ABSTRACT
Let be an oriented graph with n vertices and m arcs having underlying graph G. The skew matrix of , denoted by is a -skew symmetric matrix. The skew eigenvalues of are the eigenvalues of and its characteristic polynomial is the skew characteristic polynomial of . The sum of the absolute values of the skew eigenvalues is the skew energy of and is denoted by . 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 and B.R. Rakshith, More skew-equienergetic digraphs, Commun. Comb. Optim., 1(1) (2016) 55–71].
MATHEMATICS SUBJECT CLASSIFICATION:
1. Introduction
Let G be a simple graph having n vertices and m edges. The vertex set is . Let be a digraph, where edge is assigned arbitrarily a direction. The digraph is said to be an orientation of G or oriented graph associated with G. The graph G is viewed as the underlying graph of . Let be the out-degree, be the in-degree and be the degree of the vertex . Let be the set of out-neighbours, be the set of in-neighbours and be the set of neighbours of the vertex vi in . The adjacency matrix of a graph G is a n-square matrix with , if there is an edge between the vertices vi and vj and , 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 . The sum of the absolute values of the eigenvalues of G is called energy of G and is denoted by . 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 Akbari et al. (Citation2022) and the book Li et al. (Citation2012).
The skew adjacency matrix of an oriented graph is an n × n matrix with when there is an arc from vi to vj, and when there is an arc from vj to vi, and otherwise. It is clear that the matrix is a skew symmetric matrix, so all its eigenvalues are zero or purely imaginary. The characteristic polynomial of is the skew characteristic polynomial of and is denoted by . The zeros of the polynomial are the eigenvalues of the matrix and are called skew eigenvalues of . The skew spectrum of is denoted by , which describes the eigenvalues of as well as their multiplicities.
The skew energy of the oriented graph is called the energy of the matrix . It is defined by the following equation.
where are the skew eigenvalues of . 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 is called skew rank of ). One of the most studied problems in the theory of skew energy is the determination of extremal oriented graphs for 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 . This problem is the topic of many papers in literature. Some recent examples can be found in (Deng et al., Citation2018; Taghvaee & Fath-Tabar, Citation2020). We refer to (Alhevaz et al., Citation2020; Bhat, Citation2017; Ganie, Citation2019; Ganie et al., Citation2019; Ganie et al., Citation2021; Li & Lian, Citation2015; Pirzada et al., Citation2020; Qiu et al., Citation2021; Rather et al., Citation2023; Shang, Citation2018) for more development of skew energy theory.
Given a cycle , its sign is signified as . Here, sij means the entry of the skew matrix in the intersection of ui row and uj column. If the sign of an even oriented cycle Ck is positive or negative, it is referred to as evenly-oriented or oddly-oriented, respectively. We say is evenly-oriented if every even cycle in is evenly-oriented. When , the even oriented cycle 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.
2. The skew spectrum of joined union of oriented graphs
Consider an n × n complex matrix
where Xij is an block matrix for and . The element bij is the average row sum of Xij. We define an s × s matrix with elements being the average row sums of Xij and we call it the quotient matrix . The matrix B becomes an equitable quotient matrix when each block Xij has constant row sum. A complex matrix has a connection with equitable quotient matrix in terms of its spectrum as below (You et al., Citation2019).
Lemma 2.1.
The equitable quotient matrix B and the matrix M defined in (Equation1(1) (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 (Ganie, Citation2022; Rather et al., Citation2021, Citation2023). The joined union was extended to digraphs in (Ganie, Citation2022). In Ganie, (Citation2022), the author have discussed the Aα-spectrum of the joined union of diagonalizable digraphs and as applications the Aα-spectrum of various families of digraphs are found. Recently, in Ganie, Ingole, et al., (Citation0000), the authors defined generalized join of oriented graphs as follows.
