On the skew characteristics polynomial/eigenvalues of operations on bipartite oriented graphs and applications

. Let −→ G be an oriented graph with n vertices and m arcs having underlying graph G . The skew matrix of −→ G , denoted by S ( −→ G ) is a ( − 1 , 0 , 1)-skew symmetric matrix. The skew eigenvalues of −→ G are the eigenvalues of S ( −→ G ) and its characteristic polynomial is the skew characteristic polynomial of −→ G . The sum of the absolute values of the skew eigenvalues is the skew energy of −→ G and is denoted by E S ( −→ G ). In this paper, we study the skew characteristic polynomial and skew eigenvalues of joined union of oriented bipartite graphs and some of its variations. We show that the skew eigenvalues of the joined union of oriented bipartite graphs and some variations of oriented bipartite graphs is the union of the skew eigenvalues of the component oriented graphs except some eigenvalues, which are given by an auxiliary matrix associated with the joined union. As a special case we obtain the skew eigenvalues of join of two oriented bipartite graphs and the lexicographic product of an oriented graph and an oriented bipartite graph. Some examples of orientations of well-known graphs are presented to highlight the importance of the results. As applications to our result we obtain some new infinite families of skew equienergetic oriented graphs. Our results extend and generalize some of the results obtained in [C. Adiga

In this paper, we study the skew characteristic polynomial and skew eigenvalues of joined union of oriented bipartite graphs and some of its variations.We show that the skew eigenvalues of the joined union of oriented bipartite graphs and some variations of oriented bipartite graphs is the union of the skew eigenvalues of the component oriented graphs except some eigenvalues, which are given by an auxiliary matrix associated with the joined union.As a special case we obtain the skew eigenvalues of join of two oriented bipartite graphs and the lexicographic product of an oriented graph and an oriented bipartite graph.Some examples of orientations of well-known graphs are presented to highlight the importance of the results.As applications to our result we obtain some new infinite families of skew equienergetic oriented graphs.Our results extend and generalize some of the results obtained in [C.Adiga   (v i ) be the set of in-neighbours and (v i ) be the set of neighbours of the vertex v i in − → G .The adjacency matrix A(G) = (a ij ) of a graph G is a n-square matrix with a ij = 1, if there is an edge between the vertices v i and v j and a ij = 0, otherwise.All the eigenvalues of A(G) are real numbers as it is a real symmetric matrix.The eigenvalues of the matrix A(G) are called eigenvalues (or adjacency eigenvalues) of G and are denoted by λ 1 , λ 2 , . . ., λ n .The sum of the absolute values of the eigenvalues of G is called energy of G and is denoted by E(G).That is, This spectral graph invariant is one among the most studied spectral graph invariants in spectral graph theory because of its applications in mathematical and other sciences.For some recent works on energy of graphs, we refer to [3] and the book [16].
The skew adjacency matrix S = S( − → G ) = (s ij ) of an oriented graph − → G is an n × n matrix with s ij = 1 when there is an arc from v i to v j , and s ij = −1 when there is an arc from v j to v i , and s ij = 0 otherwise.It is clear that the matrix S( − → G ) is a skew symmetric matrix, so all its eigenvalues are zero or purely imaginary.The  The skew energy of the oriented graph − → G is called the energy of the matrix S( − → G ).It is defined by the following equation.
where ξ 1 , ξ 2 , . . ., ξ n are the skew eigenvalues of − → G .This type of spectral invariant appears in the literature with numerous results regarding their bounds and it has abundant connections with the different graph parameters like matching number, vertex covering number and independence number, its connections with the skew rank (the rank of the matrix S( − → G ) is called skew rank of − → G ).One of the most studied problems in the theory of skew energy is the determination of extremal oriented graphs for E s ( − → G ) in a given class of oriented graphs.In fact, due to the hardness of this problem, many researchers have started with a graph G and tried to find the orientations of G which attain the extremal value for E s ( − → G ).This problem is the topic of many papers in literature.Some recent examples can be found in [8,23].We refer to [5,10,11,14,15,[19][20][21]24] for more development of skew energy theory.Given a cycle C k = u 1 u 2 . . .u k u 1 , its sign is signified as sgn(C k ) = s 12 s 23 . . .s k−1k s k1 .Here, s ij means the entry of the skew matrix S( − → G ) in the intersection of u i row and u j column.If the sign of an even oriented cycle C k is positive or negative, it is referred to as evenly-oriented or oddly-oriented, respectively.We say − → G is evenly-oriented if every even cycle in − → G is evenlyoriented.When sgn(C 2k ) = (−1) k , the even oriented cycle C 2k becomes uniformly oriented.
