Some inequalities on the skew-spectral radii of oriented graphs
© Xu; licensee Springer 2012
Received: 11 February 2012
Accepted: 18 September 2012
Published: 2 October 2012
Let G be a simple graph and be an oriented graph obtained from G by assigning a direction to each edge of G. The adjacency matrix of G is and the skew-adjacency matrix of is . The adjacency spectral radius of G and the skew-spectral radius of are defined as the spectral radius of and respectively.
In this paper, we firstly establish a relation between and . Also, we give some results on the skew-spectral radii of and its oriented subgraphs. As an application of these results, we obtain a sharp upper bound of the skew-spectral radius of an oriented unicyclic graph.
In this paper, i will always denote an imaginary unit. The identity matrix is denoted by I and the transpose of the matrix A by . Let G be a simple graph with n vertices. The adjacency matrix is the symmetric matrix , where if is an edge of G, otherwise . We call the characteristic polynomial of G, denoted by . The adjacency spectrum of G is defined as the spectrum of . Since A is symmetric, its eigenvalues are real, and we assume that . We call the adjacency spectral radius of G.
Let be a simple graph with an orientation σ, which assigns to each edge of G a direction so that becomes a directed graph. The skew-adjacency matrix is the real skew-symmetric matrix , where and if is an arc of , otherwise . We call the skew-characteristic polynomial of , denoted by . The skew spectrum of is defined as the spectrum of . Since S is skew-symmetric, its eigenvalues are purely imaginary numbers . Also, we assume that and call the skew-spectral radius of .
In this paper, we will denote by the set of all the oriented graphs obtained from G by giving an arbitrary orientation to each edge. Also, we refer to [1–3] for more terminologies and notations not defined here.
Unlike the adjacency matrix of a graph, there is little research on the skew-adjacency matrix , except that into enumeration of perfect matchings of a graph. As early as in 1947, Tutte  derived his famous characterization of the graphs with no perfect matchings. Tutte’s result motivates a lot of work on the matchings polynomial and enumerating perfect matchings of graphs in terms of its skew-adjacency matrix; see, for example, [5–7] and references therein.
Recently, many researchers have paid a great deal of attention to the spectral properties of skew-symmetric matrices in terms of oriented graphs. IMA-ISU research group on minimum rank  studied the minimum rank of skew-symmetric matrices. In 2009, Shader and So  investigated the spectra of the skew-adjacency matrix of an oriented graph. And in 2010, Adiga et al.  discussed the properties of the skew energy of an oriented graph. In the papers  and , all the coefficients of the skew-characteristic polynomial of in terms of G were interpreted.
The motivation of this paper is to study more carefully the skew spectra of oriented graphs. In Section 2, we firstly establish a relation between and . In Section 3, we give some results on the skew-spectral radii of and its oriented subgraphs. Finally, in Section 4, we give an application of the previous results - to obtain a sharp upper bound of the skew-spectral radius of an oriented unicyclic graph.
2 A relation between and
It is interesting to discuss the relations between and . Let G be a simple graph. We denoted by the neighborhood of the vertex v in G, by the subgraph obtained from G by deleting the edge e and by the subgraph obtained from G by removing the vertex v together with all edges incident to it. A walk W of length k from u to v in G is a sequence of vertices starting at u and ending at v such that consecutive vertices are adjacent. If all vertices in a walk are distinct, then such a walk is called a path of G, denoted by P. Let be a path with . Then P together with the edge is called a cycle of G, denoted by C.
Obviously, for an even closed walk (that is to say, ), we can simply refer to it as a positive (or negative) even closed walk according to its sign, regardless of the order of its vertices. Similarly, we can define a positive (or negative) even cycle. Moreover, an even cycle with 2k vertices is said to be oriented uniformly if its sign in is .
Note that in terms of defining walks, paths, cycles etc., we focus only on the underlying undirected graph. And their signs are based on the oriented graph.
For , one reverses the orientations of all the arcs incident to a particular vertex of . We call such an operation a reversal of . Let be the digraph obtained from by doing some reversals. Then and are said to be quasi-isomorphic and denoted by . For example, if T is a tree and , then we always have .
On quasi-isomorphic oriented graphs, Adiga et al.  obtained the following result.
Lemma 2.1 ()
For oriented bipartite graphs, we have
Now, to prove the lemma, we can consider the following two cases.
