Spectral lower bounds for the orthogonal and projective ranks of a graph

The orthogonal rank of a graph $G=(V,E)$ is the smallest dimension $\xi$ such that there exist non-zero column vectors $x_v\in\mathbb{C}^\xi$ for $v\in V$ satisfying the orthogonality condition $x_v^\dagger x_w=0$ for all $vw\in E$. We prove that many spectral lower bounds for the chromatic number, $\chi$, are also lower bounds for $\xi$. This result complements a previous result by the authors, in which they showed that spectral lower bounds for $\chi$ are also lower bounds for the quantum chromatic number $\chi_q$. It is known that the quantum chromatic number and the orthogonal rank are incomparable. We conclude by proving an inertial lower bound for the projective rank $\xi_f$, and conjecture that a stronger inertial lower bound for $\xi$ is also a lower bound for $\xi_f$.


Introduction
For any graph G let V denote the set of vertices where |V | = n, E denote the set of edges where |E| = m, A denote the adjacency matrix, χ(G) denote the chromatic number, ω(G) denote the clique number, α(G) the independence number and G the complement of G. Let µ 1 ≥ µ 2 ≥ ... ≥ µ n denote the eigenvalues of A and then the inertia of G is the ordered triple (n + , n 0 , n − ), where n + , n 0 and n − are the numbers of positive, zero and negative eigenvalues of A, including multiplicities. Note that rank(A) = n + + n − and null(A) = n 0 . A graph is called non-singular if n 0 = 0.
Let D be the diagonal matrix of vertex degrees, and let L = D − A denote the Laplacian of G and Q = D + A denote the signless Laplacian of G. The eigenvalues of L are θ 1 ≥ . . . ≥ θ n = 0 and the eigenvalues of Q are δ 1 ≥ ... ≥ δ n .
Let χ v (G) denote the vector chromatic number as defined by Karger et al [13] and χ sv (G) denote the strict vector chromatic number. Karger et al [13] proved that χ sv (G) = θ(G), where θ is the Lovasz theta function [15], and let θ + denote Szegedy's [20] variant of θ. Let χ f (G) and χ c (G) denote the fractional and circular chromatic numbers and let χ q (G) and χ (r) q (G) denote the quantum and rank-r quantum chromatic numbers, as defined by Cameron et al [2].
Definition 1 (Orthogonal rank ξ(G)). The orthogonal rank of G is the smallest positive integer ξ(G) such that there exists an orthogonal representation, that is a collection of nonzero column vectors x v ∈ C ξ(G) for v ∈ V satisfying the orthogonality condition for all vw ∈ E.
The normalized orthogonal rank of G is the smallest positive integer ξ ′ (G) such that there exists an orthogonal representation, with the added restriction that the entries of each vector must all have the same modulus.
Let ξ f (G) denote the projective rank which was defined by Mancinska and Roberson [16], who showed that ω(G) ≤ ξ f (G) ≤ ξ(G). We use the definition of the r-fold orthogonal rank ξ [r] (G) due to Hogben et al. in [10,Section 2.1.] and their results in [10, Section 2.2.] to provide an equivalent and simpler definition of the projective rank.
Definition 2 (r-fold orthogonal rank ξ [r] (G) and projective orthogonal rank The r-fold orthogonal rank ξ [r] (G) is defined as follows: The projective rank, ξ f (G), is defined as follows: (G) r , and this limit exists.
The projective rank is also called the fractional orthogonal rank.
Clearly, the vectors x v ∈ C ξ(G) of an orthogonal representation correspond to the rank-1 orthogonal projectors It is also clear that ξ [1] (G) = ξ(G).
Definition 3 (Vectorial chromatic number χ vect (G)). Paulsen and Todorov [19] defined the vectorial chromatic number, χ vect (G), as follows. Let G = (V, E) be a graph and c ∈ N. A vectorial c-coloring of G is a set of vectors (x v,i : v ∈ V, i ∈ [c]) in a Hilbert space such that the following conditions are satisfied: The least integer c for which there exists a vectorial c-coloring will be denoted χ vect (G) and called the vectorial chromatic number of G.
Note that χ vect differs from χ v . Cubitt et al [4] (Corollary 16) proved the following (unexpected) equality between a chromatic number and a theta function: and provided an example of a graph with χ vect < χ q . Paulsen et al [18]

