SOME RELATIONS BETWEEN THE SKEW SPECTRUM OF AN ORIENTED GRAPH AND THE SPECTRUM OF CERTAIN CLOSELY ASSOCIATED SIGNED GRAPHS

Let RG′ be the vertex-edge incidence matrix of an oriented graph G′. Let Λ(Ḟ ) be the signed graph whose vertices are identified as the edges of a signed graph Ḟ , with a pair of vertices being adjacent by a positive (resp. negative) edge if and only if the corresponding edges of Ġ are adjacent and have the same (resp. different) sign. In this paper, we prove that G′ is bipartite if and only if there exists a signed graph Ḟ such that R G′ RG′ − 2I is the adjacency matrix of Λ(Ḟ ). It occurs that Ḟ is fully determined by G′. As an application, in some particular cases we express the skew eigenvalues of G′ in terms of the eigenvalues of Ḟ . We also establish some upper bounds for the skew spectral radius of G′ in both the bipartite and the non-bipartite case.


Introduction
For a finite simple graph G = (V, E), an oriented graph G is a pair (G, σ ), where σ is the orientation satisfying σ (ij) ∈ {i, j}, for every ij ∈ E. If σ (ij) = j, we say that the edge ij is oriented from i to j and designate this by i → j (or j ← i); an oriented edge is also known as an arc. Similarly, a signed graphĠ is a pair (G, σ), where σ is the signature satisfying σ(ij) ∈ {1, −1}, for every ij ∈ E. The edge set ofĠ consists of positive and negative edges, and we interpret a graph as a signed graph with all edges being positive. In both cases we denote n = |V | and say that G is the underlying graph (of G orĠ).
The skew adjacency matrix (of G ) S G = (s ij ) is the n × n matrix defined by: s ij = 1 if there is an edge oriented from j to i, s ij = −1 if there is an edge oriented from i to j, and s ij = 0 otherwise. The eigenvalues of S G are called the skew eigenvalues of G and they form the skew spectrum of G , which consists of purely imaginary numbers. It is easy to verify that non-zero skew eigenvalues come in pairs µ and −µ (with equal algebraic multiplicity). Consequently, the rank of S G 42 ZORAN STANIĆ is even. The largest modulus of skew eigenvalues of G is called the (skew) spectral radius and denoted by ρ(G ).
The vertex-edge incidence matrix R G of G is the matrix whose rows and columns are indexed by V (G ) and E(G ), respectively, such that its (i, e)-entry is 1 if e is oriented to i, −1 if e is oriented from i, and 0 otherwise.
The adjacency matrix (ofĠ) AĠ = (a ij ) is obtained from the (0, 1)-adjacency matrix of G by reversing the sign of all 1s which correspond to negative edges. The eigenvalues and the spectrum ofĠ are identified as the eigenvalues and the spectrum of AĠ, respectively. Since AĠ is symmetric, its eigenvalues are real. The spectral radius is defined as in the case of oriented graphs and denoted by the same symbol.
For a signed graphĠ, we denote by Λ(Ġ) the signed graph whose vertices are the edges ofĠ, such that a pair of vertices is joined by a positive (resp. negative) edge if and only if the corresponding edges ofĠ are adjacent and have the same (resp. different) sign. In relation to this definition, we remark that there are several definitions of a signed line graph. A combinatorial one can be found in works of Zaslavsky [8], while the one tailored for the spectral theory can be found in [1,3,6]. These definitions differ in sign and the corresponding underlying graph is also the underlying graph of Λ(Ġ). In case of unsigned graphs, the line graph L(G) of G is the graph whose vertices are identified with the edges of G, with two vertices being adjacent whenever the corresponding edges are adjacent.
To visualize the previous definitions, in Figure 1 we illustrate an underlying graph G, an oriented graph G = (G, σ ) (in which the edge orientation is designated by arrows), a signed graphĠ = (G, σ) (in which negative edges are dashed), the line graph L(G) and the signed graph Λ(Ġ). Enumerations of the edges in G andĠ correspond to enumerations of the vertices in L(G) and Λ(Ġ). This figure will also help us to visualize some forthcoming results.
Since the skew adjacency matrix is asymmetric with non-real eigenvalues (unless G has no edges), in our recent work [7] we considered the existence of a related signed graph whose eigenvalues fully determine the skew eigenvalues of G . It occurs that in such cases the entire theory of real symmetric matrices can be used in the study of skew spectra of oriented graphs. Motivated by these results, in this paper we consider relations between the skew spectrum of an oriented graph G and the spectrum of a signed graph whose adjacency matrix is given by R G R G −2I. In particular, we establish some properties of such a signed graph and prove that G is bipartite if and only if there exists a signed graphḞ such that R G R G − 2I is the adjacency matrix of Λ(Ḟ ). It occurs thatḞ is fully described by G . In certain cases we express the characteristic polynomial of Λ(Ḟ ) in terms of the characteristic polynomial of G . The paper concludes with some upper bounds for the spectral radius of G expressed in terms of the spectral radius of L(G) and the spectral radius of L(bd(G)), where bd(G) denotes the bipartite double of G, defined in the next section.
Some terminology, notation and necessary results are given in Section 2. In particular, one can find some recently established relations between the skew spectrum of an oriented graph and the spectrum of an associated (for the specific definition of 'associated') signed graph. Results that include similar relations are reported in Section 3.

