On the N-spectrum of oriented graphs

Abstract Given any digraph D, its non-negative spectrum (or N-spectrum, shortly) consists of the eigenvalues of the matrix AA T , where A is the adjacency matrix of D. In this study, we relate the classical spectrum of undirected graphs to the N-spectrum of their oriented counterparts, permitting us to derive spectral bounds. Moreover, we study the spectral effects caused by certain modifications of a given digraph.

On the other hand, research on the spectrum of directed graphs is relatively rare compared to that of undirected graphs. The main reason is that the adjacency matrix of a digraph in general is difficult to work with. It is not symmetric (so we lose the property that eigenvalues are real) and not diagonalizable (hence the dimension of an eigenspace may differ from the algebraic multiplicity of the corresponding eigenvalue in the characteristic polynomial).
Many researchers have proposed definitions of Hermitian matrices to represent digraphs. For example, Guo and Mohar (see [9]) defined the so-called Hermitian adjacency matrix of a digraph D as follows: given a digraph D with vertex set V and arc set E, the Hermitian adjacency matrix of D is the matrix H(D) If D is an oriented graph, then the above definition can be tailored to the matrix iA − iA T , where A is the well-known adjacency matrix of D.
Another way of defining an alternative version of the adjacency matrixas it was done by Jovanović in [10] is to consider the eigenvalues of the matrix ( ) ≔ N D AA T out for any given digraph D, where A is the classical adjacency matrix of D. N out (D) is symmetric, so it has many of the well-known benefits of the classical adjacency matrix of an undirected graph. Moreover, N out (D) is positive semi-definite, hence Jovanović called the spectrum of this matrix the non-negative spectrum of a digraph or, shortly, the N-spectrum. Note the relation of this definition to the singular values of a digraph D, defined to be the square roots of the eigenvalues of the matrix AA T .
In what follows, we will use the N-spectrum of oriented graphs as a tool to find upper bounds on the spectral radius, in particular of bipartite graphs. Moreover, we investigate how modifying a digraph will affect the N-spectrum of this digraph.

Preliminaries
A directed graph (or digraph) D = (V, E) consists of a set of vertices V = {v 1 , v 2 ,…,v n } and a set of arcs E ⊆ V × V. Hereafter, we assume that all digraphs considered are loopless, i.e., E does not contain any entries of the form Conversely, we call v i an outgoing neighbor (or out-neighbor) of v j . The set of outgoing (resp. incoming) neighbors of a vertex v i is denoted by N + (v i ) (resp. N − (v i )). Moreover, we define the out-degree deg A vertex without any incoming neighbors is called a source, whereas a vertex without outgoing neighbors is termed a sink.
A digraph with symmetric relation E can be interpreted as an (undirected) graph G, by identifying arc pairs (v i , v j ) and (v j , v i ) and interpreting them as a single edge between the two vertices. Hence, the two notions of neighborhood (resp. degree) coincide so that we write N(v i ) for the set of neighbors of v i in G and deg(v i ) = |N(v i )|. The multiset σ(G) of eigenvalues λ 1 (G) ≥…≥ λ n (G) of the (symmetric) adjacency matrix of G (in which the multiplicity of each eigenvalue matches the dimension of its eigenspace) is called the spectrum of G. The largest eigenvalue is called the spectral radius of G.
Given a digraph D with adjacency matrix ∈ × A n n , the eigenvalues μ 1 (D) ≥…≥ μ n (D) of the symmetric are called the N-eigenvalues of D. The largest (N-)eigenvalue modulus is achieved by λ 1 and called the (N-)spectral radius. The multiset σ N (D) of N-eigenvalues of D is called the N-spectrum of D. As mentioned in Section 1, these definitions are originally motivated by the fact that AA T is positive semidefinite. Observe that an alternate definition can be obtained by considering the matrix A T A. Determining the N-eigenvalues according to one definition yields the same result as if using the alternate definition on the digraph after the directions of all of its arcs have been reversed.
