Lower bounds of the skew spectral radii and skew energy of oriented graphs
aa r X i v : . [ m a t h . C O ] J un Lower bounds of the skew spectral radii andskew energy of oriented graphs ∗ Xiaolin Chen, Xueliang Li, Huishu LianCenter for Combinatorics and LPMC-TJKLCNankai University, Tianjin 300071, P.R. ChinaE-mail: [email protected]; [email protected]; [email protected]
Abstract
Let G be a graph with maximum degree ∆, and let G σ be an oriented graph of G with skew adjacency matrix S ( G σ ). The skew spectral radius ρ s ( G σ ) of G σ isdefined as the spectral radius of S ( G σ ). The skew spectral radius has been stud-ied, but only few results about its lower bound are known. This paper determinessome lower bounds of the skew spectral radius, and then studies the oriented graphswhose skew spectral radii attain the lower bound √ ∆. Moreover, we apply the skewspectral radius to the skew energy of oriented graphs, which is defined as the sumof the norms of all the eigenvalues of S ( G σ ), and denoted by E s ( G σ ). As results,we obtain some lower bounds of the skew energy, which improve the known lowerbound obtained by Adiga et al. Keywords: oriented graph, skew adjacency matrix, skew spectral radius, skew en-ergy
AMS Subject Classification 2010:
The spectral radius of a graph is one of the fundamental subjects in spectral graphtheory, which stems from the spectral radius of a matrix. Let M be a square matrix. ∗ Supported by NSFC and PCSIRT. M , denoted by ρ ( M ), is defined as the maximum normof its all eigenvalues. If G is a simple undirected graph with adjacency matrix A ( G ),then the spectral radius of G is defined to be the spectral radius of A ( G ), denoted by ρ ( G ). It is well-known that ρ ( G ) is the largest eigenvalue of A ( G ). The spectral radius ofundirected graphs has been studied extensively and deeply. For the bounds of the spectralradius, many results have been obtained. The spectral radius of a graph is related to thechromatic number, independence number and clique number of the graph; see [4] fordetails. Moreover, the spectral radius of a graph has applications in graph energy [10].Recently, the spectral radii of skew adjacency matrices of oriented graphs have beenstudied in [5, 6, 11, 14, 15]. Let G σ be an oriented graph of G obtained by assigning toeach edge of G a direction such that the induced graph G σ becomes a directed graph.The graph G is called the underlying graph of G σ . The skew adjacency matrix of G σ isthe n × n matrix S ( G σ ) = ( s ij ), where s ij = 1 and s ji = − h v i , v j i is an arc of G σ , s ij = s ji = 0 otherwise. It is easy to see that S ( G σ ) is a skew symmetric matrix. Thusall the eigenvalues of S ( G σ ) are pure imaginary numbers or 0 ′ s , which are said to be theskew spectrum Sp s ( G σ ) of G σ . The skew spectral radius of G σ , denoted by ρ s ( G σ ), isdefined as the spectral radius of S ( G σ ).There are only few results about the skew spectral radii of oriented graphs. Xu andGong [15] and Chen et al. [6] studied the oriented graphs with skew spectral radii no morethan 2. Cavers et al. [5] and Xu [14] independently deduced an upper bound, that is, theskew spectral radius of G σ is dominated by the spectral radius of the underlying graph G . In Section 2, we further investigate the skew spectral radius of G σ , and obtain somelower bounds for ρ s ( G σ ). One of the lower bounds is based on the maximum degree of theunderlying graph G , that is, ρ s ( G σ ) ≥ √ ∆. We, in Section 3, study the oriented graphswhose skew spectral radii attain the lower bound √ ∆. In Section 4, we apply the skewspectral radius to the skew energy of oriented graphs.The skew energy E s ( G σ ) of G σ , first introduced by Adiga et al. [1], is defined as thesum of the norms of all the eigenvalues of S ( G σ ). They obtained that for any orientedgraph G σ with n vertices, m arcs and maximum degree ∆, q m + n ( n −