Preliminaries
We first specify some cycles of an oriented and a signed graph. In the forthcoming notation, the subscript denotes the number of vertices of a cycle under consideration. We say that an even oriented cycle C 2l , considered as a subgraph of an oriented graph, is oriented uniformly if by traversing the cycle we pass through an odd (resp. even) number of edges oriented in the route direction for l odd (even). A cycle in a signed graph is called positive if it contains an even number of negative edges; otherwise, it is said to be negative.
For a subset U ⊆ V (Ġ), letĠ U be the signed graph obtained fromĠ by reversing the sign of every edge between a vertex in U and a vertex in V (Ġ) \ U . We say thatĠ andĠ U are switching equivalent. Switching equivalent signed graphs share the same spectrum. We recall from [8] that a signed graph is switching equivalent to its underlying graph if and only if all its cycles are positive. The negation −Ġ is obtained by reversing the sign of every edge ofĠ.
For an oriented graph G , the bipartite double bd(G ) is the oriented graph whose skew adjacency matrix is defined by the Kronecker product We proceed with some theoretical results. For an oriented graph G = (G, σ ) and a signed graphĠ = (G, σ), we say that the signature σ is associated with the orientation σ if σ(ik)σ(jk) = s ik s jk holds for every pair of edges ik and jk. (2.1) Being associated is a symmetric relation. We know from [7] that, for a graph G and an orientation σ , there exists a signature σ associated with σ if and only if G is bipartite. The orientation σ and the signature σ are associated in the following sense: σ induces two switching equivalent signed graphs (one with signature σ and the other being the negation of the first one) and the signature σ induces two oriented graphs (one with orientation σ and the other obtained by reversing the orientation of every edge of the first one). Further, if σ and σ are associated, then we also say that G andĠ are associated. The reader may observe that the oriented graph G and the signed graphĠ of Figure 1 are mutually associated.
In the following theorems the exponent stands for the multiplicity of the corresponding eigenvalue.
are the eigenvalues ofḢ.

