b) . Let A(G e ) be the skew adjacency matrix of G e .  ...

### Skew Spectra of Oriented Graphs

$G$ is called the underlying graph of $G^{\sigma}$, and we denote by $Sp(G)$ the adjacency spectrum of $G$.  ...  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.  ...  Let Y be obtained from X by changing the (1,1) entry to −1. Consider the orientation σ of G such that S(G σ ) = 0 Y −Y T 0 .  ...
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