1) (det( S )) /n ≤ E s ( G σ ) ≤ n √ ∆ , (1.1)where S is the skew adjacency matrix of G σ and det( S ) is the determinant of S .The upper bound that E s ( G σ ) = n √ ∆ is called the optimum skew energy. Theyfurther proved that an oriented graph G σ has the optimum skew energy if and only ifits skew adjacency matrix S ( G σ ) satisfies that S ( G σ ) T S ( G σ ) = ∆ I n , or equivalently, all2igenvalues of G σ are equal to √ ∆, which implies that G is a ∆-regular graph.From the discussion of Section 3, it is interesting to find that if the underlying graph isregular, the oriented graphs with ρ s ( G σ ) = √ ∆ have the optimal skew energy. Moreover,by applying the lower bounds of the skew spectral radius obtained in Section 2, we derivesome lower bounds of the skew energy of oriented graphs, which improve the lower boundin (1.1).Throughout this paper, when we simply mention a graph, it means a simple undirectedgraph. When we say the maximum degree, average degree, neighborhood, etc. of anoriented graph, we mean the same as those in its underlying graph, unless otherwisestated. For terminology and notation not defined here, we refer to the book of Bondy andMurty [3]. In this section, we deduce some lower bounds of the skew spectral radii of orientedgraphs, which implies the relationships between skew spectral radius and some graphparameters.We begin with some definitions. Let G σ be an oriented graph of a graph G with vertexset V . Denote by S ( G σ ) and ρ s ( G σ ) the skew adjacency matrix and the skew spectralradius of G σ , respectively. For any two disjoint subsets A, B of V , we denote by γ ( A, B )the number of arcs whose tails are in A and heads in B . Let G σ [ A ] be the subgraph of G σ induced by A , where G σ [ A ] has vertex set A and contains all arcs of G σ which joinvertices of A . Then we can deduce a lower bound of ρ s ( G σ ) as follows. Theorem 2.1
Let G be a graph with vertex set V , and let G σ be an oriented graph of G with skew adjacency matrix S ( G σ ) and skew spectral radius ρ s ( G σ ) . Then for any twononempty subsets A, B ⊆ V , ρ s ( G σ ) ≥ | γ ( A, B ) − γ ( B, A ) | p | A || B | , where | A | and | B | are the number of elements of A and B , respectively.Proof. Suppose that | A | = k and | B | = l , and let C = A ∩ B with order t , A = A − C withorder k − t , B = B − C with order l − t , and D = V − A ∪ B with order n + t − k − l . Withsuitable labeling of the vertices of V , the skew adjacency matrix S can be formulated as3ollows: S S S S − S T S S S − S T − S T S S − S T − S T − S T S , where S = S ( G σ [ A ]) is the skew adjacency matrix of the induced oriented graph G σ [ A ]with order k − t , S = S ( G σ [ C ]) is the skew adjacency matrix of G σ [ C ] with order t and S = S ( G σ [ B ]) is the skew adjacency matrix of G σ [ B ] with order l − t , S = S ( G σ [ D ])is the skew adjacency matrix of the induced oriented graph G σ [ D ] with order n + t − k − l .Let H = ( − i ) S . Since S is skew symmetric, H is an Hermitian matrix. By Rayleigh-Ritz theorem, we obtain