Results
We first consider the matrix R G R G − 2I.
The following statements hold: (i) A is the adjacency matrix of a signed graph, sayḢ.
(ii) Every triangle ofḢ arises either from a triplet of edges incident with the same vertex of G or a triangle of G . Every induced cycle of length ≥ 4 arises from an induced cycle of the same length of G . (iii) Every triangle ofḢ that arises from a triplet of edges incident with the same vertex is positive. An induced cycle that arises from an induced cycle of G of an odd (resp. even) length is negative (resp. positive). (iv) The spectrum ofḢ is determined by the spectrum of R G R G .
Proof. (i): Since every column of R G contains exactly two non-zero entries, one being 1 and the other −1, we get that every entry of the main diagonal of R G R G is 2. An off-diagonal (i, j)-entry of R G R G is equal to r i · r j (where r i is the ith column of R G and the dot stands for the standard inner product), and thus it belongs to {−1, 0, 1}. It follows that R G R G − 2I is a symmetric (−1, 0, 1)-matrix with zero diagonal, i.e., it is the adjacency matrix of a signed graph.
(ii): If i, j, k are the vertices of a triangle inḢ then we have r i · r j , r j · r k , r k · r i ∈ {−1, 1}, which means that the corresponding edges of G are either incident with the same vertex or form a triangle.
LetĊ l be an induced cycle of length l ≥ 4 with vertices i 1 , i 2 , . . . , i l (indexed in the natural order). Then for the corresponding rows of R G , we have r i · r i+1 ∈ {1, −1}, for 1 ≤ i ≤ l − 1 and r 1 · r l ∈ {1, −1}, which means that they determine the edges of a cycle of G . If this cycle is not an induced one, then we have r j · r k ∈ {1, −1} for an additional (non-consecutive) pair of columns, which implies It follows immediately that every triplet of edges with a common vertex gives rise to a positive triangle ofḢ. Let C l be an induced cycle of length l in G , and assume first that its edges are oriented in the route direction. Then, for l odd (resp. even), the corresponding cycle ofḢ is negative (resp. positive). By observing that reversing of the orientation of a single edge of C l causes reversing of the sign of exactly two edges in the corresponding cycle ofḢ, we conclude that, in fact, its signature remains unchanged. Since every orientation of C l is obtained by a successive reversing of the orientation of a single edge, we arrive at the desired result.
(iv): This follows since R G R G and R G R G share the same non-zero eigenvalues along with their multiplicities. Now, we consider Λ(Ġ) in more detail. Similarly to the previous theorem, a triangle in Λ(Ġ) arises either from a triplet of edges with a common vertex or from a triangle (ofĠ). Other induced cycles arise from induced cycles of the same length. We immediately get that the triangle obtained from a triplet of edges with a common vertex is positive. Let furtherĊ l be an induced cycle inĠ. If all the edges ofĊ l are negative, then the corresponding cycle of Λ(Ġ) is positive (by definition of Λ(Ġ)). For otherwise, observe that every maximal sequence of consecutive negative edges ofĊ l gives rise to exactly 2 negative edges in the corresponding cycle of Λ(Ġ): they are obtained from the first and the last edge of such a sequence and their neighbouring positive edges. Since all the negative edges ofĊ l can be partitioned into the described maximal sequences, we get that the number of negative edges in the corresponding cycle of Λ(Ġ) is even, and so this cycle is positive. Summa summarum, all induced cycles of Λ(Ġ) are positive.
Assume now that Λ(Ġ) contains a non-induced negative cycle, and letĊ be such a cycle which is shortest in length. Obviously,Ċ contains a pair of adjacent vertices such that the edge between them does not belong toĊ. Then the edges ofĊ and the mentioned additional edge gives 2 cycles such that exactly one of them is negative. The existence of the negative one contradicts either the fact that all induced cycles are positive or the assumption on the length ofĊ. Therefore, all cycles of Λ(Ġ) are positive, which by [8] means that this signed graph is switching equivalent to its underlying graph, and we are done.

Corollary 3.3.
If G is the underlying graph ofĠ, then L(G) and Λ(Ġ) share the same spectrum.
Proof. By definitions of the line graph L(G) and the signed graph Λ(Ġ), we conclude that if G is the underlying graph ofĠ, then L(G) is the underlying graph of Λ(Ġ), and then the result follows by Theorem 3.2 (ii).
To visualize the previous proof, the reader can see Figure 1. We proceed by the question concerning the existence of a signed graphḞ such that R G R G − 2I is the adjacency matrix of Λ(Ḟ ).

Theorem 3.4. An oriented graph G is bipartite if and only if there exists a signed graphḞ such that
Proof. Assume that R G R G − 2I is the adjacency matrix of Λ(Ḟ ) for someḞ . By Theorem 3.2 (ii), Λ(Ḟ ) is switching equivalent to its underlying graph, and consequently it does not contain a negative cycle. By Theorem 3.1 (iii), G does not contain an induced cycle of an odd length, and then the same holds for noninduced ones; hence, G is bipartite.
Assume now that G is bipartite. By Theorem 2.1, there exists a signed graphĠ associated with G in the sense of (2.1). We claim thatĠ appears in the role ofḞ , that is, R G R G − 2I is the adjacency matrix of Λ(Ġ). Observe that the signature ofĠ is determined by R G in exactly the same way as the signature of the signed graph determined by R G R G − 2I. Namely, two adjacent edges ofĠ are of the same sign if and only if the corresponding vertices of the signed graph determined by R G R G − 2I are joined by a positive edge. Therefore, by definition of Λ(Ġ), R G R G − 2I is its adjacency matrix.
We record that, by the previous proof, if G is bipartite andĠ is associated with G , then R G R G − 2I is the adjacency matrix of Λ(Ġ). For example, by an appropriate labelling of vertices of G of Figure 1, one can get that R G R G − 2I is the adjacency matrix of Λ(Ġ) of the same figure.
In the remainder of the paper we use Φ to denote the characteristic polynomial of an (oriented or signed) graph under consideration. It coincides with the characteristic polynomial of the corresponding (skew) adjacency matrix. To avoid possible confusion, the graph is indicated in the subscript. Recall that a graph is said to be bipartite semiregular if it is bipartite and the vertices belonging to the same part have equal degree.
We start with a simple lemma.
Lemma 3.5. If a bipartite signed graphĠ is associated with an oriented graph G , thenĠ is switching equivalent to its underlying graph if and only if every cycle of G is oriented uniformly.

