Skew-adjacency Matrix Research Articles

Skew Spectra of Oriented Graphs

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 ${\bf i}=\sqrt{-1}$.

Pfaffian Orientation and Enumeration of Perfect Matchings for some Cartesian Products of Graphs

The importance of Pfaffian orientations stems from the fact that if a graph $G$ is Pfaffian, then the number of perfect matchings of $G$ (as well as other related problems) can be computed in polynomial time. Although there are many equivalent conditions for the existence of a Pfaffian orientation of a graph, this property is not well-characterized. The problem is that no polynomial algorithm is known for checking whether or not a given orientation of a graph is Pfaffian. Similarly, we do not know whether this property of an undirected graph that it has a Pfaffian orientation is in NP. It is well known that the enumeration problem of perfect matchings for general graphs is NP-hard. L. Lovász pointed out that it makes sense not only to seek good upper and lower bounds of the number of perfect matchings for general graphs, but also to seek special classes for which the problem can be solved exactly. For a simple graph $G$ and a cycle $C_n$ with $n$ vertices (or a path $P_n$ with $n$ vertices), we define $C_n$ (or $P_n)\times G$ as the Cartesian product of graphs $C_n$ (or $P_n$) and $G$. In the present paper, we construct Pfaffian orientations of graphs $C_4\times G$, $P_4\times G$ and $P_3\times G$, where $G$ is a non bipartite graph with a unique cycle, and obtain the explicit formulas in terms of eigenvalues of the skew adjacency matrix of $\overrightarrow{G}$ to enumerate their perfect matchings by Pfaffian approach, where $\overrightarrow{G}$ is an arbitrary orientation of $G$.

On the matching polynomials of graphs with small number of cycles of even length

AbstractSuppose that G is a simple graph. We prove that if G contains a small number of cycles of even length then the matching polynomial of G can be expressed in terms of the characteristic polynomials of the skew adjacency matrix A(Ge) of an arbitrary orientation Ge of G and the minors of A(Ge). In addition to a formula previously discovered by Godsil and Gutman, we obtain a different formula for the matching polynomial of a general graph. © 2005 Wiley Periodicals, Inc. Int J Quantum Chem, 2005

On the Permanental Polynomials of Some Graphs

Let G be a simple graph with adjacency matrix A(G) and π(G,x) the permanental polynomial of G. Let G × H denotes the Cartesian product of graphs G and H. Inspired by Klein’s idea to compute the permanent of some matrices (Mol. Phy. 31 (3) (1976) 811–823), in this paper in terms of some orientation of graphs we study the permanental polynomial of a type of graphs. Here are some of our main results. 1.If G is a bipartite graph containing no subgraph which is an even subdivision of K 2,3, then G has an orientation G e such that π(G,x) = det (xI-A(G e )), where A(G e ) denotes the skew adjacency matrix of G e. 2.Let G be a 2-connected outerplanar bipartite graph with n vertices. Then there exists a 2-connected outerplanar bipartite graph $$\overline G$$ with 2n+2 vertices such that π(G,x) is a factor of $$\pi (\overline G ,x)$$ . 3.Let T be an arbitrary tree with n vertices. Then $$\pi (T \times K_2 ,x)\prod _{i = 1}^n (x^2 + 1 + \alpha _i^2 )$$ , where α 1 , α 2 , ..., α n are the eigenvalues of T.