that ρ s ( G σ ) = ρ ( S ) = ρ ( H ) = max x ∈ C n x ∗ Hxx ∗ x , where C n is the complex vector space of dimension n and x ∗ is the conjugate transposeof x . We take x = (cid:16) √ k , . . . , √ k | {z } k − t , √ k + i √ l , . . . , √ k + i √ l | {z } t , i √ l , . . . , i √ l | {z } l − t , , . . . , (cid:17) T = (cid:16) √ k Tk − t , ( √ k + i √ l ) Tt , i √ l Tl − t , T (cid:17) T in the above equality and derive that ρ s ( G σ ) ≥ x ∗ Hxx ∗ x = − i (cid:18) k Tk − t S k − t + (cid:18) k + 1 l (cid:19) Tt S t + 1 l Tl − t S l − t + (cid:18) k + i √ kl (cid:19) Tk − t S t − (cid:18) k − i √ kl (cid:19) Tt S T k − t + i √ kl Tk − t S l − t − − i √ kl Tl − t S T k − t + (cid:18) i √ kl + 1 l (cid:19) Tt S l − t − (cid:18) − i √ kl + 1 l (cid:19) Tl − t S T t (cid:19) . Note that Tk − t S k − t = Tt S t = Tl − t S l − t = 0, since S , S and S are bothskew symmetric. Moreover, it can be verified that Tk − t S t = Tt S T k − t = γ ( A , C ) − γ ( C, A ) , Tk − t S l − t = Tl − t S T k − t = γ ( A , B ) − γ ( B , A ) , and Tt S l − t = Tl − t S T t = γ ( C, B ) − γ ( B , C ) . Note that γ ( A, B ) − γ ( B, A ) = γ ( A , C ) + γ ( A , B ) + γ ( C, B ) − γ ( C, A ) − γ ( B , A ) − ( B , C ).Combining the above equalities, it follows that ρ s ( G σ ) ≥ γ ( A, B ) − γ ( B, A ) √ kl . Similarly, we can derive that ρ s ( G σ ) ≥ γ ( B,A ) − γ ( A,B ) √ kl . The proof is thus complete.The above theorem implies a lower bound of ρ s ( G σ ) in terms of the maximum degreeof G . Before proceeding, it is necessary to introduce the notion of switching-equivalent [9]of two oriented graphs. Let G σ be an oriented graph of G and W be a subset of its vertexset. Denote W = V ( G σ ) \ W . Another oriented graph G τ of G , obtained from G σ byreversing the orientations of all arcs between W and W , is said to be obtained from G σ by switching with respect to W . Two oriented graphs G σ and G τ are called switching-equivalent if G τ can be obtained from G σ by a sequence of switchings. The followinglemma shows that the switching operation keeps skew spectrum unchanged. Lemma 2.2 [9] Let G σ and G τ be two oriented graphs of a graph G . If G σ and G τ areswitching-equivalent, then G σ and G τ have the same skew spectra. Corollary 2.3
Let G σ be an oriented graph of G with maximum degree ∆ . Then ρ s ( G σ ) ≥ √ ∆ . Proof.
Let G τ be an oriented graph of G obtained from G σ by switching with respect toevery neighbor of v if necessary, such that all arcs incident with v have the common tail v . Then G σ and G τ are switching-equivalent. By Lemma 2.2, G σ and G τ have the sameskew spectra. Consider the oriented graph G τ and let A = { v } and B = N ( v ). Obviously, γ ( A, B ) − γ ( B, A ) = ∆. By Theorem 2.1, ρ s ( G σ ) = ρ s ( G τ ) ≥ ∆ √ ∆ = √ ∆.It is known [4] that for any undirected tree T , ρ ( T ) ≥ ¯ d , where ¯ d is the averagedegree of T . Moreover, all oriented trees of T have the same skew spectra which areequal to i times the spectrum of T ; see [11]. Therefore, for any oriented tree T σ of T , ρ s ( T σ ) = ρ ( T ) ≥ ¯ d . We next consider a general graph. The following lemma [2] isnecessary. Lemma 2.4 [2] Let G = ( V, E ) be a graph with n vertices and m edges. Then G containsa bipartite subgraph with at least m/ edges. Then we obtain the following result for a general graph by applying Theorem 2.1.5 orollary 2.5
For any simple graph G with average degree ¯ d , there exists an orientedgraph G σ of G such that ρ s ( G σ ) ≥ ¯ d . Proof.
