Skew-adjacency Matrix Research Articles

The skew spectral radius and skew Randić spectral radius of general random oriented graphs

Let G be a simple connected graph on n vertices, and let Gσ be an orientation of G with skew adjacency matrix S(Gσ). Let di be the degree of the vertex vi in G. The skew Randić matrix of Gσ is the n×n real skew symmetric matrix RS(Gσ)=[(RS)ij], where (RS)ij=−(RS)ji=(didj)−12 if (vi,vj) is an arc of Gσ, and (RS)ij=(RS)ji=0 otherwise. The skew spectral radius ρS(Gσ) and the skew Randić spectral radius ρRS(Gσ) of Gσ are defined as the spectral radius of S(Gσ) and RS(Gσ) respectively. In this paper we give upper bounds for the skew spectral radius and skew Randić spectral radius of general random oriented graphs.

Relations between the skew spectrum of an oriented graph and the spectrum of an associated signed graph

We say that an oriented graph G′=(G,σ′) and a signed graph G˙=(G,σ˙) are mutually associated if σ˙(ik)σ˙(jk)=siksjk holds for every pair of edges ik and jk, where (sij) is the skew adjacency matrix of G′. We prove that this occurs if and only if the underlying graph G is bipartite. On the basis of this result we prove that, in the bipartite case, the skew spectrum of G′ can be obtained from the spectrum of an associated signed graph G˙, and vice versa. In the non-bipartite case, we prove that the skew spectrum of G′ can be obtained from the spectrum of a signed graph associated with the bipartite double of G′. In this way, we show that the theory of skew spectra of oriented graphs has a strong relationship with the theory of spectra of signed graphs. In particular, we demonstrate how some problems concerning oriented graphs can be considered in the framework of signed graphs.

Skew-adjacency matrices of tournaments with bounded principal minors

Let T be a tournament with n vertices v1,…,vn. The skew-adjacency matrix of T is the n×n zero-diagonal matrix S=[sij] in which sij=−sji=1 if vi dominates vj. It is well-known that the determinant of S is zero or the square of an odd integer. Moreover, the principal minors of S are at most 1 if and only if T is a local order. In this paper, we characterize the class of tournaments for which the principal minors of the skew-adjacency matrix do not exceed 9.

Smith Normal Form and the generalized spectral characterization of oriented graphs

Spectral characterization of graphs is an important topic in spectral graph theory. An oriented graph Gσ is obtained from a simple undirected graph G by assigning to every edge of G a direction so that Gσ becomes a directed graph. The skew-adjacency matrix of an oriented graph Gσ is a real skew-symmetric matrix S(Gσ)=(sij), where sij=−sji=1 if (i,j) is an arc; sij=sji=0 otherwise. Let Gσ and Hτ be two oriented graphs whose skew-adjacency matrices are S(Gσ) and S(Hτ), respectively. We say Gσ is R-cospectral to Hτ if tJ−S(Gσ) and tJ−S(Hτ) have the same spectrum for any t∈R, where J is the all-ones matrix. An oriented graph Gσ is said to be determined by the generalized skew spectrum (DGSS for short), if any oriented graph which is R-cospectral to Gσ is isomorphic to Gσ. Let W(Gσ)=[e,S(Gσ)e,S2(Gσ)e,…,Sn−1(Gσ)e] be the skew-walk-matrix of Gσ, where e is the all-ones vector. A theorem of Qiu, Wang and Wang [9] states that if Gσ is a self-converse oriented graph and 2−⌊n2⌋det⁡W(Gσ) is odd and square-free, then Gσ is DGSS. In this paper, based on the Smith Normal Form of the skew-walk-matrix of Gσ we obtain our main result: Let q be a prime and Gσ be a self-converse oriented graph on n vertices with det⁡W(Gσ)≠0. Assume that rankq(W(Gσ))=n−1 if q is an odd prime, and rankq(W(Gσ))=⌈n2⌉ if q=2. If Q is a regular rational orthogonal matrix satisfying QTS(Gσ)Q=S(Hτ) for some oriented graph Hτ, then the level of Q divides dn(W(Gσ))q, where dn(W(Gσ)) is the n-th invariant factor of W(Gσ). Consequently, it leads to an easier way to prove Qiu, Wang and Wang's theorem above.

