New feasibility conditions for directed strongly regular graphs

We prove two results for directed strongly regular graphs that have an eigenvalue of multiplicity less than k, the common out-degree of each vertex. The first bounds the size of an independent set, and the second determines an eigenvalue of the subgraph on the out-neighborhood of a vertex. Both lead to new nonexistence results for parameter sets.


Introduction
Directed strongly regular graphs were defined by Art Duval [4] in 1988, as a directed version of strongly regular graphs.Starting with results of Klin et al in 2004 [8], there has been a lot of work on both constructions and nonexistence results.The website [3] maintains a table of feasible parameters.
In particular, many properties of strongly regular graphs extend to directed strongly regular graphs: the adjacency matrix is diagonalizable, the eigenvalues are integers ( [4]), and there is a version of the absolute bound ( [7], [5]).
In this paper, we find two nonexistence conditions for directed strongly regular graphs which are related to bounds for strongly regular graphs.One gives a bound on the size of an independent set, and the second gives information about an eigenvalue of the induced subgraph of the out-neighborhood of a vertex.For each, we give examples of feasible parameter sets ruled out by the condition.
We start with definitions and standard results.For a directed graph Γ and vertices i and j, write i → j if there is an edge from i to j (and this will include the case that there is also an edge from j to i).If there are edges both from i to j and from j to i, we say that i and j are adjacent vertices and write i ∼ j.As usual, the adjacency matrix of Γ is the matrix A whose i, j entry is 1 if i → j and 0 otherwise.
A directed graph Γ on v vertices is a directed strongly regular graph with parameters (v, k, t, λ, µ) if 0 < t < k and A satisfies the matrix equations where J is the all 1's matrix.In this case, we will say that Γ is a DSRG(v, k, t, λ, µ).Note that if k = t and these matrix conditions are satisfied, then Γ is an (undirected) strongly regular graph, while if t = 0 then Γ is a doubly regular tournament.
The matrix conditions are equivalent to the following structural conditions (which are often given as the definition).A digraph Γ is a DSRG(v, k, t, λ, µ) if and only if (a) Every vertex has in-degree and out-degree k, and is adjacent to t vertices.(b) Let i and j be distinct vertices.The number of vertices The adjacency matrix of a directed strongly regular graph is diagonalizable but not unitarily diagonalizable.However the all 1's vector j is an eigenvector with eigenvalue k, and all eigenvectors for other eigenvalues are orthogonal to j.The following theorem gives basic information about eigenvalues.
We will use θ, τ for the eigenvalues when we don't want to order them, and m θ , m τ for the corresponding multiplicities.
We will also need the following technical lemma about general digraphs.Recall that an independent set is a set of vertices such that the induced subgraph has no directed edges.
Lemma 2. Let Γ be a digraph with v vertices, with every vertex of in-degree 1.Then Γ has an independent set of size at least v 3 .
Proof.It is not hard to see that the strong components of Γ are all directed cycles (including those of length 2) or isolated vertices.Each weak component must be an isolated vertex, a directed cycle, a directed tree (an oriented tree with all directions away from the root), or a cycle with directed trees rooted at one or more of its vertices.We can take at least half of the vertices of the directed trees (excluding the root).For a directed cycle of length c, we can take c 2 of the vertices.The worst case occurs when all of the vertices (save 1 or 2, depending on v (mod 3)) are in directed cycles of length three.

Independent Sets
For a strongly regular graph which is not multipartite, interlacing shows that the size of an independent set is bounded above by the multiplicity of the negative eigenvalue, see [1], Theorem 9.4.1.We can derive a similar result for directed strongly regular graphs using the fact that A is diagonalizable.In fact, the proof also works for strongly regular graphs since the eigenvalues are given by the same formulas.Theorem 3. Let Γ be a DSRG(v, k, t, λ, µ) with eigenvalues k, θ, τ and multiplicities 1, m θ , m τ .Suppose Γ has an independent set Y of size c.If θ = 0, then c m τ .
Proof.Let B = J − I − A (the adjacency matrix of the complement of Γ), and let Using the fact that eigenvectors for τ and θ are orthogonal to j, easy calculations show that E τ is the projection onto the eigenspace for τ and hence is an idempotent of rank m τ .
The principal submatrix of E τ indexed by Y equals Since θ = 0 and k > 0, this matrix has full rank, namely rank c.Therefore c m τ .
We get the following immediate corollary, which surprisingly rules out parameter sets whose existence was open.
The corollary follows from the fact that in such a graph, the out-neighborhood of a vertex must be an independent set.This result can also be derived from Corollary 7.
We note that this rules out the following parameter sets.
Proof.The out-neighborhood of a vertex has k vertices, and it is easy to see that λ = 1 implies that each of these has in-degree 1.By Lemma 2 this neighborhood must contain an independent set of size at least k 3 .The result then follows from Theorem 3. It is less clear how to apply Theorem 3 for larger λ.For particular parameters, one may be able to apply the theorem directly.
The smallest parameter set ruled out by Corollary 5 has 585 points, and is thus too large to be listed in [3].Here's a list of some feasible parameter sets are ruled out by Corollary 5: For any nonzero vector z in the right nullspace of N M , N (N M ), M z is an eigenvector of A with eigenvalue τ .Let M z = w = (w i ); we will investigate the entries of w.
Since N w = N M z = 0, we have that w i = 0 for i ∈ {0, k + 1, . . ., v − 2}.Also, (Aw) 0 = τ w 0 = 0, and the first row of A is e i ) T w + w v−1 = j T w = 0 and hence w v−1 = 0.This shows that the support of w is contained entirely in X 1 , and hence restricting w to X 1 gives an eigenvector of A 1 with eigenvalue τ .If z 1 , . . ., z t is a basis for N (N M ), then M z 1 , . . ., M z t are also linearly independent since M has independent columns.Hence the right τ eigenspace of A 1 has dimension greater than or equal to the nullity of N M , which is greater than or equal to k − m θ .Proof.Use the same notation as Theorem 6, and note that the definition of parameter λ implies that the column sums of A 1 are λ, constant for all columns.Since A is a (0, 1) matrix, we can apply Perron-Frobenius theory, and hence λ is the spectral radius of A 1 .Therefore |τ | λ.