Let be an oriented graph of order n and let be oriented graphs of order where . The joined union of the oriented graphs with respect to oriented graph is denoted by and is defined as the oriented graph with vertex set and arc set
In other words, the joined union is the union of oriented graphs together with the arcs , where and , whenever is an arc in . Clearly, the usual join of two oriented graphs and defined in Ramane et al., (Citation2016) is a special case of the joined union where 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 be an oriented graph of order n and let , be a bipartite oriented graph with partite sets Vi and Ui, for all . Let be the joined union of the oriented graphs with respect to oriented graph . That is, . Note that if there is an arc between the vertices vi and vj in , then there are arcs between all the vertices of Vi and Vj; between all the vertices of Vi and Uj; between all the vertices of Ui and Vj and between all the vertices of Ui and Uj. Let be the oriented graph obtained from by deleting all the arcs between Ui and Vj and all the arcs between Ui and Uj. Let be the oriented graph obtained from by deleting all the arcs between Vi and Vj and all the arcs between Vi and Uj.
A digraph D is said to be Eulerian if the out-degree of any vertex in D is same as its in-degree, that is, , for all . The following theorem was obtained in Ganie, Ingole, et al., (Citation0000) and gives the skew spectrum of the joined union of Eulerian oriented graphs , in terms of the skew spectrum of the component oriented graphs and the eigenvalues of an auxiliary matrix determined by the joined union.
Theorem 2.2.
Let be an oriented graph of order having m arcs. Let be Eulerian oriented graph of order ni having skew characteristic polynomial where . Then the skew characteristic polynomial of the oriented graph of order is
where is the characteristic polynomial of the matrix
where , if there is an arc from vi to vj; , if there is an arc from vj to vi and , if there is no arc between vi and vj.
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 , let , be a bipartite oriented graph with partite sets Vi and Ui of same cardinality ni, having the skew adjacency matrix , where Xi is a -matrix satisfying 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 , in terms of the skew characteristic polynomial of the component oriented graphs and the eigenvalues of an auxiliary matrix determined by the joined union.
Theorem 2.3.
Let be an oriented graph of order having m arcs. For , let , be a bipartite oriented graph with , having the skew adjacency matrix , where Xi is a -matrix satisfying . Let where be the skew characteristic polynomial of . Then the skew characteristic polynomial of the oriented graph of order is
where is the characteristic polynomial of the matrix
where ; , if there is an arc from vi to vj; , if there is an arc from vj to vi and , if there is no arc between vi to vj.
Proof. Let be the vertex set of and let be the vertex set of , for . Let . Let us label the vertices in H1 in such a way that the vertices in are labelled first, the vertices of are labelled after the vertices in , and so on. With this labelling, the skew matrix of takes the form
where,
and , with and , if ; and , if and and , if . Note that is the all one matrix of order and is the zero matrix of order .
By assumption is a bipartite oriented graph for all i with partite sets Vi and Ui of same cardinality ni and skew matrix, , where Xi is a -matrix satisfying . It is easy to verify that is an eigenvalue of with corresponding eigenvector . Similarly, we can verify that is an eigenvalue of with corresponding eigenvector . Since is a skew symmetric matrix, so it is a diagonalisable matrix with its eigenvectors forming an orthogonal set. Let λik be an eigenvalue of other than with the corresponding eigenvector satisfying That is, . Now, consider the vector , where
As gives that and coordinates of the vector Y corresponding to vertices of which are not in are zeros, we have
This shows that Y is an eigenvector of corresponding to the eigenvalue λik and so every eigenvalue λik (other than ) of is an eigenvalue of . So, using this process we will obtain eigenvalues of . To determine the remaining 2 n eigenvalues of , we use the equitable quotient matrix. The equitable quotient matrix of is
where ; , if ; , if and , if . Since by Lemma 2.1, the eigenvalues of M are the eigenvalues of , the result follows.□
The lexicographic product of graphs G and H is the graph with vertex set and edge whenever . It is interesting to see that the lexicographic product can be constructed by joined union where for . Note that in the case we get .
If in particular the oriented bipartite graphs in Theorem 2.2 are same, say , for , then we obtain the following Theorem, which gives the skew spectrum of the joined union , which represents an orientation of the lexicographic product .
Theorem 2.4.