Hierarchies of graph parameters
There are numerous graph parameters that lie between the clique number and the chromatic number. The following chains of inequalities summarise the relationships between many of them, and combines results in Cameron et al [2], Mancinska and Roberson ([17] and [16]), Paulsen et al [18] and Elphick and Wocjan [6]. The chains are broken into two parts so the rightmost ends of (6) and leftmost ends of (7) coincide.
As illustrated above, Mancinska and Roberson ( [17] and [16]) demonstrated that ξ and χ q are incomparable, as are χ f and χ q ; and also χ f and ξ. They also proved that ξ f is a lower bound for ξ, χ q and χ f . Cubitt et al [4] demonstrated that χ vect and ξ f are incomparable. We can also demonstrate that χ vect and χ f are incomparable as follows. It is straightforward that for C 5 , χ vect > χ f . However if we consider the disjunctive product C 5 * K 3 , then from [4] χ vect (C 5 * K 3 ) ≤ 7 but χ f (C 5 * K 3 ) = 7.5, because χ f is multiplicative for the disjunctive product. Note that ξ, ξ ′ , χ vect , χ q , χ (1) q are integers, χ f is rational but ξ f may be irrational.
These hierarchies of parameters resolve a question raised by Wocjan and Elphick (see Section 2.4 of [21]) of whether χ v ≤ ξ ′ .
Wocjan and Elphick [22] proved that many spectral lower bounds for χ(G) are also lower bounds for χ q (G). In this paper we prove that many spectral lower bounds for χ(G) are also lower bounds for ξ(G). In Theorem 1 we prove an inertial lower bound for ξ(G) by strengthening a proof in [6]. In Theorem 2 we prove several eigenvalue lower bounds for ξ(G) by proving lower bounds for χ vect (G). We conjecture that all of these bounds are also lower bounds for ξ f (G), and make limited progress in this direction in Theorem 3.
Theorem 1 (Inertial lower bound for orthogonal rank). Let ξ(G) be the orthogonal rank of a graph G with inertia (n + , n 0 , n − ). Then Theorem 2 (Eigenvalue lower bounds for vectorial chromatic number). Let ξ(G) be the orthogonal rank and χ vect (G) be the vectorial chromatic number of a graph G. Then These bounds, reading from left to right, have been proved to be lower bounds for χ(G) by Hoffman [11], Lima et al [14] and Kolotilina [12].
Theorem 3 (Inertial lower bound for projective rank). Let ξ f (G) be the projective rank of a graph G with inertia (n + , n 0 , n − ). Then, In particular, when the graph G is non-singular the lower bounds in Theorems 1 and 3 coincide.

Remark 1.
All results also apply to weighted adjacency matrices W • A, where W is an arbitrary Hermitian matrix and • denotes the Hadamard product (also called Schur product).

Proof of the inertial lower bound on the orthogonal rank ξ(G)
Let f 1 , . . . , f n ∈ C n denote the eigenvectors of unit length corresponding to the eigenvalues Note that B and C are positive semidefinite and that rank(B) = n + and rank(C) = n − . Let denote the orthogonal projectors onto the subspaces spanned by the eigenvectors corresponding to the positive and negative eigenvalues respectively. Note that B = P + AP + and Proof. Assume to the contrary that rank(X) < rank(Y ). Then, there exists a non-trival vector v in the range of Y that is orthogonal to the range of X. Consquently, be an orthogonal representation. Note that we may assume that the first entries of these vectors are all equal to 1, that is, for the following reason. If we apply any unitary transformation U ∈ C ξ×ξ to x v we obtain an equivalent orthogonal representation y v = Ux v . Clearly, there must exist a unitary matrix U such that the resulting orthogonal representation y v = (y 1 v , . . . , y ξ v ) T satisfies the condition y 1 v = 0 for all v ∈ V due to a simple parameter counting argument. We can now rescale each vector to additionally achieve y 1 v = 1. We now have all the tools to prove Theorem 1.
be an orthogonal representation satisfying the additional condition x 1 v = 1 as in the remark above. We define ξ diagonal matrices We see that this sum is the zero matrix because all its entries s vw are zero either due to the orthogonality condition of the orthogonal representation x † v x w = 0 for vw ∈ E or due to a vw = 0 for vw ∈ E. Observe that D 1 = I due to the above remark. We obtain Equation (11) can be rewritten as Multiplying both sides by P − from left and right yields: Using that Then using that the rank of a sum is less than or equal to the sum of the ranks of the summands, that the rank of a product is less than or equal to the minimum of the ranks of the factors, and Lemma 1, we have that (ξ − 1)n + ≥ n − . Similarly (ξ − 1)n − ≥ n + is obtained by multiplying equation (11) by -1 and repeating the arguments (but multiplying by P + instead of P − from the left and right).

