Posted on 4th July 2023

Recently Clive Elphink gave us a nice seminar where he discussed some of his "conjectures in spectral graph theory". Clive has an interesting history: after a career in business, he is now semi-retired and returned to research mathematics. By his own admission, he is more on the experimental side, and his talk said almost nothing about proofs. Here is a old result in this area:

Theorem [Hoffman]:We have that \( \chi(G) \geq 1 + \frac{\lambda_1}{-\lambda_n} \).

I explain the notation shortly, but in words, this result relates a vertex colouring of a graph to the eigenvalues of the adjacency matrix. To me, this seems hugely surprising, as why would the spectrum of \( A \) have anything to do with a vertex colouring? I want to explain the elegant arguments of Elphick and Wocjan, and also some generalisations to quantum colourings (in part 2).

Let us set the scene. Here a *graph* is \( G=(V,E) \) where \( V \) is a finite set of vertices, and \( E \) is a set of undirected edges. We disallow loops, and between a pair of vertices there can be at most one edge, so our graph is *simple*. Often we write \( u\sim v \) to indicate there is an edge between vertices \( u \) and \( v \). Set \( n=|V| \) and \( m=|E| \); sometimes we identify \( V \) with \( [n]=\{1,2,\cdots,n\} \). A *colouring* of \( G \) is an assignment \( f:V\rightarrow[c] \) such that \( u\sim v \implies f(u)\not=f(v) \), that is, adjacent vertices are coloured distinctly. The minimal \( c \) we can choose is the *chromatic number* \( \chi(G) \).

The *adjacency matrix* of \( G \) is \( A=A_G \), the \( \{0,1\} \)-valued matrix indexed by \( V\times V \) with \( A_{u,v}=1 \) if and only if \( u\sim v \). As \( G \) is undirected, \( A \) is symmetric/hermitian, and so has real eigenvalues and a complete set of orthogonal eigenvectors. Let \( \lambda_1 \geq \lambda_2 \geq \cdots \geq \lambda_n \) be the eigenvalues listed in non-increasing order.

Lemma:If \( G \) is non-trivial, then \( \lambda_1>0 \). Further, \( \sum_i \lambda_i=0 \) and so \( \lambda_n < 0 \).

Proof:Let \( (e_u) \) be the standard unit vector basis of \( \mathbb C^V \), choose \( u\sim v \), and set \( \xi=e_u+e_v \). Then \( (\xi|A\xi) = A_{u,u} + A_{u,v} + A_{v,u} + A_{v,v} = 2 \), so \( -A \) is not positive. Thus there must be some strictly positive eigenvector, so \( \lambda_1>0 \).The sum of the eigenvalues is the trace of \( A \), which is \( 0 \).

That \( \lambda_1>0 \) is also a consequence of the Perronâ€“Frobenius theorem.

As this is an informal writeup, I will not give precise references, but instead list some further reading at the end. For now, I follow a couple of papers of Elphick and Wocjan. The first hint of a link between colourings and matrices is the observation that given a colouring \( f:V\rightarrow [c] \) (wlog every colour is used), we can permute the vertices to list them in colour order, that is, find \( 1=n_0 < n_1 < \cdots < n_{c-1} < n_{c}=n \) so that \( f(u)=i \) exactly when \( n_{i-1} \leq u < n_i \). We partition the standard basis \( (e_i)_{i=1}^n \) of \( \mathbb C^n \) according to this partition, and hence view matrices in \( \mathbb M_n \) as \( c\times c \) block matrices. As vertices which share a colour cannot be adjacent, if we view \( A \) as such a block matrix, the diagonal blocks are all zero. \[ A = \begin{pmatrix} 0_{n_1} & * & \cdots & * \\ * & 0_{n_2-n_1} & \vdots & * \\ \vdots & \ddots & \ddots & \vdots \\ * & * & \cdots & 0_{n-n_{c-1}} \end{pmatrix} \]

Letting \( P_i \) be the orthogonal projection on the \( i \)th block, that is, the span of \( e_{n_{i-1}},\cdots,e_{n_i-1} \), we equivalently have that \( P_i A P_i = 0 \) for each \( i\in [c] \). This sort of operation has a name in Quantum Information Theory (QIT).

Definition:Let \( (P_i)_{i=1}^c \) be orthogonal projections which sum to \( 1 \in \mathbb M_n \). The operation \[ \mathcal C: \mathbb M_n\rightarrow \mathbb M_n; \quad x\mapsto \sum_{i=1}^c P_i x P_i \] is called apinching.

There is a seemingly unrelated operation using unitary matrices.