Let be an oriented graph of order having m arcs. Let , be a bipartite oriented graph with , having the skew adjacency matrix , where X1 is a -matrix satisfying . Let be the skew characteristic polynomial of . Then the skew characteristic polynomial of the oriented graph of order is
where is the characteristic polynomial of the matrix
where ; , if there is an arc from vi to vj; , if there is an arc from vj to vi and , if there is no arc between vi to vj, where J2 is the all one matrix of order and 02 is the zero matrix of order .
Proof. If , then from Theorem 2.3 the 2 n eigenvalues of are given by the matrix
where ; , if there is an arc from vi to vj; , if there is an arc from vj to vi and , if there is no arc between vi to vj. With this the result now follows.□
In the next theorem we determine the skew characteristic polynomial of the oriented graph , when the component oriented graphs are bipartite with partite sets of same cardinality ni, having the skew adjacency matrix , where Xi is a -matrix satisfying .
Theorem 2.5.
Let be an oriented graph of order having m arcs. For , let , be a bipartite oriented graph with , having the skew adjacency matrix , where Xi is a -matrix satisfying . Let where be the skew characteristic polynomial of . Then the skew characteristic polynomial of the oriented graph of order is
where is the characteristic polynomial of the matrix
where ; , if there is an arc from vi to vj; , if there is an arc from vj to vi and , if there is no arc between vi to vj.
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 , when the component oriented graphs are bipartite with partite sets of same cardinality ni, having the skew adjacency matrix , where Xi is a -matrix satisfying .
Theorem 2.6.
Let be an oriented graph of order having m arcs. For , let , be a bipartite oriented graph with , having the skew adjacency matrix , where Xi is a -matrix satisfying . Let where be the skew characteristic polynomial of . Then the skew characteristic polynomial of the oriented graph of order is
where is the characteristic polynomial of the matrix
where ; , if there is an arc from vi to vj; , if there is an arc from vj to vi and , if there is no arc between vi to vj.
Proof. The proof follows on similar lines as in Theorem 2.3 and is therefore omitted.□
Let and be two oriented bipartite graphs of order and , respectively. Let be the join of and . Clearly, . 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 (Adiga & Rakshith, Citation2016).
Theorem 2.7.
For , let , be a bipartite oriented graph with , having the skew adjacency matrix , where Xi is a -matrix satisfying . Let where be the skew characteristic polynomial of . Then the skew characteristic polynomial of the oriented graph is
Proof.
The proof follows from Theorem 2.3 by taking and using the fact that the characteristic polynomial of the matrix
is .
Let and be three oriented bipartite graphs of order and , respectively. Let be the join of with the union of and . It is easy to see that , where is the orientation of the star graph with arcs directed from vertex of degree 2. The following consequence of Theorem 2.3, gives the skew spectrum of the oriented graph .
Theorem 2.8.
For , let , be a bipartite oriented graph with , having the skew adjacency matrix , where Xi is a -matrix satisfying . Let where be the skew characteristic polynomial of . Then the skew characteristic polynomial of the oriented graph is
where is the characteristic polynomial of matrix M given by
Proof.
The proof follows from Theorem 2.3 by taking , where is the orientation of star graph with arcs directed from vertex of degree 2.
Let and be oriented bipartite graphs with partite sets and V2, respectively. The join-1 of and , denoted by , is defined in (Adiga & Rakshith, Citation2016) as the oriented graph obtained from and by joining arcs from all the vertices of U1 to each the vertex of U2 and V2. The next Theorem was obtained as part first of Theorem 8 in (Adiga & Rakshith, Citation2016) and gives the skew characteristic polynomial of join-1 of the oriented graphs and .
Theorem 2.9.
For , let , be a bipartite oriented graph with , having the skew adjacency matrix , where Xi is a -matrix satisfying . Let where be the skew characteristic polynomial of . Then the skew characteristic polynomial of join-1 is
Proof.
The proof follows from Theorem 2.5 by taking and using the fact that the characteristic polynomial of matrix M given by
is .
This shows that Theorem 2.9 is a generalization of the part first of Theorem 8 in (Adiga & Rakshith, Citation2016). In fact, the operation defined to obtain the oriented graph is actually the generalization of the join-1 operation defined in (Adiga & Rakshith, Citation2016).