Note that the entry of AA T found at position (i, j) represents the number of common out-neighbors of the vertices v i and v j in D. In particular, the entry found at position (i, i) represents the out-degree of the vertex v i . We immediately deduce that the trace of AA T equals |E|, which in turn equals the sum of all N-eigenvalues of D (where we count each eigenvalue according to its multiplicity). As a consequence, D contains only isolated vertices if and only if all of its N-eigenvalues are zero. When it comes to analyzing the N-spectrum of a digraph, it is possible to conveniently choose a vertex order such that the matrix AA T assumes the block diagonal form. Following [11], we define a zig-zag trail (of length 2l) between two vertices x and y to be a sequence of arcs such that it starts with a forward arc from x, then a reverse arc, then again a forward arc, and so on, until we reach y with a backward arc. A path of length zero is also considered a zig-zag trail. Next, two vertices shall be related to each other if they have a zig-zag trail between them. It is easy to see that this relation is an equivalence relation. Therefore, it partitions the vertex set V into subsets { … } , , , k . This partition is called common out-neighbor partition. Renumbering the vertices of V starting with vertices in 1 then 2 and so on will result in the desired block diagonal matrix form. This follows directly from the interpretation of the entries of AA T that we gave in the previous paragraph.
As an example, consider the digraph in Figure 1. The grey vertices form the largest block of the common out-neighbor partition. The black vertices form a block of two vertices, whereas each of the white vertices forms a singleton block. The chosen vertex order is consistent with the proposed way of numbering the vertices according to the common out-neighbor partition. Hence, we get the following matrix: Using the common out-neighbor partition, it is possible to relate the N-eigenvalues of certain directed bipartite graphs to the classical eigenvalues of their undirected counterparts. A bipartite graph is a graph whose vertex set V can be partitioned into two subsets V 1 , V 2 such that neither of the two subgraphs induced by V 1 resp. V 2 contains any edges. Theorem 1. (Square Theorem [11]). Let D be a bipartite digraph such that each vertex is either a source or a sink. For any vector ∈ x n , we may construct its source part by setting all those entries of x to zero which correspond to the non-sources (i.e. sinks) of D. Likewise, we construct the sink part of x. Let k be the number of sources and l be the number of sinks in D. Furthermore, let G be the underlying undirected graph of D.
(i) Given an eigenspace basis for eigenvalue λ ≠ 0 of G, the source parts of these vectors form an N-eigenspace basis for N-eigenvalue λ 2 of D and their sink parts are all N-eigenvectors for N-eigenvalue 0 of D. (ii) Every eigenvector for eigenvalue 0 of G is also an N-eigenvector for N-eigenvalue 0 of D. (iii) If the source part of any N-eigenvector for N-eigenvalue 0 of D is not null, then this source part is an eigenvector for eigenvalue 0 of G. (iv) Given a basis of + k l of eigenvectors of G, an N-eigenspace basis for N-eigenvalue 0 of D can be constructed as follows. Collect the sink parts of all the vectors associated with positive eigenvalues of G, together with the vectors associated with eigenvalue 0. Alternatively, collect the source parts of all vectors of the given basis and determine a maximal linearly independent subset of resulting set, together with l unit vectors, one for each sink (such that it is non-zero exactly on the considered sink).

Upper bounds for the spectrum of a graph
Study of the spectral radius of graphs has been a prominent research topic in the spectral graph theory since its inception in the 1950s to this day. In 1990, Cvetković and Rowlinson (see [12]) surveyed numerous results on the spectral radius. This is complemented by a more recent survey by Stevanović (see [13]). In this section, we introduce new upper bounds for the spectral radius of graphs. We use the N-spectra of certain oriented graphs to establish these bounds.
Recall that a graph is bipartite if and only if its vertex set can be partitioned into two sets such that the end points of each edge belong to different sets. For what follows, we introduce the notion of zig-zag orientation of a bipartite graph. To this end, we simply choose one set of its vertex bipartition and orient all arcs of the graph toward this set. As a result, we obtain an oriented graph in which each vertex is either a source or a sink.
Proof. Let D be a digraph obtained by zig-zag orientation of the given graph G such that the vertices are numbered according to the common out-neighbor partition. Let N out (D) = [a ij ]. Using Theorem 1 and observing that the largest singular value of a matrix M is equal to its spectral matrix norm ∥M∥ (see [14]), we obtain Note that the row in N out (D) corresponding to the vertex u has row sum equal to deg + (u) plus the number of common out-neighbor vertices with vertices in the same partite class as u. However, the number of common out-neighbor vertices with vertices in the same partite class of u is exactly Since D has a zig-zag orientation, we have Let us mention that the above upper bound is related to the Brualdi-Hoffman conjecture, which states for any graph G (see [15]). This conjecture has already been settled by Bhattacharya et al. [16]. In the special case of bipartite graphs, the upper bound given in Theorem 2 is tighter than the upper bound given by the Brualdi-Hoffman conjecture.