Case 1. ;
Let , . Obviously, we can take some reversals to obtain a new oriented graph such that all the arcs (, ). Now, if , then there exists an arc , where and . Thus, for the cycle , we have . Contradicting the condition of this lemma that each even cycle of G is oriented uniformly in , , and then .
Case 2. ;
Hence, the closed -walk of satisfies in . Thus, each even cycle of is oriented uniformly in . It tells us that the graph satisfies the conditions of this lemma too. Furthermore, if the result holds for , we can get immediately the same result for the graph G. By taking some similar operations, we can obtain the graph . And then, by the Case 1 and the above discussion, we know the result holds. □
Moreover, the following result tells us that for a bipartite graph G and its oriented graph D defined as in Lemma 2.2.
Lemma 2.3 ()
be two real matrices. and are denoted by the spectrum of A and B respectively. Then .
To obtain the main result of this section, we also need the following result.
Lemma 2.4 ()
Let A be an irreducible nonnegative matrix and B be a complex matrix such that (entry-wise). Then for each eigenvalue of B; and iff , where and , the identity matrix.
Now, by the above lemmas, we can give the relation between and as follows.
Theorem 2.1 Let G be a connected graph, . Then with equality if and only if G is a bipartite graph and each even cycle of G is oriented uniformly in .
Proof By Lemma 2.4, we know that .
Since or −1, r must be an even number. Hence, G is a bipartite graph, and each even cycle of G is oriented uniformly in .
Thus, by Lemma 2.1 and Lemma 2.3, we have . □
Remark 2.1 By this theorem and those known results on the adjacency spectral radii of bipartite graphs, we can obtain similar results on the skew-spectral radii of oriented bipartite graphs.
For example, we can give the following sharp upper bound on the skew-spectral radii of oriented bipartite graphs.
where is the matrix with all entries equal to 1.
Proof It is well known that with equality if and only if . Thus, by Lemma 2.2 and Theorem 2.1, we know the result holds. □
3 Some relations on the skew-spectral radii of an oriented graph and its subgraphs
In this section, we will discuss the relations on the skew-spectral radii of an oriented graph and its subgraphs due to vertex (or edge) deletion. Certainly, in this paper, each subgraph of an oriented graph is also referred to as an oriented graph and preserves the direction of each arc, even if we do not indicate specially.
Let G be a simple graph and . A subgraph H of G is called a basic subgraph if each component of H is an edge or an even cycle. Of course, is an even number.
where the summation is over all the basic subgraphs of G having 2r vertices (), and c are respectively the number of positive even cycles and the number of even cycles contained in .
As an application of this lemma, we can obtain the following results (the result (b) of the following theorem was also obtained by Hou and Lei ) which can be used to find recursions for the skew-characteristic polynomial of some oriented graphs.
where and the summation is over all the even cycles of G containing the edge e.
the summation is over all the even cycles of G containing the vertex u. And , where the summation is over all the basic subgraphs of having vertices.
When j is an odd number, we write ().
Thus, the result holds.
(b) The proof is similar to that of (a). □
Now, we consider the relations on the skew-spectral radii of an oriented graph and its subgraphs due to vertex (or edge) deletion. Firstly, according to a classical result in the matrix theory, we have
Lemma 3.2 (Interlacing of eigenvalues)
Proof Let be the skew-adjacency matrix of . Since is a Hermitian matrix, by the well-known Cauchy-Poincare theorem, we know the result holds. □
Now, we can get the following strict inequalities on the skew-spectral radii of an oriented graph and its subgraphs.
, where u is an arbitrary vertex of G.
, where e is an arbitrary edge of G. Moreover, for , we have .
Proof (a) We prove the statement by induction on n. Obviously, the result holds for .
Let and .
Case 1. ;
Case 2. ;
where . Thus the result holds.
Furthermore, we know . □
Moreover, by this theorem, we have the following
for . In particular, we have .
Remark 3.1 Note that the above results will help us to compare the skew-spectral radii of two oriented graphs and then to clarify the skew-spectral properties of some oriented graphic classes. In the next section, we will just give such an example.
4 An application of the previous results
A so-called unicyclic graph is a connected simple graph in which the number of edges equals the number of vertices. In this section, we will give an application of the previous results - to obtain a sharp upper bound of the skew-spectral radius of an oriented unicyclic graph.
Also, we denoted by the unicyclic graph obtained from by joining a vertex of with .
Regarding the adjacency spectral radii of unicyclic graphs, Hong  obtained the following results.