Let and be oriented bipartite graphs with partite sets and V2, respectively. We define the join-1 of and , denoted by as the oriented graph obtained from and by joining arcs from all the vertices of V1 to each vertex of U2 and V2. In the next Theorem we obtain the skew characteristic polynomial of join-1, of oriented graphs and .
Theorem 2.10.
For , let , be a bipartite oriented graph with , having the skew adjacency matrix , where Xi is a -matrix satisfying . Let where be the skew characteristic polynomial of . Then the skew characteristic polynomial of join-1, is
Proof.
The proof follows from Theorem 2.6 by taking and using the fact that the characteristic polynomial of matrix M is given by
is .
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 Kn be a complete graph on n vertices. Any orientation of Kn is said to be a tournament. Consider the complete t-partite graph , it is easy to verify that . Let us orient the edges in Kt arbitrarily to obtain the oriented graph , then oriented graph gives an orientation of the complete t-partite graph , which we denote by . In the following result we obtain the skew characteristic polynomial of .
Corollary 3.1.
The skew characteristic polynomial of , where with each and , is given by
where is the characteristic polynomial of the matrix
with or , according to as there is an arc from vi to vj or from vj to vi in and , for all i.
Proof.
Taking , a tournament on n vertices and , an empty graph, for all in Theorem 2.3 and using the fact that the skew characteristic polynomial of is , for all i, the result follows. Note that the empty graph can be considered as the bipartite graph with partite sets Vi and Ui of same cardinality ni and the skew adjacency matrix of given by , which satisfies .
If , then by Corollary 3.1, it follows that the skew eigenvalues of consist of the eigenvalue 0 with multiplicity and the t eigenvalues , where are the skew eigenvalues of tournament . This follows from Corollary 3.1 and the fact that the quotient matrix M for the oriented graph is . Now using the Theorem 4.2.12 from (Horn & Johnson, Citation1985) the result follows. In fact, if is any oriented graph of order t, then the skew eigenvalues of oriented graph consists of the eigenvalue 0 with multiplicity and the t eigenvalues , where are the skew eigenvalues of .
Taking , a tournament on n vertices and , an orientation of the complete bipartite graph with all edges oriented from one partite set to another, for all , in Theorem 2.3 and using the fact that the skew characteristic polynomial of is , we obtain the following consequence of Theorem 2.3.
Corollary 3.2.
The skew characteristic polynomial of , where with each , is given by
where is the characteristic polynomial of the matrix
with or , according to as there is an arc from vi to vj or from vj to vi in and .
Proof.
Note that the skew adjacency matrix of given by , satisfies .
In particular, if then by Corollary 3.2, it follows that the skew eigenvalues of consists of the eigenvalue 0 with multiplicity and the 2 n eigenvalues of the matrix , where I is the identity matrix of order n and is the skew matrix of . In fact, if is any oriented graph of order n, then the skew eigenvalues of oriented graph consists of the eigenvalue 0 with multiplicity and the 2 n eigenvalues of the matrix .
Let be a ni-regular bipartite graph with partite sets . Let us orient the edges of Bi from Vi to Ui, then it is clear that the resulting oriented graph is evenly-oriented. So, using the fact (see Theorem 5.4 in (Li & Lian, Citation2015)) that the skew spectrum of is i times the adjacency spectrum of Bi. It follows that the skew eigenvalues of are , where are the eigenvalues (adjacency eigenvalues) of Bi. Also, it is clear that the skew adjacency matrix satisfies . Therefore, it follows from Theorem 2.3 that the skew spectrum of the oriented graph consists of the eigenvalues , for , the remaining 2 n eigenvalues are given by the matrix with or or , according to as there is an arc from vi to vj or from vj to vi or there is no arc between vi and vj in and .
Consider the complete bipartite graphs and . Let and be the partite sets of and . Let us orient the edges of and in such a way that all the edges are directed from one partite set to another. Let and be the resulting oriented graphs. Since the oriented graphs and are evenly-oriented, it follows that their skew spectrum is ι times their adjacency spectrum. Therefore, the skew spectrum of and are and , respectively. Moreover, their skew adjacency matrices and satisfies and , respectively. We have the following consequence of Theorem 2.7, Theorem 2.9 and Theorem 2.10.