The rest of the papers is organized as follows.In Section 2, we study the joined union of oriented bipartite graphs and some of its variations.We obtain the skew spectrum of joined union of oriented bipartite graphs and its some of its variations, in terms of the component oriented graphs and an auxiliary matrix determined by the operation.In Section 3, we use the results obtained in Section 2 to obtain the skew spectrum of various families of oriented graphs.As applications to results obtained in Section 2 and 3, we construct various new families of skew equienergetic oriented graphs in Section 4.

The skew spectrum of joined union of oriented graphs
Consider an n × n complex matrix where X ij is an n i × n j block matrix for 1 ≤ i, j ≤ s and n = s i=1 n i .The element b ij is the average row sum of X ij .We define an s × s matrix with elements being the average row sums of X ij and we call it the quotient matrix B = (b ij ).The matrix B becomes a equitable quotient matrix when each block X ij has constant row sum.A complex matrix has a connection with equitable quotient matrix in terms of its spectrum as below [26].
Lemma 2.1 The equitable quotient matrix B and the matrix M defined in (2.1) share the same eigenvalues.
The generalized join (also called joined union) of graphs has different versions of definition.The spectrum of generalized join of graphs in terms of different matrices has been investigated in [9,18,24].The joined union was extended to digraphs in [9].In [9], the author have discussed the A α -spectrum of the joined union of diagonalizable digraphs and as applications the A α -spectrum of various families of digraphs are found.Recently, in [12], the authors defined generalized join of oriented graphs as follows: Let − → G (V, E) be an oriented graph of order n and let − → G i (V i , E i ) be oriented graphs of order n i , where i = 1, . . ., n.The joined union of the oriented graphs with vertex set W = n i=1 V i and arc set In other words, the joined union is the union of oriented graphs is the oriented graph corresponding to the complete graph of order 2. By taking each of the component in joined union as bipartite oriented graphs, we can define the following variations of the joined union of the oriented graphs.
Let − → G (V, E) be an oriented graph of order n and let ), be a bipartite oriented graph with partite sets V i and U i , for all i = 1, 2, . . ., n.Let − → H 1 be the joined union of the oriented graphs Note that if there is an arc between the vertices v i and v j in − → G , then there are arcs between all the vertices of V i and V j ; between all the vertices of V i and U j ; between all the vertices of U i and V j and between all the vertices of U i and U j .Let − → H 2 be the oriented graph obtained from − → H 1 by deleting all the arcs between U i and V j and all the arcs between U i and U j .Let − → H 3 be the oriented graph obtained from − → H 1 by deleting all the arcs between V i and V j and all the arcs between V i and U j .
A digraph D is said to Eulerian if the out-degree of any vertex in D is same as its in-degree, that is, d + i = d − i , for all v i ∈ V (D).The following theorem was obtained in [12]  Then the skew characteristic polynomial of the oriented graph where ϕ(M, x) is the characteristic polynomial of the matrix , where ψ ij = n j , if there is an arc from v i to v j ; ψ ij = −n j , if there is an arc from v j to v i and ψ ij = 0, if there is no arc between v i and v j .
It is clear that Theorem 2.2 is applicable to Eulerian oriented graphs only.However, in the next theorem we will show that for the bipartite oriented digraphs, the condition of being Eulerian can be relaxed.
For i = 1, 2, . . ., n, let ), be a bipartite oriented graph with partite sets V i and U i of same cardinality n i , having the skew adjacency matrix S( , where X i is a (0, 1)-matrix satisfying X i e n i = r i e n i and e is the all one column vector.
In the next theorem we determine the skew characteristic polynomial of the joined union of oriented bipartite graphs , where X i is a (0, 1)-matrix satisfying X i e n i = r i e n i .Let , where i = 1, 2, . . ., n be the skew characteristic polynomial of − → B i .Then the skew characteristic polynomial of the oriented graph where ϕ(M, x) is the characteristic polynomial of the matrix , there is an arc from v j to v i and ϕ ij = 0 0 0 0 , if there is no arc between v i to v j .
, where, and ).Note that J n i ×n j is the all one matrix of order n i × n j and 0 n i ×n j is the zero matrix of order n i × n j .