By Lemma 2.4, G contains a bipartite subgraph H = ( A, B ) with at least m/ G such that all arcs between A and B go from A to B and the directions of the other arcs are arbitrary. By Theorem 2.1 and Lemma 2.4, ρ s ( G σ ) ≥ | γ ( A, B ) − γ ( B, A ) | p | A || B | ≥ n | γ ( B, A ) − γ ( A, B ) | ≥ mn = ¯ d . Hofmeister [7] and Yu et al. [16] deduced a lower bound of the spectral radius of agraph in terms of its degree sequence. Specifically, let G be a connected graph with degreesequence d , d , . . . , d n . Then ρ ( G ) ≥ q n P ni =1 d i . Similarly, for an oriented graph, weconsider the relation between its skew spectral radius and vertex degrees, where it shouldbe taken into account the out-degree and in-degree of every vertex.Let G σ be an oriented graph with vertex set { v , v , . . . , v n } . Denote by d + i and d − i theout-degree and in-degree of v i in G σ , respectively. Let ˜ d i = d + i − d − i . Then we establish alower bound of the skew spectral radius of G σ as follows. Theorem 2.6
Let G σ be an oriented graph with vertex set { v , v , . . . , v n } and skew spec-tral radius ρ s ( G σ ) . Then ρ s ( G σ ) ≥ s ˜ d + ˜ d + · · · + ˜ d n n . Proof. If G σ is an Eulerian digraph, then the right-hand of the above inequality is 0, andthe inequality is obviously true since ρ s ( G σ ) ≥ G σ is not Eulerian in the following. Let S = [ s ij ] be the skew adjacency matrix of G σ .Then ρ s ( G σ ) = ρ ( S ) = p ρ ( S T S ). We consider the spectral radius of S T S . Since S T S issymmetric, by Rayleigh-Ritz theorem, ρ ( S T S ) = max x ∈ R n x T ( S T S ) xx T x . We take x = √ ˜ d + ˜ d + ··· + ˜ d n (cid:16) ˜ d , ˜ d , . . . , ˜ d n (cid:17) T in the above equality and obtain that ρ ( S T S ) ≥ x T ( S T S ) x = ( Sx ) T ( Sx ) .
6t is easy to compute that Sx = 1 q ˜ d + ˜ d + · · · + ˜ d n n X j =1 s j ˜ d j , n X j =1 s j ˜ d j , . . . , n X j =1 s nj ˜ d j ! T . Applying the Cauchy-Schwarz’s inequality, we obtain that( Sx ) T Sx = 1˜ d + ˜ d + · · · + ˜ d n n X j =1 s j ˜ d j ! + n X j =1 s j ˜ d j ! + · · · + n X j =1 s nj ˜ d j ! ≥ n ˜ d + ˜ d + · · · + ˜ d n P nj =1 s j ˜ d j + P nj =1 s j ˜ d j + · · · + P nj =1 s nj ˜ d j n ! . Note that n X j =1 s j ˜ d j + n X j =1 s j ˜ d j + · · · + n X j =1 s nj ˜ d j = (1 , , . . . , Sx = − ˜ d − ˜ d − · · · − ˜ d n . Therefore, ( Sx ) T ( Sx ) ≥ ˜ d + ˜ d + · · · + ˜ d n n . We thus conclude that ρ s ( G σ ) ≥ p ( Sx ) T ( Sx ) ≥ s ˜ d + ˜ d + · · · + ˜ d n n . This completes the proof.
Remark 2.1
Theorem 2.6 can also implies Corollary 2.3.
Another proof of Corollary 2.3 . Suppose that d G ( v ) = ∆ and N ( v ) = { v , v , . . . , v ∆+1 } .Let G τ be an oriented graph of G obtained from G σ by a sequence of switchings such thatall arcs incident with v have the common tail v . Then by Lemma 2.2, ρ s ( G σ ) = ρ s ( G τ ).Let H τ be the subgraph of G τ induced by v and its all adjacent vertices. Let H =( − i ) S ( G τ ) and H = ( − i ) S ( H τ ). Then ρ s ( G τ ) = ρ ( H ) and ρ s ( H τ ) = ρ ( H ). Notethat H and H are both Hermitian matrices. By interlacing of eigenvalues ( [8], 4.3.16Corollary), ρ ( H ) = λ ( H ) ≥ λ ( H ) = ρ ( H ).Suppose that for any 1 ≤ i ≤ ∆, the vertex v i has out-degree t + i and in-degree t − i in H τ . Let ˜ t i = t + i − t − i . It can be found that ˜ t + ˜ t + · · · + ˜ t ∆+1 = 0 and ˜ t = ∆. It follows7hat ˜ t + ˜ t + · · · + ˜ t ≥ ∆ + ∆. Then by Theorem 2.6, ρ s ( H τ ) ≥ s ˜ t + ˜ t + · · · + ˜ t ∆ + 1 ≥ √ ∆ . Now we conclude that ρ s ( G σ ) = ρ s ( G τ ) ≥ ρ s ( H τ ) ≥ √ ∆, which implies Corollary 2.3.It is known that Wilf [13] considered the relation between the spectral radius and thechromatic number of a graph. As for the oriented graphs, Sopena [12] introduced thenotion of oriented chromatic number, which motivates us to consider the relation betweenthe skew spectral radius and the oriented chromatic number of an oriented graph.Let G σ be an oriented graph with vertex set V . An oriented k -coloring of G σ is apartition of V into k color classes such that no two adjacent vertices belong to the samecolor class, and all the arcs between two color classes have the same direction. Theoriented chromatic number of G σ , denoted by χ o ( G σ ), is defined as the smallest number k satisfying that G σ admits an oriented k -coloring. The following theorem presents a lowerbound of the skew spectral radius in terms of the average degree and oriented chromaticnumber. Theorem 2.7
Let G be a graph with average degree ¯ d . Let G σ be an oriented graph of G with skew spectral radius ρ s ( G σ ) and oriented chromatic number χ o ( G σ ) . Then ρ s ( G σ ) ≥ ¯ dχ o ( G σ ) − . Proof.
