On the N-spectrum of oriented graphs

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 ∈ ℝ 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.

1. Given an eigenspace basis for eigenvalue λ ≠ 0 of G, the source parts of these vectors form an N-eigenspace basis for N-eigenvalue λ² of D and their sink parts are all N-eigenvectors for N-eigenvalue 0 of D.

2. Every eigenvector for eigenvalue 0 of G is also an N-eigenvector for N-eigenvalue 0 of D.

3. 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.

4. Given a basis of ℝ 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).

Theorem 2

For any bipartite graph G = (V, E), we have

where λ ₁(G) is the largest eigenvalue of G.

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

where ∥N _out(D)∥_∞ is the maximum row sum norm of the matrix N _out(D).

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 deg D − ( v ) = deg G ( v ) and deg D + ( u ) = deg G ( u ) . Therefore,

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

Consequently, λ 1 2 ( G ) ≤ 4 μ 1 ( D ) and finally λ 1 ( G ) ≤ 2 μ 1 ( D ) .□

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

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

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:

1. Any path induced by an odd column C 2 l + 1 has zig-zag orientation, with starting arc (2, 2l + 1)(1, 2l + 1).

1. Any path induced by an even column C 2 l has zig-zag orientation, with starting arc (1, 2l)(2, 2l).

1. Any path induced by an odd row ℛ 2 k + 1 has zig-zag orientation, with starting arc (2k + 1, 1)(2k + 1, 2).

2. Any path induced by an even row ℛ 2 k has zig-zag orientation, with starting arc (2k, 2)(2k, 1).

Then,

where d = ( ⌈ m 2 ⌉ − ⌊ m 2 ⌋ ) ⌊ n 2 ⌋ + ( ⌈ n 2 ⌉ − ⌊ n 2 ⌋ ) ⌈ m 2 ⌉ .

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 sink – a 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 D 1 = { ( i , j ) :   i − j is even } and D 2 = { ( i , j ) :   i − j is odd } .

Observe that ℬ 1 , k ≔ D 1 ∩ ℛ k contains exactly the sources of the subgraph of D induced by the vertex set ℛ k . Moreover, ℬ 2 , l ≔ D 2 ∩ C l contains exactly the sources of the subgraph of D induced by the vertex set C l . Since no other zig-zag paths exist in D apart from those induced by the separate rows and columns of the grid it follows that the sets

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). For each column that contains ⌈ n 2 ⌉ local sinks we need to add a zero N-eigenvalue. This happens ⌈ m 2 ⌉ times. For the case when m is odd we need to consider that the zig-zag paths contained in the grid rows, alternatingly, contain ⌊ m 2 ⌋ and ⌈ m 2 ⌉ sinks. A zero N-eigenvalue needs to be added whenever there are ⌈ m 2 ⌉ local sinks per row. Observing the slightly different pattern, we see that this happens only ⌊ n 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

but by Eq. (1) and Lemma 2 we have

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 .

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,

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,

Theorem 7

Let D = (V, E) be a digraph and I = {i ₁, i ₂,…,i _r } ⊂ V. Suppose that λ ₁ ≥ λ ₂ ≥ … ≥ λ _n are the N-eigenvalues of the digraph D and μ ₁ ≥ μ ₂ ≥…≥ μ _n+1 are the N-eigenvalues of D ^−I (resp. D ^+I ). Then,

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,

Note that A ^T A is the principal submatrix of (A ^−I ) ^T (A ^−I ). Applying the interlacing Theorem 5 on (A ^−I ) ^T (A ^−I ) and using the fact that the matrices (A ^−I ) ^T (A ^−I ) and (A ^−I )(A ^−I ) ^T have the same spectra (and likewise AA ^T and A ^T A), we get

It remains to prove μ ₁ ≤ λ ₁ + r. In order to do that consider

Now observe ∑ j = 1 r e i j ∑ k = 1 r e i k T = [ b s t ] , where

This implies that ∑ j = 1 r e i j ∑ k = 1 r e i k T has a maximum row sum of r. Now, since (A ^−I )(A ^−I ) ^T is a positive semi-definite matrix, we have

The proof for the case D ^+I works analogously.□

Theorem 8

Let D = (V, E) be a digraph, I = { v i 1 , v i 2 , … , v i r } ⊂ V any set of source vertices in D and D ^±I the digraph obtained from D by adding a new vertex η and connecting it to each vertex of I (with an arc of arbitrary orientation). Then,

where λ ₁ is the N-spectral radius of D and μ ₁ is the N-spectral radius of D ^±I .

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

so it follows that

where

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

Using the interlacing Theorem 5, we therefore obtain

but ∑ v i j ∈ N − ( η ) e i j ∑ v i j ∈ N − ( η ) e i j T is the matrix that describes the new common out-neighbors of the vertices of I. In fact, any two vertices in N⁻(η) have common out-neighbors, therefore, the matrix ∑ v i j ∈ N − ( η ) e i j ∑ v i j ∈ N − ( η ) e i j T has a maximum row sum of deg⁻(η). This implies λ ₁ − deg⁻(η) ≤ μ ₁. The other inequality can be proved in the same way.□

Corollary 1

Let D be a digraph, I = { v i 1 , v i 2 , … , v i r } ⊂ V ( D ) any set of sink vertices in D and D ^±I the digraph obtained from D by adding a new vertex η with incoming and outgoing arcs attached to the vertices in I. Then,

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,

where λ ₁(G) denotes the largest eigenvalue of G and μ ₁(D − u) denotes the largest N-eigenvalue of D − u.

Proof

Using Theorem 3, we get λ 1 ( G ) ≤ 2 μ 1 ( D ) . But Corollary 2 implies μ ₁(D) = μ ₁(D − u), hence the proof is complete.□