Corollary 3.3.
Let and be the orientations of the complete bipartite graphs and defined above.
The skew spectrum of oriented graph consists of the eigenvalues 0 with multiplicity , the remaining four eigenvalues are the zeros of the polynomial .
The skew spectrum of oriented graph consists of the eigenvalues 0 with multiplicity , the remaining four eigenvalues are the zeros of the polynomial .
The skew spectrum of oriented graph consists of the eigenvalues 0 with multiplicity , the remaining four eigenvalues are the zeros of the polynomial .
Let Cn be the cycle of order , where n is even. Since Cn is a bipartite graph, let us orient all the edges with direction from one partite set to another and let be the resulting oriented graph. It is shown in (Adiga et al., Citation2010) that the skew spectrum of the oriented graph is . Let us orient the edges of the cycles and in such a way that the orientations and 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 be an evenly oriented cycle of order , for . Then the skew spectrum of the oriented graph consists of the eigenvalues , for , where , the remaining six eigenvalues are given by the matrix M in Theorem 2.8.
In particular, if , then the skew spectrum of the oriented graph consists of the eigenvalues , for , each with multiplicity three and the six zeros of the polynomial . In fact, if B is a r-regular bipartite graph with partite sets U and V of same cardinality n1 and an orientation of B with all edges directed from U to V, then it is clear that the skew adjacency matrix of satisfies . Therefore, using Theorem 2.8 the skew eigenvalues of the oriented graph consists of the eigenvalues each with multiplicity three, where are the adjacency eigenvalues of B and the remaining six eigenvalue are given by the zeros of the polynomial .
Let and , where is the empty oriented graph of order , is the evenly-oriented complete bipartite oriented graph of order and is the directed cycle. Then from Theorem 2.7 and Theorem 2.8, we obtain the skew spectrum of the oriented graphs , , , and .
Corollary 3.5.
The skew spectrum of consists of the eigenvalue 0 with multiplicity and the remaining two eigenvalues are .
The skew spectrum of consists of the eigenvalue 0 with multiplicity , the eigenvalues , for and the remaining two eigenvalues are .
The skew spectrum of consists of the eigenvalue 0 with multiplicity , the eigenvalues , for and the remaining four eigenvalues are the zeros of the polynomial .
The skew spectrum of oriented graph consists of the eigenvalue 0 with multiplicity , the eigenvalues , for and the remaining six eigenvalues are given by the matrix M given by Theorem 2.8 with and .
The skew spectrum of oriented graph consists of the eigenvalue 0 with multiplicity , the eigenvalues , for and the remaining six eigenvalues are given by the matrix M given by Theorem 2.8 with and .
The skew spectrum of oriented graph consists of the eigenvalue 0 with multiplicity , the eigenvalues , for and the remaining six eigenvalues are given by the matrix M in Theorem 2.8 with and .
Similarly, we can obtain the skew spectrum of the oriented graphs , etc.
4. 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 D1 and D2 are said to be skew equienergetic if they have same skew energy, that is, . 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 (Li & Lian, Citation2015) by Li and Lian. 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. (Ramane et al., Citation2016), Adiga et al. (Adiga & Rakshith, Citation2016) and Liu et al. (Liu et al., Citation2019). In (Ramane et al., Citation2016) 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 and with the property that the out-degree and in-degree of each vertex in and is same (that is the oriented graphs and are Eulerian digraphs). Using their results they have constructed some infinite families of non-cospectral skew equienergetic oriented graphs. In (Adiga & Rakshith, Citation2016) 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 (Liu et al., Citation2019) 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 (Ganie, Ingole, et al., Citation0000) 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 (Ramane et al., Citation2016) 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 of oriented bipartite graphs .
Theorem 4.1.
Let be an oriented graph of order . For , let be an oriented bipartite graph with partite sets of same cardinality ni having the skew adjacency matrix , where Xi is a -matrix satisfying . Then
where are the eigenvalues of the matrix M given in Theorem 2.3.
Proof.