Spectral Slater index of tournaments

The Slater index $i(T)$ of a tournament $T$ is the minimum number of arcs that must be reversed to make $T$ transitive. In this paper, we define a parameter $\Lambda(T)$ from the spectrum of the skew-adjacency matrix of $T$, called the spectral Slater index. This parameter is a measure of remoteness between the spectrum of $T$ and that of a transitive tournament. We show that $\Lambda(T)\leq8\, i(T)$ and we characterize the tournaments with maximal spectral Slater index. As an application, an improved lower bound on the Slater index of doubly regular tournaments is given.

On unimodular tournaments

A tournament is unimodular if the determinant of its skew-adjacency matrix is 1. In this paper, we give some properties and constructions of unimodular tournaments. A unimodular tournament T with skew-adjacency matrix S is invertible if S−1 is the skew-adjacency matrix of a tournament. A spectral characterization of invertible tournaments is given. Lastly, we show that every n-tournament can be embedded in a unimodular tournament by adding at most n−⌊log2⁡(n)⌋ vertices.

Oriented graphs determined by their generalized skew spectrum

Spectral characterization of graphs is a well-studied topic in spectral graph theory which has received a lot of attention from researchers. The spectral characterization of oriented graphs, however, is less studied so far. Given a simple undirected graph G with an orientation σ, the oriented graphGσ is a digraph obtained from G by assigning to every edge of G a direction according to σ. An oriented graph Gσ is self-converse if it is isomorphic to its converse (Gσ)T, a graph obtained from Gσ by reversing each directed edge in Gσ. We denote by S(Gσ) the skew-adjacency matrix of Gσ. For two oriented graphs Gσ and Hτ with skew-adjacency matrices S(Gσ) and S(Hτ), respectively, we say Gσ is R-cospectral to Hτ, if for any t∈R, two matrices tJ−S(Gσ) and tJ−S(Hτ) have the same spectrum, where J is the all-one matrix. An oriented graph Gσ is said to be determined by the generalized skew spectrum (DGSS for short) if, any oriented graph R-cospectral to Gσ is isomorphic to Gσ.Oriented graphs that are DGSS must be self-converse. In this paper, we give a simple arithmetic condition for a self-converse oriented graph being DGSS, which provides an analogue of a similar result for ordinary graphs; see [15]. More precisely, let Gσ be a self-converse oriented graph with skew-adjacency matrix S(Gσ), and W(Gσ)=[e,S(Gσ)e,⋯,Sn−1(Gσ)e] (e is the all-one vector) be the skew-walk-matrix. We show that if 2−⌊n2⌋det⁡W(Gσ) is odd and square-free, then Gσ is DGSS. Moreover, we also illustrate that our result is the best possible in certain sense.

Upper Bounds for the Largest Singular Value of Certain Digraph Matrices

In this paper, we consider digraphs with possible loops and the particular case of oriented graphs, i.e. loopless digraphs with at most one oriented edge between every pair of vertices. We provide an upper bound for the largest singular value of the skew Laplacian matrix of an oriented graph, the largest singular value of the skew adjacency matrix of an oriented graph and the largest singular value of the adjacency matrix of a digraph. These bounds are expressed in terms of certain parameters related to vertex degrees. We also consider some bounds for the sums of squares of singular values. As an application, for the skew (Laplacian) adjacency matrix of an oriented graph and the adjacency matrix of a digraph, we derive some upper bounds for the spectral radius and the sums of squares of moduli of eigenvalues.

The number of the skew-eigenvalues of digraphs and their relationship with optimum skew energy

Let Gσ be an oriented graph of simple graph G with the orientation σ and vertex set V(G)={v1,v2,…,vn}. The skew-adjacency matrix of Gσ is the {0,1,−1}-matrix S=S(Gσ)=[sij], such that sij=1 if (vi,vj) is an arc in Gσ, sij=−1 if (vj,vi) is an arc in Gσ and sij=0, otherwise. In this paper, all oriented graphs with two and three skew-eigenvalues are characterized. Also we determine the relationship between the number of skew-eigenvalues and optimum skew energy of oriented graphs.