Definition:Let \( (U_i)_{i=1}^d \) be a collection of unitary matrices in \( \mathbb M_n \). The operation \[ \mathcal D: \mathbb M_n\rightarrow \mathbb M_n; \quad x\mapsto \frac{1}{d} \sum_{i=1}^d U_i x U_i^* \] is called atwirling.

In fact, any pinching can be expressed as a twirling. Let \( (P_i)_{i=1}^c \) be orthogonal projections which sum to \( 1 \) and let \( \zeta \) be a \( c^{\text{th}} \) root of unity (so \( \zeta^c=1 \) and \( \zeta^k\not=1 \) for \( 1\leq k < c \); e.g. we could have \( \zeta = e^{2\pi i/c} \)). Set \[ U = \sum_{k=1}^c \zeta^k P_k \quad\implies\quad U^*U = \sum_{k,l=1}^c \zeta^{-l} \zeta^k P_l P_k = \sum_{k=1}^c \zeta^{k-k} P_k = 1, \] and similarly \( UU^*=1 \), so \( U \) is unitary. Here we used that the projections are orthogonal and sum to \( 1 \). Similarly, that the projections are orthogonal shows that \( U^t = \sum_{k=1}^c \zeta^{kt} P_k \). For any matrix \( x \), \[ \mathcal D(x) = \frac1c \sum_{k=1}^c U^k x (U^*)^k = \frac1c \sum_{k,t,s=1}^c \zeta^{k(t-s)} P_txP_s = \sum_{t=1}^c P_txP_t = \mathcal C(x), \] using that \( \frac{1}{c} \sum_{k=1}^c \zeta^{kr} = \delta_{r,0} \).

We can now prove Hoffman's bound.

Proof (of Hoffman's Theorem):As argued above, if \( G \) admits a vertex colouring using \( c \) colours then, after permuting the vertices, we can find "coordinate" projections \( (P_i)_{i=1}^c \) such that the pinching of the adjacency matrix satisfies \( \mathcal C(A)=0 \). (Here "coordinate projection" means a projection onto the span of some of standard unit vector basis elements \( (e_i)_{i=1}^n \).)Form \( U \) as above, so the twirling also satisfies \( \mathcal D(A)=0 \). As \( \zeta^c=1 \) we see that \( U^c = \sum_{k=1}^c P_k = 1 \) and hence \[ \mathcal D(A)=0 \implies A = \sum_{k=1}^{c-1} U^k(-A)(U^*)^k. \] Let \( v \) be a unit length eigenvector for the eigenvalue \( \lambda_1 \), so \( (v|Av) = (v|\lambda_1 v) = \lambda_1 \) while \[ \sum_{k=1}^{c-1} (v|U^k(-A)(U^*)^kv) = \sum_{k=1}^{c-1} ( (U^*)^k v | (-A) (U^*)^k v ) \leq (c-1) \lambda_{\max}(-A), \] where \( \lambda_{\max}(-A) = -\lambda_n \) is the greatest eigenvalue of \( -A \). Here we used that for each \( (U^*)^k v \) is a unit vector, for any \( k \).

We have hence shown that \( \lambda_1 \leq (c-1) (-\lambda_n) \) or equivalently, \( c \geq 1 + \frac{\lambda_1}{-\lambda_n} \). As \( \chi(G) \) is the minimal choice for \( c \), this establishes Hoffman's bound.

Many other spectral bounds on \( \chi(G) \) can be established in a rather similar way; for details see the papers by Elphick and Wocjan listed below.

Bhatia, Rajendra "Pinching, trimming, truncating, and averaging of matrices." Am. Math. Mon. 107, No. 7, 602-608 (2000). Zbl 0984.15024

Elphick, Clive; Wocjan, Pawel "Spectral lower bounds for the quantum chromatic number of a graph." J. Comb. Theory, Ser. A 168, 338-347 (2019). Zbl 1421.05042

Elphick, Clive; Wocjan, Pawel "An inertial lower bound for the chromatic number of a graph." Electron. J. Comb. 24, No. 1, Research Paper P1.58, 9 p. (2017). Zbl 1358.05104

Elphick, Clive; Wocjan, Pawel "Unified spectral bounds on the chromatic number." Discuss. Math., Graph Theory 35, No. 4, 773-780 (2015). Zbl 1326.05080

Wocjan, Pawel; Elphick, Clive "New spectral bounds on the chromatic number encompassing all eigenvalues of the adjacency matrix." Electron. J. Comb. 20, No. 3, Research Paper P39, 18 p. (2013). Zbl 1295.05112