Suppose that G σ contains m arcs and χ o ( G σ ) = k . Let { V , V , . . . , V k } be anoriented k -coloring of G σ . Denote a ij = | γ ( V i , V j ) − γ ( V j , V i ) | . By the definition of anoriented k -coloring, we get that either γ ( V i , V j ) = 0 or γ ( V j , V i ) = 0. It follows that P i Oriented graphs with skew spectral radius √ ∆ From the previous section, we know that for any oriented graph G σ , ρ s ( G σ ) ≥ √ ∆.In this section, we investigate the oriented graphs with ρ s ( G σ ) = √ ∆.We first recall the following proposition on the skew spectra of oriented graphs. Proposition 3.1 Let { iλ , iλ , . . . , iλ n } be the skew spectrum of G σ , where λ ≥ λ ≥· · · ≥ λ n . Then (1) λ j = − λ n +1 − j for all ≤ j ≤ n ; (2) when n is odd, λ ( n +1) / = 0 andwhen n is even, λ n/ ≥ ; and (3) P nj =1 λ j = 2 m . An oriented regular graph is an oriented graph of a regular graph. We consider thecase of oriented regular graphs, which is associated with optimum skew energy orientedgraphs. Theorem 3.2 Let G σ be an oriented graph of a ∆ -regular graph G with skew adjacencymatrix S . Then ρ s ( G σ ) = √ ∆ if and only if S T S = ∆ I n , i.e., G σ has the optimum skewenergy.Proof. Let { iλ , iλ , . . . , iλ n } be the skew spectrum of G σ with λ ≥ λ ≥ · · · ≥ λ n . ByProposition 3.2, we get that λ + λ + · · · + λ n = 2 m = n ∆ and λ ≥ | λ i | for any 2 ≤ i ≤ n .It follows that λ = ρ s ( G σ ) and λ + λ + · · · + λ n ≤ n ( ρ s ( G σ )) . If ρ s ( G σ ) = √ ∆, then wecan conclude that λ = | λ | = · · · = | λ n | = √ ∆, that is to say, S T S = ∆ I n . The converseimplication follows easily.The above theorem gives a good characterization for an oriented ∆-regular graph with ρ s ( G σ ) = √ ∆, which says that any two rows and any two columns of its skew adjacencymatrix are all orthogonal. For any oriented graph G σ with ρ s ( G σ ) = √ ∆, we also considerthe orthogonality of its skew adjacency matrix and obtain an extended result as follows. Theorem 3.3 Let G be a graph with vertex set V = { v , v , . . . , v n } and maximum degree ∆ . Let G σ be an oriented graph of G with ρ s ( G σ ) = √ ∆ and skew adjacency matrix S = ( S , S , . . . , S n ) . If d G ( v i ) = ∆ , then ( S i , S j ) = S Ti S j = 0 for any j = i .Proof. Without loss of generality, suppose that d G ( v ) = ∆. It is sufficient to considerthe matrix S T S and prove that for any j = 1, ( S T S ) j = 0. Notice from ρ s ( G σ ) = √ ∆that ∆ is the maximum eigenvalue of S T S . Suppose that ∆ is an eigenvalue of S T S with multiplicity l . Since S T S is a real symmetric matrix, there exists an orthogonalmatrix P such that S T S = P T DP , where D is the diagonal matrix with the form D =diag(∆ , . . . , ∆ , u l +1 , . . . , u n ) with 0 ≤ u i < ∆. Denote P = ( P , P , . . . , P n ) = ( p ij ).9ote that ( S T S ) = ∆ since d ( v ) = ∆. It follows that P T DP = ∆, that is,∆ p + · · · + ∆ p l + u l +1 p l +1 , + · · · + u n p n = ∆ . Since P T P = 1, we derive that p l +1 , = p l +2 , = · · · = p n = 0. Then for any j = 1, wecompute the (1 , j )-entry of S T S as follows.