ZORAN STANIĆ
Proof. The result follows since a cycle inĠ is positive if and only if the same cycle in G is oriented uniformly.
We now determine the characteristic polynomial of Λ(Ġ) in some particular cases.
Theorem 3.6. If G is a bipartite semiregular oriented graph with n 1 vertices of degree r 1 and n 2 (n 2 ≤ n 1 ) vertices of degree r 2 , such that all its cycles are oriented uniformly, then whereĠ is associated with G , β = n 1 r 1 − n 1 − n 2 and λ 1 , λ 2 , . . . , λ n2 are the first n 2 largest imaginary parts of the skew eigenvalues of G .
Proof. Since every cycle of G is oriented uniformly, from Lemma 3.5 we get thatĠ is switching equivalent to its underlying graph G, and therefore Φ G is its characteristic polynomial. By Corollary 3.3, Φ L(G) is the characteristic polynomial of Λ(Ġ). Further, we know form [2, Proposition 1.2.18] that the characteristic polynomial of the line graph of a bipartite semiregular graph is given by where β is defined in this theorem and ν 1 , ν 2 , . . . , ν n2 are the first n 2 largest eigenvalues of G. By the previous part of the proof, we may replace Φ L(G) with Φ Λ(Ġ) in (3.1). Finally, since G shares the spectrum withĠ, by Theorem 2.1, the first n 2 largest eigenvalues of G coincide with the first n 2 largest imaginary parts of the skew eigenvalues of G , and we are done.
Here is an immediate corollary concerning bipartite regular oriented graphs.

Corollary 3.7.
If G is a bipartite regular oriented graph with n vertices and m edges, such that all its cycles are oriented uniformly, then whereĠ is associated with G , r is the vertex degree and λ 1 , λ 2 , . . . , λ n are the imaginary parts of the skew eigenvalues of G .
In the last two statements we give some upper bounds on ρ(G ) expressed in terms of ρ(L(G)) (if G is bipartite) and ρ(L(bd(G))) (if G is non-bipartite). Proof. By Theorem 2.1, we have ρ(G ) = ρ(Ġ), where as beforeĠ is associated with G . Observe that the largest eigenvalue ofĠ does not exceed the largest eigenvalue of G, along with equality if and only ifĠ and G are switching equivalent; this fact is explicitly proved in [5]. Now, sinceĠ is bipartite, its spectrum is symmetric about the origin, and so we have that ρ(Ġ) ≤ ρ(G), with equality if and only if they are switching equivalent. By the result of Shi [4], we have ρ(G) ≤ 1 2 (ρ(L(G)) + 2) (with equality if and only if G is regular) and, if G is not a path, ρ(G) ≤ ρ(L(G)) (with equality if and only if G is a cycle). Thus, we get the inequalities of (i) and (ii). Equality cases follow by ρ(G ) = ρ(G) if and only ifĠ is switching equivalent to G, that is, by Lemma 3.5, if and only if the cycles of G are oriented uniformly.
Proof. Following the proof of the previous theorem, we get ρ(G ) = ρ(bd(Ġ)) ≤ ρ(bd(G)), (3.2) and then we get ρ(G ) ≤ 1 2 (ρ(L(bd(G))) + 2) and ρ(G ) ≤ ρ(L(bd(G))). It remains to show that, in these upper bounds, equality cannot occur. Assuming the contrary, we get that the inequality of (3.2) must reduce to equality, i.e., we get ρ(bd(Ġ)) = ρ(bd(G)), which implies that bd(Ġ) and bd(G) are switching equivalent. Let C l be an odd cycle (of G ) with vertices i 1 , i 2 , . . . , i l (indexed in the natural order), and let C 2l be the corresponding cycle of bd(G ). Then every vertex i u of C l produces its two copies in C 2l , say i u 1 and i u 2, and (according to the definition of a bipartite double) the vertices of C 2l are ordered as i 1 1, i 2 2, i 3 1, . . . , i l 1, i 1 2, i 2 1, i 3 2, . . . , i l 2. It follows also by definition that an edge i u i v of C l gives its two copies in C 2l , i u 1i v 2 and i u 2i v 1, whose orientation is determined by the orientation of i u i v in the sense that if i u → i v , then i u 1 → i v 2 and i u 2 → i v 1. This, together with the vertex ordering of C 2l , implies that C 2l contains an even number of edges oriented in the route direction. In other words, C 2l is not oriented uniformly, and thus the corresponding cycle of bd(Ġ) is negative. Hence, by Lemma 3.5, bd(Ġ) is not switching equivalent to its underlying graph, which contradicts our assumption.