By assumption ) is a bipartite oriented graph for all i with partite sets V i and U i of same cardinality n i and skew matrix, S( , where X i is a (0, 1)-matrix ) is a skew symmetric matrix, so it is a diagonalisable matrix with its 2n i eigenvectors forming an orthogonal set.Let λ ik be an eigenvalue of S( t ik = 0. Now, consider the vector Y = (y 1 , y 2 , . . ., y N ) T , where

As e T
2n i X = 0 gives that Γ ij X = 0 and coordinates of the vector Y corresponding to vertices of − → H 1 which are not in − → B i are zeros, we have This shows that Y is an eigenvector of S( − → H 1 ) corresponding to the eigenvalue λ ik and so every eigenvalue λ ik (other than ±ιr . So, using this process we will obtain , we use the equitable quotient matrix.The equitable quotient matrix of , Since by Lemma 2.1, the eigenvalues of , the result follows.
The lexicographic product G H of graphs G and H is the graph with vertex set V (G)×V (H) and edge (a, x)(b, y If in particular the oriented bipartite graphs then we obtain the following Theorem, which gives the skew spectrum of the joined union , where X 1 is a (0, 1)-matrix satisfying where ϕ(M, x) is the characteristic polynomial of the matrix , there is an arc from v j to v i and ϕ ij = 0 2 , if there is no arc between v i to v j , where J 2 is the all one matrix of order 2 × 2 and 0 2 is the zero matrix of order 2 × 2. Proof.
] are given by the matrix , there is an arc from v j to v i and ϕ ij = 0 0 0 0 , if there is no arc between v i to v j .With this the result now follows.
In the next theorem we determine the skew characteristic polynomial of the oriented graph − → H 2 , when the component oriented graphs are bipartite − → B i (V i , U i ) with partite sets of same cardinality n i , having the skew adjacency matrix S( , where X i is a (0, 1)-matrix satisfying , where X i is a (0, 1)-matrix satisfying X i e n i = r i e n i .Let , where i = 1, 2, . . ., n be the skew characteristic polynomial of − → B i .Then the skew characteristic polynomial of the oriented graph where ϕ(M, x) is the characteristic polynomial of the matrix , there is an arc from v j to v i and ϕ ij = 0 0 0 0 , if there is no arc between v i to v j .
Proof.The proof follows on similar lines as in Theorem 2.3 and is therefore omitted.
In the next theorem we determine the skew characteristic polynomial of the oriented graph − → H 3 , when the component oriented graphs are bipartite − → B i (V i , U i ) with partite sets of same cardinality n i , having the skew adjacency matrix S( , where X i is a (0, 1)-matrix satisfying X i e n i = r i e n i .
Theorem 2.6 Let − → G be an oriented graph of order n ≥ 2 having m arcs.
, where X i is a (0, 1)-matrix satisfying X i e n i = r i e n i .Let , where i = 1, 2, . . ., n be the skew characteristic polynomial of − → B i .Then the skew characteristic polynomial of the oriented graph where ϕ(M, x) is the characteristic polynomial of the matrix , there is an arc from v j to v i and ϕ ij = 0 0 0 0 , if there is no arc between v i to v j .
Proof.The proof follows on similar lines as in Theorem 2.3 and is therefore omitted.
2 ) be two oriented bipartite graphs of order 2n 1 and 2n 2 , respectively.Let The following consequence of Theorem 2.3, gives the skew spectrum of the join of two oriented bipartite graphs.We note that Theorem 2.7 is Theorem 5 obtained in [2].
, where X i is a (0, 1)-matrix , where i = 1, 2 be the skew characteristic polynomial of − → B i .Then the skew characteristic polynomial of the oriented graph . (2.7) Proof.The proof follows from Theorem 2.3 by taking − → G = − → K 2 and using the fact that the characteristic polynomial of the matrix ) be three oriented bipartite graphs of order 2n 1 , 2n 2 and 2n 3 , respectively.Let is the orientation of the star graph K 1,2 with arcs directed from vertex of degree 2. The following consequence of Theorem 2.3, gives the skew spectrum of the oriented graph , where X i is a (0, 1)- , where i = 1, 2, 3 be the skew characteristic polynomial of − → B i .Then the skew characteristic polynomial of the oriented graph , where ϕ(M, x) is the characteristic polynomial of matrix M given by Proof.The proof follows from Theorem 2.3 by taking is the orientation of star graph K 1,2 with arcs directed from vertex of degree 2.