Since the skew adjacency matrix of has the property that . It is easy to verify that is eigenvalue of . Let be the skew eigenvalues of , for . Then, it is clear from Theorem 2.3 that the skew eigenvalues of the oriented graph are for and the remaining 2 n eigenvalues are the eigenvalues of the matrix M. Now, using the definition of skew energy the result follows.
Taking in particular in Theorem 4.1, we obtain the skew energy of the oriented graph .
Corollary 4.2.
Let be an oriented graph of order . Let be an oriented bipartite graph with partite sets of same cardinality n1 having the skew adjacency matrix , where X1 is a -matrix satisfying . Then
where are the eigenvalues of the matrix .
Since the matrix M is determined by the structure of and the orders ni of the oriented bipartite graphs , for . We have the following observation from Theorem 4.1.
Corollary 4.3.
Let be an oriented graph of order . Let and be oriented bipartite graphs of order ni, for and let and be their skew adjacency matrices with . If the oriented bipartite graphs and are non-cospectral with , for all , then the oriented graphs and are non-cospectral with
For , let Bi be a bipartite graph with partite sets Vi and Ui of same cardinality ni. Let us orient the edges of Bi in such a way that the resulting orientation is uniformly oriented. Then, using the fact (see Theorem 5.4 in (Li & Lian, Citation2015)) that the skew spectrum of is ι times the adjacency spectrum of Bi. It follows that the skew energy of is same as the energy of Bi, that is, , for all . Therefore, we have following observation which gives the construction of skew equienergetic oriented graphs from the equienergetic graphs.
Corollary 4.4.
Let be an oriented graph of order . Let and be ri-regular bipartite equienergetic graphs with partite sets of same cardinality ni, for . If the orientations and are uniformly oriented, then the oriented graphs and are non-cospectral with
A lot of papers can be found in the literature regarding the construction of equienergetic graphs, see the book (Li et al., Citation2012) and the references therein. Let be the duplication digraph of a digraph defined in (Adiga & Rakshith, Citation2016). Since, the graph D(G) is always a bipartite graph with , giving that if Gi and Hi are equienergetic graphs then the bipartite graphs and 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 in Theorem 4.1 and using Theorem 2.7, we obtain the following result which is Theorem 6 in (Adiga & Rakshith, Citation2016), and gives the skew energy of the join of oriented bipartite graphs and .
Corollary 4.5.
For , let be an oriented bipartite graph with partite sets Vi and Ui of same cardinality ni and skew adjacency matrix , where Xi is a -matrix satisfying . Then
where are the zeroes of the polynomial .
If and 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.
Corollary 4.6.
Let and be two non-cospectral oriented graphs of order . Let be an oriented bipartite graph with partite sets V1 and U1 of same cardinality n1 and skew adjacency matrix , where X1 is a -matrix satisfying . If
where . Then
Let 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 .
Theorem 4.7.
Let be an oriented graph of order . For , let be a bipartite oriented graph with partite sets of same cardinality ni having the skew adjacency matrix , where Xi is a -matrix satisfying . Let be the oriented graph defined in Section 2, then
where are the eigenvalues of the matrix M given in Theorem 2.5.
Since the matrix M is determined by the structure of and the orders ni of the oriented bipartite graphs , for . We have the following observation from Theorem 4.7.
Corollary 4.8.
Let be an oriented graph of order . Let and be oriented bipartite graphs with partite sets and of same cardinality ni, for . Let and be their skew adjacency matrices with . Let be the oriented graph obtained from by deleting all the arcs between Ui and , and let be the oriented graph obtained from by deleting all the arcs between Qi and , . If the oriented bipartite graphs and are non-cospectral with , for all , then the oriented graphs and are non-cospectral with
Taking in Theorem 4.7, we obtain the following result obtained in (Adiga & Rakshith, Citation2016) as part first of Theorem 9.
Corollary 4.9.
For , let be an oriented bipartite graph with partite sets of same cardinality ni having the skew adjacency matrix , where Xi is a -matrix satisfying . Then
where are the zeroes of the polynomial .
5. 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 Adiga & Rakshith, (Citation2016).
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