To motivate a question about the nonnegativity of the Dirichlet energy of a weighted graph, we first discuss some background on curvatures and flows on complex manifolds.

Definition 1. Let $(M^n, \omega )$ be a Hermitian manifold. The quadratic orthogonal bisectional curvature (from now on, QOBC) is the function

$$ \begin{align*}\mathrm{QOBC}_{\omega} : \mathcal{F}_M \times \mathbb{R}^n \backslash \{ 0 \} \to \mathbb{R}, \quad \mathrm{QOBC}_{\omega}: (\vartheta, v) \mapsto \frac{1}{| v |_{\omega}^2} \sum_{\alpha, \gamma =1}^n R_{\alpha \overline{\alpha} \gamma \overline{\gamma}} (v_{\alpha} - v_{\gamma})^2,\end{align*} $$

where the $R_{\alpha \overline {\alpha } \gamma \overline {\gamma }}$ denote the components of the Chern connection of $\omega $ with respect to the unitary frame $\vartheta $ (a section of the unitary frame bundle $\mathcal {F}_M$ ).

This curvature first appeared implicitly in [Reference Bishop and Goldberg1] and is the Weitzenböck curvature operator (in the sense of [Reference Petersen12, Reference Petersen and Wink13, Reference Petersen and Wink14]) acting on real $(1,1)$ -forms. (See [Reference Broder3] for alternative descriptions of the QOBC.) From [Reference Li, Wu and Zheng9], the QOBC is strictly weaker than the orthogonal bisectional curvature $\text {HBC}_{\omega }^{\perp }$ (the restriction of the holomorphic bisectional curvature $\text {HBC}_{\omega }$ to pairs of orthogonal $(1,0)$ -tangent vectors). From [Reference Gu and Zhang8], the Kähler–Ricci flow on a compact Kähler manifold with $\text {HBC}_{\omega }^{\perp } \geq 0$ converges to a Kähler metric $\text {HBC}_{\omega } \geq 0$ . Hence, Mok’s extension [Reference Mok10] of the solution of the Frankel conjecture [Reference Mori11, Reference Siu and Yau16] shows that all compact Kähler manifolds with $\text {HBC}_{\omega }^{\perp } \geq 0$ are biholomorphic to a product of Hermitian symmetric spaces (of rank $\geq 2$ ) and projective spaces. In particular, although $\text {HBC}_{\omega }^{\perp }$ is an algebraically weaker curvature notion than $\text {HBC}_{\omega }$ , the positivity of $\text {HBC}_{\omega }^{\perp }$ does not generate new examples.

Wu et al. [Reference Wu, Yau and Zheng17] showed that if $(M, \omega )$ is a compact Kähler manifold with $\text {QOBC}_{\omega } \geq 0$ , then every non-Kähler class on the boundary of the Kähler cone affords a semi-positive representative (which is certainly not true in general; see [Reference Demailly, Peternell and Schneider7]). Further, Chau and Tam [Reference Chau and Tam4] showed that all harmonic $(1,1)$ -forms are parallel on compact Kähler manifolds with $\text {QOBC}_{\omega } \geq 0$ .

In [Reference Broder2], I showed that the real bisectional curvature of Yang and Zheng [Reference Yang and Zheng18] was best understood as a Rayleigh quotient. Continuing this program, we find that the QOBC is best understood as the Dirichlet energy of a certain weighted graph. In particular, we realise the difference of the Hodge and metric Laplacians, acting on $(1,1)$ -forms, as the Dirichlet energy of a weighted graph.

To analyse the QOBC, we recall the following definition. Let G be a finite weighted graph, with vertices $V(G) = \{ x_1, \ldots , x_n \}$ , and weighting specified by its adjacency matrix $A \in \mathbb {R}^{n \times n}$ . The Dirichlet energy for a weighted graph is defined by

$$ \begin{align*}\mathcal{E}(f) : = \sum_{i,j=1}^n A_{ij}(f(x_i) - f(x_j))^2,\end{align*} $$

where $f : V(G) \to \mathbb {R}$ is a function defined on the vertices of G.

To understand when the QOBC is nonnegative, we can ask the following (equally natural) question.

The main theorem of this note gives an answer to this problem. To this end, let us recall some terminology arising from distance geometry.