, is defined in [2] as the oriented graph obtained from − → B 1 and − → B 2 by joining arcs from all the vertices of U 1 to each the vertex of U 2 and V 2 .The next Theorem was obtained as part first of Theorem 8 in [2] and gives the skew characteristic polynomial of join-1 , where X i is a (0, 1)-matrix , where i = 1, 2 be the skew characteristic polynomial of − → B i .Then the skew characteristic polynomial of join-1 .
Proof.The proof follows from Theorem 2.5 by taking − → G = − → K 2 and using the fact that the characteristic polynomial of matrix M given by This shows that Theorem 2.9 is a generalization of the part first of Theorem 8 in [2].In fact, the operation defined to obtain the oriented graph − → H 2 is actually the generalization of the join-1 operation defined in [2].
2 ) be oriented bipartite graphs with partite sets U 1 , V 1 , U 2 and V 2 , respectively.We define the join-1 as the oriented graph obtained from − → B 1 and − → B 2 by joining arcs from all the vertices of V 1 to each vertex of U 2 and V 2 .In the next Theorem we obtain the skew characteristic polynomial of join-1 , where X i is a (0, 1)-matrix .
Proof.The proof follows from Theorem 2.6 by taking − → G = − → K 2 and using the fact that the characteristic polynomial of matrix M is given by 3 Skew spectrum of some oriented graphs As applications to the resulted obtained in Section 2, we obtain the skew spectrum of some special classes of oriented graphs.Let K n be a complete graph on n vertices.Any orientation of K n is said to be a tournament.Consider the complete t-partite graph K 2n 1 ,2n 2 ,...,2nt , it is easy to verify that K 2n 1 ,2n where ϕ(M, x) is the characteristic polynomial of the matrix with ϕ ij = n j J 2 or −n j J 2 , according to as there is an arc from v i to v j or from v j to v i in − → K n and ϕ i = 0 2 , for all i.

Proof. Taking
K n , a tournament on n vertices and − → G i = K 2n i , an empty graph, for all i = 1, 2, . . ., n in Theorem 2.3 and using the fact that the skew characteristic polynomial of K 2n i is P s (K 2n i , x) = x 2n i , for all i, the result follows.Note that the empty graph K 2n i can be considered as the bipartite graph with partite sets V i and U i of same cardinality n i and the skew adjacency matrix S(K 2n i ) of K 2n i given by S( , which satisfies If i , an orientation of the complete bipartite graph K n i ,n i with all edges oriented from one partite set to another, for all i = 1, 2, . . ., n, in Theorem 2.3 and using the fact that the skew characteristic polynomial of , we obtain the following consequence of Theorem 2.3. where where ϕ(M, x) is the characteristic polynomial of the matrix with ϕ ij = n j J 2 or −n j J 2 , according to as there is an arc from v i to v j or from Proof.Note that the skew adjacency matrix S( In particular, if Let us orient the edges of B i from V i to U i , then it is clear that the resulting oriented graph − → B i is evenlyoriented.So, using the fact (see Theorem 5.4 in [15]) that the skew spectrum of − → B i is i times the adjacency spectrum of B i .It follows that the skew eigenvalues of − → B i are ιλ i1 , ιλ i2 , . . ., ιλ in , where n i = λ i1 , λ i2 , . . ., −n i = λ in are the eigenvalues (adjacency eigenvalues) of B i .Also, it is clear that the skew adjacency matrix S( with ϕ ij = n j J 2 or −n j J 2 or 0J 2 , according to as there is an arc from v i to v j or from v j to v i or there is no arc between v i and v j in − → G and Consider the complete bipartite graphs K a,a and K b,b .Let V 1 , U 1 and V 2 , U 2 be the partite sets of K a,a and K b,b .Let us orient the edges of K a,a and K b,b in such a way that all the edges are directed from one partite set to another.Let Let C n be the cycle of order n ≥ 3, where n is even.Since C n is a bipartite graph, let us orient all the edges with direction from one partite set to another and let − → C n be the resulting oriented graph.It is shown in [1] that the skew spectrum of the oriented graph − → C n is {±ι2, 2ι sin 2π(j−1) n : j = 1, 2, . . ., n − 2}.Let us orient the edges of the cycles C 2n 1 , C 2n 2 and C 2n 3 in such a way that the orientations 3 are evenly-oriented, then using Theorem 2.8, we have the following observation, which gives the skew spectrum of the oriented graph Corollary 3.4 Let − → C 2n i be an evenly oriented cycle of order 2n i , for i = 1, 2, 3. Then the skew spectrum of the oriented graph 3 ) consists of the eigenvalues 2ι sin 2π(j−1) 2n k , for j = 1, 2, . . ., 2n k − 1, where k = 1, 2, 3, the remaining six eigenvalues are given by the matrix M in Theorem 2.8.