The following theorem gives an upper bound for the spectral radius of any graph G in terms of the N-spectral radius of some orientation of this graph into a digraph D.
Theorem 3. Let G be a graph and D an oriented graph whose unoriented counterpart is G. Then, where λ 1 (G) is the largest eigenvalue of G and μ 1 (D) is the largest N-eigenvalue of D.
Proof. Suppose that the adjacency matrix of D is A. Then, the adjacency matrix of G is A + A T . Now observe that It follows that and finally ( ) ≤ ( ) λ G μ D 2 1 1 . □ An immediate question arises when thinking about applying Theorem 3: which orientation yields the lowest upper bound? In the following we turn our attention to grid graphs and show that the inequality mentioned in Theorem 3 is sharp.
Let P i denote the path with i vertices. A grid is a graph that is isomorphic to the Cartesian product P n × P m . For the sake of simplicity, we shall assume that its vertex set is the set of all pairs (i, j) such that i ∈ {1,…,n} and j ∈ {1,…,m}. Thus, two vertices (i, j), (k, l) of a grid are adjacent if and only if either |i − k| = 1 and j = l or if i = k and |j − l| = 1. Figure 2 depicts a grid P 5 × P 6 that has been oriented. Note that the chosen layout gives us a notion of the vertical/horizontal direction. We find that vertex (i, j) lies in row number i and column number k of the grid. With this image before our eyes, we define the rows = {( )} = … k j , Lemma 1. Let D be an oriented graph obtained from zig-zag orientation of a given path P n such that at least one of its leaves becomes a source. Then, the characteristic polynomial of the principal submatrix of N out (D) corresponding to the sources of D is x πj n 4 cos 1 . Proof. The claim is implicit in the proof of Corollary 4 in [11]. To prove it, we recall the well-known fact (see e.g. [4]) that the eigenvalues of P n are       + = … πj n j n 2 cos 1 , for 1, , .
For     = … j 1, , n 2 these eigenvalues are distinct positive numbers. Using Theorem 1, they can be directly attributed to the sources of D. If n is odd, one needs to add a zero eigenvalue, namely the one obtained for     = j n 2 . Thus, we get the formula which then can be simplified to get the desired result. □ Lemma 2. Let D be the oriented graph obtained from orienting a given grid P n × P m as follows: -Any path induced by an odd column + l 2 1 has zig-zag orientation, with starting arc (2, 2l + 1)(1, 2l + 1). − Any path induced by an even column l 2 has zig-zag orientation, with starting arc (1, 2l)(2, 2l). -Any path induced by an odd row + k 2 1 has zig-zag orientation, with starting arc (2k + 1, 1)(2k + 1, 2). -Any path induced by an even row k 2 has zig-zag orientation, with starting arc (2k, 2)(2k, 1).
Then, where Proof. First observe that, by the definition of the particular orientation, there exist neither sources nor sinks in D. Now suppose that (i,j) and (i + 1, j + 1) have a common out-neighbor, say (i, j + 1). Since both the path induced by the vertices (k, j + 1) with k ∈ {1,…,n} and the path induced by the vertices (i, k) with k ∈ {1,…,m} have zig-zag orientation, it actually follows that (i, j + 1) must be a sinka contradiction. As a result, common neighbors can only occur along those zig-zag paths that were explicitly mentioned in the definition of the chosen orientation. No other zig-zag paths exist in D.
Let us divide the vertices according to whether they belong to an even or odd diagonal of the grid. To this end, let is even is odd 2 . Observe that ≔ ∩ form the common out-neighbor partition of D. Hence, we may use Lemma 1 to determine the characteristic polynomial of N out (D) on a per-block basis, where each block can also be obtained "locally" by considering a suitable zig-zag-oriented path and determining the block of its N out matrix corresponding to the sources of that path. Moreover, keep in mind that the zig-zag paths of two neighboring columns or rows can be transformed into each other by reversing the orientation. Hence, their N-spectrum is the same. Let us define If both n and m are even, then it is a direct consequence of Lemma 1 that χ(D;x) = S n (x) m S m (x) n . In the case of odd n we need to consider that the zig-zag paths contained in the grid columns, alternatingly, contain     n 2 and     n 2 sinks (as can easily be seen from Figure 2 times. All in all, we obtain χ(D;x) = S n (x) m S m (x) n x d . □ Theorem 4. The minimum N-spectral radius of any oriented grid P n × P n can be achieved by using the orientation described in Lemma 2.