The set of all $n \times n$ Euclidean distance matrices (EDMs) forms a convex cone that we denote by $\mathbb {EDM}^n$ . Recall that the Frobenius inner product of two matrices $A,B \in \mathbb {R}^{n \times n}$ is defined by

$$ \begin{align*}(A,B)_{\text{F}} : = \text{tr}(AB^{t}).\end{align*} $$

This dual pairing allows us to define the dual EDM cone $(\mathbb {EDM}^n)^{\ast }$ .

Definition 4. The dual EDM cone $(\mathbb {EDM}^n)^{\ast }$ is given by

$$ \begin{align*}(\mathbb{EDM}^n)^{\ast} : = \{ A \in \mathbb{R}^{n \times n} : (A,B)_{\mathrm{F}} \geq 0\ \mathrm{for}\ \mathrm{all}\ B \in \mathbb{EDM}^n \}.\end{align*} $$

Proof.

If $V(G) = \{ x_1, \ldots , x_n \}$ is the vertex set of some graph, then we may construct a Euclidean distance matrix $B(f)$ from a graph function $f : V(G) \to \mathbb {R}$ by setting $B(f)_{ij} = (f(x_i) - f(x_j))^2$ . In particular, since

$$ \begin{align*} \text{tr}(A B(f)) = \sum_{i,j=1}^n A_{ij} B(f)_{ij} = \sum_{i,j=1}^n A_{ij}(f(x_i) - f(x_j))^2, \end{align*} $$

we see that the Dirichlet energy $\mathcal {E}$ of a weighted graph $(G, A)$ is nonnegative if and only if $\text {tr}(AB) \geq 0$ for all Euclidean distance matrices $B \in \mathbb {EDM}^n$ .

It is natural to ask what is the relation (if any) between the EDM cone (and its dual) and the $\mathbb {PSD}^n$ cone, that is, the cone of (symmetric) positive semi-definite matrices. Dattorro [Reference Dattorro5] has shown that

$$ \begin{align*}\mathbb{EDM}^n = \mathbb{S}_{\text{H}}^n \cap ( ( \mathbb{S}_{\text{C}}^n)^{\perp} - \mathbb{PSD}^n) \subset \mathbb{R}_{\geq 0}^{n \times n}.\end{align*} $$

Here, $\mathbb {S}_{\text {H}}^n$ denotes the space of symmetric $n \times n$ hollow matrices, that is, symmetric matrices with no nonzero entries on the diagonal, and $\mathbb {S}_{\text {C}}^n$ denotes the geometric centring subspace

$$ \begin{align*}\mathbb{S}_{\text{C}}^n : = \{ A \in \mathbb{S}^n : A \textbf{e} = 0 \},\end{align*} $$

where $\mathbf {e} = (1, \ldots , 1)^{t}$ . It is more natural to refer to $\mathbb {S}_{\text {C}}^{n}$ as the annihilator of $\mathbf {e} = (1, \ldots , 1)^{t} \in \mathbb {R}^{n}$ . The orthogonal complement of $\mathbb {S}_{\text {C}}^{n}$ is then

$$ \begin{align*}(\mathbb{S}_{\text{C}}^n)^{\perp} = \{ u \mathbf{e}^{t} + \mathbf{e} u^{t} : u \in \mathbb{R}^{n} \}.\end{align*} $$

In particular, standard properties of cones (see [Reference Dattorro6, page 434]) give the next proposition.

Proposition 6. Let $(G,A)$ be a weighted finite graph. The Dirichlet energy $\mathcal {E}$ is nonnegative if and only if A lies in

$$ \begin{align*} (\mathbb{EDM}^{n})^{\ast} = \mathbb{D}^{n} - \mathbb{S}_{\text{C}}^{n} \cap \mathbb{PSD}^{n}, \end{align*} $$

where $\mathbb {D}^n$ is the cone of diagonal matrices.

With this terminology in mind, appealing to the eigenvalue characterisation of the dual EDM cone given in [Reference Broder3] yields the following corollary.

Corollary 9. Let $A \in \mathbb {R}^{n \times n}$ be a real symmetric matrix with eigenvalues $\lambda _1 \geq \lambda _2 \geq \cdots \geq \lambda _n$ . Then $A \in (\mathbb {EDM}^n)^{\ast }$ if and only if, for every Euclidean distance matrix $\Sigma $ , the Perron weights $r_2, \ldots , r_k$ of $\Sigma $ satisfy

$$ \begin{align*}\lambda_1 \ge \sum^n_{k=1} r_k\lambda_k.\end{align*} $$