Lemma 4.1 ()
with equality if and only if .
Lemma 4.2 ()
Now, we begin to discuss the skew-spectral radii of unicyclic graphs. Firstly, we consider the bipartite unicyclic graphs. For this case, we have
The equality holds if and only if and the sign of the cycle in is positive.
Proof By Lemma 2.1, we know and the equality holds if and only if the sign of the 2k-cycle in is . Moreover, by Lemmas 4.1 and 4.2, and the equality holds if and only if .
Thus, with equality if and only if and the sign of the cycle in is . That is to say the sign of the cycle in is positive.
This completes the proof. □
Now, we begin to consider the non-bipartite unicyclic graphs. Let and . We can easily know that and . Moreover, we have
where . The above two equalities hold if and only if .
Proof We prove the statement by induction on . Obviously, the result holds for .
Hence, the result follows. □
Furthermore, we have
In particular, .
Proof We prove the result by induction on n. Obviously, the result holds for .
Thus, similar to the proof of Lemma 4.4, we know the result holds. □
Hence, for the non-bipartite unicyclic graphs, we can give
The equality holds if and only if .
Proof By Lemmas 4.4 and 4.5, we know , and the equality holds if and only if .
Thus the result holds. □
Now, it is sufficient to compare the skew-spectral radius of and the adjacency spectral radius of .
Lemma 4.7 If , then .
It is easy to know that when . □
Finally, by the above results, we can get the main result of this section.
The equality holds if and only if .
Proof By Lemmas 4.3, 4.6 and 4.7, we know the result holds. □
Remark 4.1 For the adjacency spectral radius of a unicyclic graph, it is well known that for . It is interesting that for the skew-spectral radius of an oriented unicyclic graph, we also have () for . But for , the inequality does not hold. In fact, by making a simple comparison, we know that and .
This work was supported by the National Natural Science Foundation of China (No. 11171373).
- Cvetković D, Doob M, Sachs H: Spectra of Graphs. Academic Press, New York; 1980.MATHGoogle Scholar
- Godsil CD: Algebraic Combinatorics. Chapmam & Hall, London; 1993.MATHGoogle Scholar
- Horn R, Johnson C: Matrix Analysis. Cambridge University Press, Cambridge; 1989.Google Scholar
- Tutte WT: The factorization of linear graphs. J. Lond. Math. Soc. 1947, 22: 107–111. 10.1112/jlms/s1-22.2.107MathSciNetView ArticleMATHGoogle Scholar
- Lovász L, Plummer M Ann. of Discrete Math. 29. In Matching Theory. North-Holland, New York; 1988.Google Scholar
- Robertson N, Seymour PD, Thomas R: Permanents, Pfaffian orientations and even directed circuits. Ann. Math. 1999, 150: 929–975. 10.2307/121059MathSciNetView ArticleMATHGoogle Scholar
- Yan WG, Zhang FJ: Enumeration of perfect matchings of a type of Cartesian products of graphs. Discrete Appl. Math. 2006, 154: 145–157. 10.1016/j.dam.2005.07.001MathSciNetView ArticleMATHGoogle Scholar
- IMA-ISU research group on minimum rank: Minimum rank of skew-symmetric matrices described by a graph. Linear Algebra Appl. 2010, 432: 2457–2472. 10.1016/j.laa.2009.10.001MathSciNetView ArticleGoogle Scholar
- Shader B, So WS: Skew spectra of oriented graphs. Electron. J. Comb. 2009., 16: Article ID N32Google Scholar
- Adiga C, Balakrishnan R, So WS: The skew energy of a graph. Linear Algebra Appl. 2010, 432: 1825–1835. 10.1016/j.laa.2009.11.034MathSciNetView ArticleMATHGoogle Scholar
- Gong SC, Xu GH: The characteristic polynomial and the matchings polynomial of a weighted oriented graph. Linear Algebra Appl. 2012, 436: 3597–3607. 10.1016/j.laa.2011.12.033MathSciNetView ArticleMATHGoogle Scholar
- Hou YP, Lei TG: Characteristic polynomials of skew-adjacency matrices of oriented graphs. Electron. J. Comb. 2011., 18: Article ID P156Google Scholar
- Hong Y: Bounds on the spectra of unicyclic graphs. J. East China Norm. Univ. 1986, 1: 31–34. in ChineseMathSciNetMATHGoogle Scholar
This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.