( S T S ) j = P T DP j = ∆ p p j + · · · + ∆ p l p j + u l +1 p l +1 , p l +1 ,j + · · · + u n p n p nj = ∆( p p j + · · · + p l p lj ) = ∆ P T P j = 0 . The last equality holds due to the orthogonality of P . The proof is now complete.Comparing Theorem 3.2 with Theorem 3.3, it is natural to ask whether the converseof Theorem 3.3 holds, that is, whether the condition that d G ( v i ) = ∆ and ( S i , S j ) = 0for every vertex v i with maximum degree and j = i implies that ρ s ( G σ ) = √ ∆. In whatfollows, we show that it is not always true by constructing a counterexample, but we canstill obtain that i √ ∆ is an eigenvalue of G σ . Theorem 3.4 Let G be a graph with vertex set V = { v , v , . . . , v n } and maximum degree ∆ . Let G σ be an oriented graph of G with skew adjacency matrix S = ( S , S , . . . , S n ) . Ifthere exists a vertex v i with maximum degree ∆ such that for any j = i , ( S i , S j ) = 0 , then i √ ∆ is an eigenvalue of G σ .Proof. Without loss of generality, suppose that d G ( v ) = ∆. Since ( S , S j ) = 0 for any j = 1, we obtain that SS T = ∆ T ∗ ! . Then ∆ is an eigenvalue of SS T , which followsthat i √ ∆ is an eigenvalue of G σ . The proof is thus complete. Example 3.1 Let G σ be the oriented graph depicted in Figure 3.1, which has the maxi-mum degree 6. It can be verified that G σ satisfies the conditions of Theorem 3.4 and √ i is an eigenvalue of G σ . But we can compute that ρ s ( G σ ) ≈ . > √ G σ Lower bounds of the skew energy of G σ Similar to the McClelland’s lower bound for the energy of undirected graphs, Adigaet al. in [1] got a lower bound for the skew energy of oriented graphs, that is, E s ( G σ ) ≥ q m + n ( n − 1) (det( S )) /n , where S is the skew adjacency matrix of G σ . This bound isalso called the McClelland’s lower bound of skew energy. In this section, we obtain somenew lower bounds for the skew energy of oriented graphs.In view of Proposition 3.1, we reconsider the McClelland’s lower bound and establisha new lower bound of E s ( G σ ). Theorem 4.1 Let G σ be an oriented graph with n vertices, m arcs and skew adjacencymatrix S . Then E s ( G σ ) ≥ q m + n ( n − 2) (det( S )) /n . (4.1) Proof. By Proposition 3.1, we have( E s ( G σ )) = ⌊ n/ ⌋ X j =1 | λ j | = 4 ⌊ n/ ⌋ X j =1 λ j + 4 X ≤ i = j ≤⌊ n/ ⌋ | λ i || λ j | . If n is odd, det( S ) = 0 and ( E s ( G σ )) ≥ P ⌊ n/ ⌋ j =1 λ j = 4 m . If n is even, by the arithmetic-geometric mean inequality, we have that( E s ( G σ )) = 4 ⌊ n/ ⌋ X j =1 λ j + 4 X ≤ i = j ≤⌊ n/ ⌋ | λ i || λ j | ≥ m + n ( n − S )) /n . The proof is thus complete. Remark 4.1 The lower bound in the above theorem is better than the McClelland’sbound. In fact, we find that(det( S )) n = n Y j =1 | λ j | ! n ≤ s P nj =1 λ j n = r mn . (4.2)Therefore, we deduce that4 m + n ( n − 2) (det( S )) n ≥ m + n ( n − 1) (det( S )) n . An oriented graph G σ is said to be singular if det( S ) = 0 and nonsingular otherwise.In what follows, we only consider the oriented graphs with det( S ) = 0. Note that if G σ is11onsingular, then n must be even and det( S ) is positive. We next derive a lower boundof the skew energy for nonsingular oriented graphs in terms of the order n , the maximumdegree ∆ and det( S ). Theorem 4.2 Let G σ be a nonsingular oriented graph with order n , maximum degree ∆ and skew adjacency matrix S . Then E s ( G σ ) ≥ √ ∆ + ( n − (cid:18) det( S )∆ (cid:19) n − (4.3) and equality holds if and only if λ = √ ∆ and λ = · · · = λ n/ .Proof. Using the arithmetic-geometric mean inequality, we obtain that E s ( G σ ) = n X j =1 | λ j | = 2 λ + n − X j =2 | λ j | ≥ λ + ( n − n − Y j =2 | λ j | ! n − = 2 λ + ( n − (cid:18) det( S ) λ (cid:19) n − with equality if and if | λ | = · · · = | λ n − | .Let f ( x ) = 2 x + ( n − (cid:16) det( S ) x (cid:17) n − . Then f ′ ( x ) = 2 − S )) n − x − nn − .It is easy to see that the function f ( x ) is increasing for x ≥ (det( S )) n . By Inequality(4.2), we have (det( S )) n ≤ p m/n ≤ √ ∆.Therefore, E s ( G σ ) ≥ f ( √ ∆) = 2 √ ∆ + ( n − (cid:18) det( S )∆ (cid:19) n − . and equality holds if and only if λ = √ ∆ and λ = · · · = λ n/ . Now the proof is complete.By expanding the right of Inequality (4.3), we obtain a simplified lower bound, but itis a little weaker than the bound (4.3). Corollary 4.3 Let G σ be a nonsingular oriented graph with order n , maximum degree ∆ and skew adjacency matrix S . Then E s ( G σ ) ≥ √ ∆ + n − S )) − ln ∆ . (4.4) Equality holds if and only if G σ is a union of n/ disjoint arcs. roof. Note that e x ≥ x for all x , where the equality holds if and only if x = 0.Combining the above inequality and Theorem 4.2, we obtain that E s ( G σ ) ≥ √ ∆ + ( n − (cid:18) det( S )∆ (cid:19) n − = 2 √ ∆ + ( n − e ln (det( S ) / ∆) n − ≥ √ ∆ + ( n − (cid:18) S ) / ∆) n − (cid:19) = 2 √ ∆ + n − S )) − ln ∆ . The equality holds in (4.4) if and only if all the inequalities in the above considerationmust be equalities, that is, λ = √ ∆, λ = · · · = λ n/ and det( S ) = ∆.It is easy to see that a union of n/ λ = √ ∆, λ = · · · = λ n/ and det( S ) = ∆, then we find that λ = √ ∆ and λ = · · · = λ n/ = 1. If ∆ = 1, then G σ is a union of n/ ≥ 2. Then we get that n ≥ 4. Considering the matrix M = S T S = ( m ij ), where S = ( S , . . . , S n ). By Theorem 3.3, we know that M is either a diagonal matrix withdiagonal entries { ∆ , ∆ , , . . . , } or a matrix of form ∆ T M ! .If M is a diagonal matrix, then the graph G must contain two vertices with maximumdegree ∆ and the other vertices with degree 1. Since ∆ ≥ 2, there must exist a path v i v j v k satisfying d G ( v i ) = 1. Then no matter how to orient the graph G , we always getthat ( S i , S k ) = 0 and m ik = 0, which is a contradiction.If M is a matrix of form ∆ T M ! , then M must be symmetric and have spectrum { ∆ , , . . . , } . Denote by A the vertex set of the connected component of G which containsthe vertex v . Suppose A = { v , v , . . . , v t +1 } . Obviously, t ≥ ∆. By the spectraldecomposition, we deduce that M = I n − + (∆ − pp T , where I n − is a unit matrix oforder n − p is a unit eigenvector of M corresponding to the eigenvalue ∆. Therefore,we get S T S = ∆ T I n − + (∆ − pp T ! . (4.5)Suppose that p = ( p , p , . . . , p n − ) T . Then we claim that p satisfies following propositions:(1) p + · · · + p n − = 1,(2) The degree sequence of G is { ∆ , − p , . . . , − p n − } ,133) ( S i +1 , S j +1 ) = (∆ − p i p j for any two distinct integer 1 ≤ i, j ≤ n − p i = 0 for 1 ≤ i ≤ t .The first proposition is trivial. The second and third propositions follow from directcalculation of Equality (4.5). Now it remains to prove the fourth proposition. If not,without loss of generality, we suppose p = 0. Then from the propositions (2) and (3), wehave d G ( v ) = 1 and ( S , S j ) = 0 for all 3 ≤ j ≤ t + 1, which is a contradiction.From the above propositions, we observe that (∆ − p j ≥ ≤ j ≤ t . Then∆ − − P n − j =1 p j ≥ t ≥ ∆, which is a contradiction.Now we conclude that G σ must be a union of n/ G σ has n vertices, 3 n/ − n/ 2. By calculation, we havedet( S ) = n / · · ·· · · Figure 4.2: The oriented graph G σ Moreover, we find a class of oriented graphs which illustrate the superiority of thebound (4.4). Let Γ be the class of connected oriented graphs of order n ≥ n ≤ ∆ ≤ det( S ) ≤ n , m ≤ n. (4.6)Obviously, the oriented graph in Figure 4.2 belongs to Γ. Theorem 4.4 The bound (4.4) is better than (4.1) for any oriented graph in Γ .Proof. For any oriented graph G σ in Γ, we get thatln (det( S )) − ln ∆ = ln (cid:18) det( S )∆ (cid:19) ≥ . 14o prove the theorem, it is sufficient to prove that2 √ ∆ + n − ≥ q m + n ( n − S )) n . (4.7)Notice that ln(det( S )) ≤ n ≤ n for n ≥ S )) n = e S )) n = 1 + 2 ln(det( S )) n + 2 e t (cid:18) ln(det( S )) n (cid:19) , where the number t satisfies that 0 ≤ t ≤ S )) n ≤ nn ≤ 1. It follows that e t ≤ e .We immediately obtain the following inequality:(det( S )) n ≤ S )) n + 2 e (cid:18) ln(det( S )) n (cid:19) . (4.8)To prove Inequality (4.7), we demonstrate that4 m + n ( n − S )) n ≤ m + n ( n − S )) n + 2 e (cid:18) ln(det( S )) n (cid:19) ! ≤ n + n + 2( n − 2) ln(det( S )) + 2 e (cid:18) n − n (cid:19) (ln(det( S ))) ≤ n + n + 4( n − 2) ln( n ) + 8 e (cid:18) n − n (cid:19) (ln( n )) ≤ n − n + 4 + 2 √ n ( n − . The last inequality follows for n ≥ √ ∆ + n − ≥ n − n + 4 +2 √ n ( n − Remark 4.2 By Theorem 4.1, we know that the bound (4.1) is always superior to theMcClelland’s bound obtained by Adiga et al. By Theorem 4.2 and Corollary 4.3, thebound (4.3) is always superior to the bound (4.4). For some cases, we obtain fromTheorem 4.4 that the bound (4.4) is better than the bound (4.1). References [1] C. Adiga, R. Balakrishnan, W. So, The skew energy of a digraph, Linear AlgebraAppl. The Probabilistic Method , Second Edition, Wiley, New York,2000.[3] J.A. Bondy, U.S.R. Murty, Graph Theory , Springer-Verlag, Berlin, 2008.[4] A.E. Brouwer, W.H. Haemers, Spectra of Graphs , Springer, Berlin, 2012.[5] M. Cavers, S.M. Cioabˇa, S. Fallat, D.A. Gregory, W.H. Haemers, S.J. Kirkland,J.J. McDonald, M. Tsatsomeros, Skew-adjacency matrices of graphs, Linear AlgebraAppl. J. Inequal. Appl. Math. Nachr. Matrix Analysis , Cambridge University Press, 1987.[9] X. Li, H. Lian, A survey on the skew energy of oriented graphs, arXiv: 1304.5707.[10] X. Li, Y. Shi, I. Gutman, Graph Energy , Springer, New York, 2012.[11] B. Shader, W. So, Skew spectra of oriented graphs, Electron. J. Combin. J. Graph Theory J. London Math.Soc. J. Inequal.Appl. LinearAlgebra Appl.