On Eigenvalues of Hermitian-Adjacency Matrix

The graph of Hermitian-adjacency matrix is a mixed graph consisting adjacency matrix of an undirected graph and skew-adjacency matrix of a digraph. In this paper we discuss eigenvalues of Hermitian-adjacency matrix. Then we use the eigenvalues to determine the possible Hamiltonian cycles of its graph.

Oriented graphs whose skew spectral radius does not exceed 2

The skew spectral radius of an oriented graph is the largest modulus of the eigenvalues of its skew adjacency matrix. We determine all maximal connected oriented graphs whose skew spectral radius does not exceed 2.

Alpha Adjacency: A generalization of adjacency matrices

B. Shader and W. So introduced the idea of the skew adjacency matrix. Their idea was to give an orientation to a simple undirected graph G from which a skew adjacency matrix S(G) is created. The -adjacency matrix extends this idea to an arbitrary field F. To study the underlying undirected graph, the average -characteristic polynomial can be created by averaging the characteristic polynomials over all the possible orientations. In particular, a Harary-Sachs theorem for the average-characteristic polynomial is derived and used to determine a few features of the graph from the average-characteristic polynomial.

Upper bound of skew energy of an oriented graph in terms of its skew rank

Let Gσ be an oriented graph with skew adjacency matrix S(Gσ). The skew energy Es(Gσ) of Gσ is the sum of the norms of all eigenvalues of S(Gσ) and the skew rank rs(Gσ) of Gσ is the rank of S(Gσ). Recently, Tian and Wong gave a lower bound of Es(Gσ) in terms of its skew rank. They proved that rs(Gσ)≤Es(Gσ) and they characterized the oriented graphs which satisfy the equality. In this paper, we aim to establish an upper bound for skew energy of an oriented graph in terms of its skew rank and maximum vertex degree. It is proved thatEs(Gσ)≤rs(Gσ)Δ for an arbitrary oriented graph Gσ with maximum degree Δ, and the upper bound is attained if and only if Gσ is the disjoint union of rs(Gσ)2 copies of (KΔ,Δ)σ together with some isolated vertices, where the orientation σ is switching-equivalent to the elementary orientation of KΔ,Δ which assigns all edges the same direction from a chromatic set to another one.

Skew-rank of an oriented graph and independence number of its underlying graph

An oriented graph $$G^\sigma $$ is a digraph which is obtained by orienting every edge of a simple graph G, where G is called the underlying graph of $$G^\sigma $$ . Let $$S(G^\sigma )$$ denote the skew-adjacency matrix of $$G^\sigma $$ and $$\alpha (G)$$ be the independence number of G. The rank of $$S(G^\sigma )$$ is called the skew-rank of $$G^\sigma $$ , denoted by $$sr(G^\sigma )$$ . Wong et al. studied the relation between the skew-rank of an oriented graph and the rank of its underlying graphs. Huang et al. recently studied the relationship between the skew-rank of an oriented graph and the independence number of its underlying graph, by giving some lower bounds for the sum, difference and quotient etc. They left over some questions for further studying the upper bounds of these parameters. In this paper, we extend this study by showing that $$sr(G^\sigma )+ 2\alpha (G) \le 2n$$ , where n is the order of G, and two classes of oriented graphs are given to show that the upper bound 2n can be achieved. Furthermore, we answer some open questions by obtaining sharp upper bounds for $$sr(G^\sigma ) + \alpha (G)$$ , $$sr(G^\sigma ) - \alpha (G)$$ and $$sr(G^\sigma ) / \alpha (G)$$ .

On the relationship between the skew-rank of an oriented graph and the rank of its underlying graph

An oriented graph Gσ is a digraph without loops and multiple arcs, where G is the underlying graph of Gσ. Let S(Gσ) denote the skew-adjacency matrix of Gσ, and A(G) be the adjacency matrix of G. The rank (resp. skew-rank) of G (resp. Gσ), written as r(G) (resp. sr(Gσ)), refers to the rank of A(G) (resp. S(Gσ)). It is natural and interesting to study the relationship between sr(Gσ) and r(G). Wong, Ma and Tian (European J. Combin. 54 (2016) 76–86) [30] determined the sharp upper bound on sr(Gσ)−r(G). As a continuance of it, in this paper, a sharp lower bound on sr(Gσ)−r(G) is determined; as well a sharp upper bound on sr(Gσ)/r(G) is determined. All the corresponding extremal oriented graphs Gσ are characterized, respectively.