Skew equienergetic oriented graphs
In this section, by using the results obtained in Section 2, we construct some new infinite families of non-cospectral skew equienergetic digraphs.
Two oriented graphs D 1 and D 2 are said to be skew equienergetic if they have same skew energy, that is, E s (D 1 ) = E s (D 2 ).If two oriented graphs are cospectral, then they are trivially skew equienergetic.Therefore, in what follows, we will be interested in finding skew equienergetic non-cospectral oriented graphs.The following problem was proposed in [15] by Li and Lian.
Problem 1 How to construct families of oriented graphs such that they have equal skew energy, but they do not have the same skew spectra?
The above problem was addressed by Ramane et al. in [22], Adiga et al. in [2] and Liu et al. in [17].In [22] the authors have extended the definition of join of graphs to oriented graphs.They obtained the skew spectrum of the join of two oriented graphs − → G 1 and − → G 2 with the property that the out-degree and in-degree of each vertex in − → G 1 and − → G 2 is same ( that is the oriented graphs − → G 1 and − → G 2 are Eulerian digraphs).Using their results they have constructed some infinite families of non-cospectral skew equienergetic oriented graphs.In [2] the authors have introduced some variations of the join of two oriented graphs for bipartite oriented graphs.They have defined four types of join operations for the bipartite oriented graphs.Using their results they were able to obtain some more infinite families of non-cospectral skew equienergetic oriented graphs.Recently, in [17] the authors have introduced the concept of corona and neighborhood corona of oriented graphs.Using these operations together with join operation they have constructed some new infinite families of non-cospectral skew equienergetic oriented graphs.Recently, the authors [12] have extended the definition of join of two oriented graph by defining the joined union of oriented graphs.They have discussed the skew spectrum of the joined union of oriented Eulerian graphs and as applications they have added some new infinite families of non-cospectral skew equienergetic oriented graphs.Moreover, the results obtained in [22] were obtained as particular cases.In the rest of this section, we aim to construct some new infinite families of non-cospectral skew equienergetic oriented graphs.
The following result gives the skew energy of the joined union Theorem 4.1 Let − → G be an oriented graph of order n ≥ 2. For i = 1, 2, . . .n, let − → B i be an oriented bipartite graph with partite sets of same cardinality n i having the skew adjacency matrix , where X i is a (0, 1)-matrix satisfying X i e n i = r i e n i .Then where ±ιx 1 , ±ιx 2 , . . ., ±ιx n are the eigenvalues of the matrix M given in Theorem 2.3.
Proof.Since the skew adjacency matrix S( and the remaining 2n eigenvalues are the eigenvalues of the matrix M .Now, using the definition of skew energy the result follows.

Taking in particular
, where X 1 is a (0, 1)-matrix satisfying X 1 e n 1 = r 1 e n 1 .Then where ±ιx 1 , ±ιx 2 , . . ., ±ιx n are the eigenvalues of the matrix S( Since be their skew adjacency matrices with For i = 1, 2, . . ., n, let B i be a bipartite graph with partite sets V i and U i of same cardinality n i .Let us orient the edges of B i in such a way that the resulting orientation − → B i is uniformly oriented.Then, using the fact (see Theorem 5.4 in [15]) that the skew spectrum of − → B i is ι times the adjacency spectrum of B i .It follows that the skew energy of ), for all i = 1, 2, . . ., n.Therefore, we have following observation which gives the construction of skew equienergetic oriented graphs from the equienergetic graphs.
A lot of papers can be found in the literature regarding the construction of equienergetic graphs, see the book [16] and the references therein.Let D( − → G ) be the duplication digraph of a digraph − → G defined in [2].Since, the graph D(G) is always a bipartite graph with E(D(G)) = 2E(G), giving that if G i and H i are equienergetic graphs then the bipartite graphs D(G i ) and D(H i ) are also equienergetic.Thus, from any given pair of equienergetic regular graphs we can construct a pair of bipartite equienergetic regular graphs which in turn can be used to construct a pair of skew equienergetic oriented graphs by Corollary 4.4.