Proof of eigenvalue lower bounds on the orthogonal rank ξ(G)
Conditions (3) and (4) in Definition 3 imply that there exist orthogonal projectors P v,i ∈ C d×d and a unit (column) vector y ∈ C d such that the P v,i form a resolution of the identity I d i∈[c] for all v ∈ V and Let e v denote the standard basis (column) vectors of C n corresponding to the vertices v ∈ V so that A = v,w∈V a vw e v e † w . For i ∈ [c], define the block-diagonal projectors P i ∈ C n×n ⊗ C d×d by They form a resolution of the identity I n ⊗ I d , which follows by applying condition (12) to each block of these projectors. Moreover, condition (5) in Definition 3 implies that To abbreviate, set P = I n ⊗ yy † . Note that the multiplication of P i (A ⊗ I d )P i by the projector P from the left and right is a so-called compression.
We now make use of [22, Lemma 1] to construct unitary matrix U from the pinching The left hand side of this equation defines a so-called twirling of the matrix A ⊗ I d , which is special because all twirling unitaries can be chosen to be powers of U = k∈[c] ω k P k , where ω = e 2πi/c .
We obtain: for any diagonal matrix E ∈ C n×n . We now have the tools to prove Theorem 2. Note that we did not make use of condition (2).

Proof of the Lima bound in Theorem 2
Proof. The proof is almost identical to the proof for the chromatic number. We use the identity D − Q = −A. We have: Define the column vector v = 1 √ n (1, 1, . . . , 1) † ⊗ y. Multiply the left and right most sides of the above matrix equation by v † from the left and by v from the right to obtain This uses that v † (A ⊗ yy † )v = v † (D ⊗ yy † )v = 2m/n, which is equal to the sum of all entries of respectively A and D divided by n due to the special form of the vector v, and that

Proof of the Hoffman and Kolotilina bounds in Theorem 2
Proof. Let E ∈ C n×n be an arbitrary diagonal matrix. Using (13) and (14), we obtain Using that λ max (X) ≥ λ max (P XP ) and λ max (X) + λ max (Y ) ≥ λ max (X + Y ) for arbitrary Hermitian matrices X and Y , we obtain [5, Corollary 5] shows that the above eigenvalue bound implies or equivalently from which the Hoffman and Kolotilina bounds are obtained by setting E = 0 and E = D, respectively.

Inertial and generalized Hoffman and Kolotilina bounds
We do not know whether the inertial bound in Theorem 1 or the generalized (multi-eigenvalue) bounds in [5] are also lower bounds for the vectorial chromatic number. The difficulty seems to be in determining what happens to the entire spectrum of the various matrices when they are compressed by P = I n ⊗ yy † . The Kolotilina and Lima bounds only use the maximal and/or minimal eigenvalues. Bilu [1] proved that the Hoffman bound is a lower bound for χ v (G). The Kolotilina and Lima bounds equal the Hoffman bound for regular graphs, but we do not know if these bounds lower bound the vector chromatic number for all graphs.