Proof. Let D be the digraph with underlying graph P n × P n and orientation as described in Lemma 2. It follows from Theorem 3 that Example 1. Consider the grid P 5 × P 5 . It has a spectral radius of approximately 3.46. As predicted by Theorem 4, this matches the bound we deduce from the N-spectral radius 3 achieved by zig-zag orientation of the given grid. Alternatively, orient all "horizontal" arcs from left to right and all "vertical" arcs from top to bottom. Then, we obtain an N-spectral radius of approximately 3.62, which only yields an upper bound of 3.81 for the spectral radius of the grid P 5 × P 5 .

Effects of modification
It is interesting to study how modifying a given digraph reflects in changes of its N-spectrum. Several modifications have already been considered in [10] and [11]. In this section, we want to focus on N-eigenvalue interlacing. Probably, the most classical theorem about eigenvalue interlacing is the following.
Theorem 5. (cf. [4]). Let G be a graph (resp. a symmetric matrix) with eigenvalues λ 1 ≥…≥ λ n and H an induced subgraph (resp. a principal submatrix) of G with eigenvalues μ 1 ≥…≥ μ m . Then, The special case m = n − 1 gives us the following inequality chain: When it comes to the N-spectrum, deriving any helpful interlacing theorems at all turns out to be quite a challenge. At least, Jovanović has been able to derive the following "edge version" of the interlacing theorem. Theorem 6. (cf. [10]). Let D be a weakly connected digraph of order n, with N-eigenvalues λ 1 ≥…≥ λ n . Assume that D has at least one vertex v with in-degree one and obtain D′ from D by removing the incoming arc at v. Now let μ 1 ≥…≥ μ n be the N-eigenvalues of D′. Then, Let us derive another N-eigenvalue interlacing theorem. To this end, let D = (V, E) be a digraph and I ⊂ V. By D −I (resp. D +I ) we denote the digraph obtained by adding a sink (resp. source) to the digraph D such that its set of in-neighbors (resp. out-neighbors) is I. Furthermore, let A −I (resp. A +I ) denote the adjacency matrix of D −I (resp. D +I ). Proof. In this proof and also later on we shall use subscripts of the form r × s to document the dimensions of some annotated matrix, in order to make the overall block structure more transparent. Moreover, if a mere number is affixed with such a dimensional annotation, then this denotes a matrix of the mentioned dimensions where all entries are equal to that number. The adjacency matrix of D −I is given by where, as usual, e k denotes the kth unit vector (of the dimension implied by the context). Therefore, The proof for the case D +I works analogously. □ The effect of adding a vertex with both incoming and outgoing arcs on the entire N-spectrum can be hardly predicted. However, the following two theorems describe the effect of such an operation on the N-spectral radius of oriented digraphs.
Proof. Observe that η only gets connected to vertices that originally were sources, but some of the old source vertices contained in I may lose their source property in the new digraph. This needs to be analyzed. The adjacency matrix of D ±I is given by Note that Q describes the new common out-neighbors of the vertices in V with the vertex η. Since all vertices in I are source vertices in D, there is no new common out-neighbor between V and η in D ±I . Therefore, Q = (0) n×1 , and by symmetry R = (0) 1×n . Furthermore, the single entry in S is equal to deg − (η), so where λ 1 is the N-spectral radius of D and μ 1 is the N-spectral radius of D ±I .
Corollary 2. Adding a new source (resp. sink) vertex to a digraph D attached to some source (resp. sink) vertices in D does not change the N-spectral radius.
Theorem 9. Let G = (V, E) be a graph and u ∈ V such that no two vertices in N(u) are adjacent. Furthermore, let D be an arbitrary orientation of G, as long as all vertices in N(u) become sinks (resp. sources) in D − u and u becomes a sink (resp. source) in D. Then, ( ) ≤ ( − ) λ G μ D u 2 , 1 1 where λ 1 (G) denotes the largest eigenvalue of G and μ 1 (D − u) denotes the largest N-eigenvalue of D − u.