Relation between the skew-rank of an oriented graph and the independence number of its underlying graph

An oriented graph $$G^\sigma $$ is a digraph without loops or multiple arcs whose underlying graph is G. Let $$S\left( G^\sigma \right) $$ be the skew-adjacency matrix of $$G^\sigma $$ and $$\alpha (G)$$ be the independence number of G. The rank of $$S(G^\sigma )$$ is called the skew-rank of $$G^\sigma $$ , denoted by $$sr(G^\sigma )$$ . Wong et al. (Eur J Comb 54:76–86, 2016) studied the relationship between the skew-rank of an oriented graph and the rank of its underlying graph. In this paper, the correlation involving the skew-rank, the independence number, and some other parameters are considered. First we show that $$sr(G^\sigma )+2\alpha (G)\geqslant 2|V_G|-2d(G)$$ , where $$|V_G|$$ is the order of G and d(G) is the dimension of cycle space of G. We also obtain sharp lower bounds for $$sr(G^\sigma )+\alpha (G),\, sr(G^\sigma )-\alpha (G)$$ , $$sr(G^\sigma )/\alpha (G)$$ and characterize all corresponding extremal graphs.

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

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.

Rank Reduction of Oriented Graphs by Vertex and Edge Deletions

In this paper we continue our study of graph modification problems defined by reducing the rank of the adjacency matrix of the given graph, and extend our results from undirected graphs to modifying the rank of skew-adjacency matrix of oriented graphs. An instance of a graph modification problem takes as input a graph G and a positive integer k, and the objective is to either delete k vertices/edges or edit k edges so that the resulting graph belongs to a particular family $$\mathcal{F}$$ of graphs. Given a fixed positive integer r, we define $$\mathcal{F}_r$$ as the family of oriented graphs where for each $$G\in \mathcal{F}_r$$ , the rank of the skew-adjacency matrix of G is at most r. Using the family $$\mathcal{F}_r$$ we do algorithmic study, both in classical and parameterized complexity, of the following graph modification problems: $$r$$ -Rank Vertex Deletion, $$r$$ -Rank Edge Deletion. We first show that both the problems are NP-Complete. Then we show that these problems are fixed parameter tractable (FPT) by designing an algorithm with running time $$2^{\mathcal{O}(k \log r)}n^{\mathcal{O}(1)}$$ for $$r$$ -Rank Vertex Deletion, and an algorithm for $$r$$ -Rank Edge Deletion running in time $$2^{\mathcal{O}(f(r) \sqrt{k} \log k )}n^{\mathcal{O}(1)}$$ . In addition to our FPT results we design polynomial kernels for these problems. Our main structural result, which is the fulcrum of all our algorithmic results, is that for a fixed integer r the size of any “reduced graph” in $$\mathcal{F}_r$$ is upper bounded by $$3^r$$ . This result is of independent interest and generalizes a similar result of Kotlov and Lovasz regarding reduced oriented graphs of rank r.

Optimum skew energy of a tournament

Let G be a simple undirected graph and Gσ the corresponding oriented graph of G with the orientation σ. The skew energy of Gσ, denoted by ε(Gσ), is defined as the sum of the singular values of its skew adjacency matrix S(Gσ). In 2010, Adiga et al. proved ε(Gσ)≤nΔ, where Δ is the maximum degree of G of order n. In this paper, we characterize the skew energy of a tournament and present some properties about an optimum skew energy tournament.

Solution to a problem on skew spectral radii of oriented graphs

Let G be a simple graph, and let Gσ be an oriented graph of G with skew adjacency matrix S(Gσ). The skew spectral radius ρs(Gσ) of Gσ is defined as the spectral radius of S(Gσ). When G is an odd-cycle graph (no even cycle), Cavers et al. (2012) [4] showed that the skew spectral radius of Gσ is the same for every orientation σ of G. They proposed a problem: If G is a connected graph and ρs(Gσ) is the same for all orientations σ of G, must G be an odd-cycle graph? In this paper, we solve this problem and give a positive answer.