Proof of the inertial lower bound on the projective rank ξ f (G)
We conjecture that for all graphs G the projective rank ξ f (G) is lower bounded by Unfortunately, we are not able to settle this question by either providing a counterexample or proving this bound for all graphs. However, we are able to prove the weaker lower bound in Theorem 3. We derive two lemmas to better organize the proof of Theorem 3.
Lemma 2. Let P be an orthogonal projector and X a positive semidefinite matrix in C m×m . Then, we have rank(P XP ) ≥ rank(P ) − null(X) .
Proof. There exist positive semidefinite matrices Y and ∆ such that Y has full rank, ∆ has rank null(X), and X + ∆ = Y . Using that rank(M + N) ≤ rank(M) + rank(N) for arbitrary matrices, we obtain rank(P XP ) + rank(P ∆P ) ≥ rank(P Y P ) .
Using that rank(MN) ≤ rank(M) for arbitrary matrices M and N, we obtain rank(P XP ) ≥ rank(P Y P ) − rank(∆) .
Lemma 3. Let P v be the projectors of a (d/r)-orthogonal representation. Define the block diagonal projector Then, we have P (A ⊗ I d )P = 0 n ⊗ 0 d and rank(P ) = nr .
Proof. This follows directly from the orthogonality condition P v P w = 0 d for all vw ∈ E. We refer the reader to [22], where a similar result is proved for the quantum chromatic number. The projectors P v have rank r for all v ∈ V so rank(P ) = nr.
We are now ready to prove Theorem 3.
Proof. Let A = B − C, defined as in Section 3, so rank(B) = n + and rank(C) = n − . Note that Lemma 3 implies Clearly, the rank of the left hand side of (15) is bounded from above by n + d = rank(B ⊗ I d ).
We now bound the rank of the right hand side of (15) from below. Observe that B + C = |A|, where |A| = n i=1 |µ i |e i e † i and µ i and e i are the eigenvalues and eigenvectors of A, respectively. Clearly, |A| is positive semidefinite, its rank is equal to rank(A) = n + + n − and its nullity is equal to null(A) = n 0 . Therefore, |A| ⊗ I d is positive semidefinite, its rank is equal to (n + + n − )d and its nullity is equal to n 0 d. We can now apply Lemma 2 to obtain rank P |A| ⊗ I d P ≥ rank(P ) − n 0 d = nr − n 0 d .
Combining the upper and lower bounds on the ranks, we obtain The result d r ≥ 1 + n + n − + n 0 is obtained by considering P (C ⊗ I d )P on the left hand side of (15).

On the equivalence of ξ [r] (G) and ξ(G [r] )
We now show that the r-fold orthogonal rank ξ [r] (G) of G is equal to the orthogonal rank ξ(G [r] ), where G [r] arises from G by a simple graph operation. The relationship between the orthogonal projectors P v and the vectors x v,i is given by so that the following two properties hold: For each v ∈ V , the vectors x v,k for k ∈ [r] form an orthonormal basis for the subspaces S v , which form a so-called (d; r) orthogonal subspace representation [10].
) is defined to be the lexicographic product of G and K r , where K r denotes the complete graph on r vertices. The adjacency matrix A [r] of G [r] is given by where J r is the all-one-matrix of size r × r and K r = J r − I r is the adjacency matrix of the complete graph on r vertices.
, and its edge set E [r] is defined as follows: for all vw ∈ E and all k, ℓ ∈ [r].
Observing that the above two conditions precisely fit the two conditions in the remark above, the following lemma is readily proved: Lemma 4. For any graph G, the r-fold orthogonal rank ξ [r] (G) of G is equal to the orthogonal rank ξ(G [r] ) of the r-fold vertex-cloned graph G [r] , that is Consequently, we have: r .
Unfortunately, we do not know how to leverage this lemma to prove a useful spectral lower bound on the projective rank.

Implications for the projective rank
The following examples demonstrate that the inertial bound is exact for ξ f for various classes of graphs. We also use Theorem 3 to derive the value of ξ f for some graphs.
The Clebsch graph is non-singular, so using Theorem 3 and that ξ f ≤ χ f , it follows that ξ f = 3.2.
More generally, if the inertial bound is exact for the fractional chromatic number of a non-singular graph, then it is also exact for the projective rank. Vertex transitive graphs have ξ f ≤ χ f = n/α, so if a non-singular vertex transitive graph has α = min (n + , n − ), then ξ f = χ f = n α .

Conclusion
We have proved that many lower bounds for χ(G) are also lower bounds for ξ(G). We have also proved that for non-singular graphs 1 + max n + n − , n − n + ≤ ξ f (G).
Elphick and Wocjan [6] proved this lower bound for χ f for non-singular graphs, using a simpler proof technique. Costello et al [3] proved that almost all (random) graphs with no isolated vertices are non-singular. This provides limited support for our conjecture that all of the spectral lower bounds described in this paper are also lower bounds for ξ f (G), and consequently for χ f (G).