Taking in particular
, where X i is a (0, 1)-matrix satisfying X i e n i = r i e n i .Then where ±ιx 1 , ±ιx 2 are the zeroes of the polynomial x 4 + (r , where X i is a (0, 1)matrix satisfying X i e n i = r i e n i .Then where ±ιx 1 , ±ιx 2 are the zeroes of the polynomial x 4 + (r 2 1 + r 2 2 + 2n 1 n 2 )x 2 + r 2 1 r 2 2 .

Conclusion
In this paper we have discussed the skew characteristic polynomial and the skew eigenvalues of the joined union and some of its variations for the oriented bipartite graphs.As applications, we have given a general method to construct infinite families of oriented graphs with same skew energy but different skew spectrum.Our ideas and results obtained generalize some of the ideas and results in [2].
→ G be an oriented graph with n vertices and m arcs having underlying graph G.The skew matrix of − → G , denoted by S( − → G ) is a (−1, 0, 1)-skew symmetric matrix.The skew eigenvalues of − → G are the eigenvalues of S( − → G ) and its characteristic polynomial is the skew characteristic polynomial of − → G .The sum of the absolute values of the skew eigenvalues is the skew energy of − → G and is denoted by E S ( − → G ).

1 Introduction
Let G be a simple graph having n vertices and m edges.The vertex set is {v 1 , v 2 , . . ., v n }.Let − → G be a digraph, where edge is assigned arbitrarily a direction.The digraph − → G is said to be an orientation of G or oriented graph associated with G.The graph G is viewed as the underlying graph of− → G .Let d + i = d + (v i ) be the out-degree, d − i = d − (v i ) be the in-degree and d i = d + i + d − i be the degree of the vertex v i ∈ V ( − → G ). Let N + − → G (v i ) be the set of out-neighbours, N − − → G characteristic polynomial of S( − → G ) is the skew characteristic polynomial of − → G and is denoted by P s ( − → G , x).The zeros of the polynomial P s ( − → G , x) are the eigenvalues of the matrix S( − → G ) and are called skew eigenvalues of − → G .The skew spectrum of − → G is denoted by Sp s ( − → G ), which describes the eigenvalues of S( − → G ) as well as their multiplicities.
b be the resulting oriented graphs.Since the oriented graphs − → K a,a and − → K b,b are evenly-oriented, it follows that their skew spectrum is ι times their adjacency spectrum.Therefore, the skew spectrum of− → K a,a and − → K b,b are {±ιa, 0 [2a−2]} and {±ιb, 0 [2b−2] }, respectively.Moreover, their skew adjacency ma-trices S( − → K a,a ) = 0 a×a J a×a −J a×a 0 a×a and S( − → K b,b ) = 0 b×b J b×b −J b×b 0 b×b satisfies J a×a e a = ae a andJ b×b e b = be b , respectively.We have the following consequence of Theorem 2.7, Theorem 2.9 and Theorem 2.10.Corollary 3.3 Let − → K a,a and − → K b,b be the orientations of the complete bipartite graphs K a,a and K b,b defined above.(i) The skew spectrum of oriented graph − → K a,a → − → K b,b consists of the eigenvalues 0 with multiplicity 2a + 2b − 4, the remaining four eigenvalues are the zeros of the polynomial x 4 + (a 2 + b 2 + 4ab)x 2 + a 2 b 2 .(ii) The skew spectrum of oriented graph − → K a,a j 1 − → K b,b consists of the eigenvalues 0 with multiplicity 2a + 2b − 4, the remaining four eigenvalues are the zeros of the polynomial x 4 + (a 2 + b 2 + 2ab)x 2 + a 2 b 2 .(iii) The skew spectrum of oriented graph − b consists of the eigenvalues 0 with multiplicity 2a+2b−4, the remaining four eigenvalues are the zeros of the polynomial x 4 +(a+b) 2 x 2 +a 2 b 2 .

Corollary 4. 4
Let − → G be an oriented graph of order n ≥ 2. Let B i and G i be r i -regular bipartite equienergetic graphs with partite sets of same cardinality n i , for i = 1, 2, . . ., n.If the orientations − → B i and − → G i are uniformly oriented, then the oriented graphs − in terms of the skew characteristic polynomial of the component oriented graphs and the eigenvalues of an auxiliary matrix determined by the joined union.
2 ,...,2nt = K t [K 2n 1 , K 2n 2 , . . ., K 2nt ].Let us orient the edges in K t arbitrarily to obtain the oriented graph − → K t , then oriented graph − → K t [K 2n 1 , K 2n 2 , . . ., K 2nt ] gives an orientation of the complete t-partite graph K 2n 1 ,2n 2 ,...,2nt , which we denote by CT (2n 1 , 2n 2 , . . ., 2n t ).In the following result we obtain the skew characteristic polynomial of CT (2n 1 , 2n 2 , . . ., 2n t ).The skew characteristic polynomial of CT (2n 1 , 2n 2 , . . ., 2n t ) = − → K t [K 2n 1 , K 2n 2 , . . ., K 2nt ], where 2n 1 + 2n 2 + • • • + 2n t = N with each n i ≥ 1 and t ≥ 2, is given by then by Corollary 3.1, it follows that the skew eigenvalues of CT (2a, 2a, . . ., 2a) = − → K t [K 2a , K 2a , . . ., K 2a ] consists of the eigenvalue 0 with multiplicity (2a − 1)t and the t eigenvalues 2aλ 1 , 2aλ 2 , . . ., 2aλ t , where λ 1 , λ 2 , . . ., λ t are the skew eigenvalues 2a , K 2a , . . ., K 2a ] consists of the eigenvalue 0 with multiplicity (2a − 1)t and the t eigenvalues 2aλ 1 , 2aλ 2 , . . ., 2aλ t , where λ 1 , λ 2 , . . ., λ t are the skew eigenvalues n , a tournament on n vertices and − ] consists of the eigenvalues ιλ i2 , ιλ i3 , . . ., ιλ in−1 , for i = 1, 2, . . ., n, the remaining 2n eigenvalues are given by the matrix M = ϕ n1 ϕ n2 . . .ϕ n      be an oriented bipartite graph with partite sets of same cardinality n 1 having the skew adjacency matrix S [2]orem 4.1 and using Theorem 2.7, we obtain the following result which is Theorem 6 in[2], and gives the skew energy of the join of oriented bipartite graphs (V i , U i ) be an oriented bipartite graph with partite sets V i and U i of same cardinality n i and skew adjacency matrix S( − → B [2]+ r22 + 4n 1 n 2 )x 2 + r 2 1 + r 2 2 .H are two oriented graphs which are non-cospectral with respect to skew matrix, then we have the following consequence of Corollary 4.2, which gives a new infinite family of non-cospectral skew equienergetic oriented graphs.(V 1 , U 1 ) be an oriented bipartite graph with partite sets V 1 and U 1 of same cardinality n 1 −X 1 0 n 1 ×n 1, where X 1 is a (0, 1)-matrix satisfyingX 1 e n 1 =r 1 e n 1 .If be the variation of the joined union of oriented bipartite graphs defined in Section 2. Proceeding similar to Theorem 4.1, we have the following result which gives the skew energy of − → H 2 .Theorem 4.7 Let − → G be an oriented graph of order n ≥ 2. For i = 1, 2, ...n, let − → B i be a bipartite oriented graph with partite sets of same cardinality n i having the skew adjacency matrixS( − → B i ) = 0 n i ×n i X i −X i 0 n i ×n i, where X i is a (0, 1)-matrix satisfying X i e n i = r i e n i .Letwhere ±ιx 1 , ±ιx 2 , ..., ±ιx n are the eigenvalues of the matrix M given in Theorem 2.5.Since the matrix M is determined by the structure of − → G and the orders n i of the oriented bipartite graphs − → B i , for i = 1, 2, ..., n.We have the following observation from Theorem 4.7.(Pi,Qi ) be oriented bipartite graphs with partite sets V i , U i and P i , Q i of same cardinality n i ,for i = 1, 2, ..., n.Let S( − → B i ) = 0 n i ×n i X i −X i 0 n i ×n i and S( − → G i ) = 0 n i ×n i Y i −Y i 0 n i ×n ibe their skew adjacency matrices with X i e n i = r i e n i = Y i e n i .Let in Theorem 4.7, we obtain the following result obtained in[2]as part first of Theorem 9. Corollary 4.9 For i = 1, 2, let − → B i be an oriented bipartite graph with partite sets of same cardinality n i having the skew adjacency matrix S( − → B n ] by deleting all the arcs between U i and V j , U j , i ̸ = j and let − → H n ] by deleting all the arcs between Q i and P j , Q j , i ̸ = j.