Squared Distance Matrix Overview

• Squared distance matrix is a symmetric, zero-diagonal matrix encoding squared pairwise distances with applications in metric geometry and graph theory.
• The analysis employs Gram matrix tests, Cayley–Menger formulations, and structured block patterns to assess Euclidean realizability and spherical constraints.
• Spectral theory reveals unique eigenvalue distributions and energy invariants in graphs and trees, guiding insights in algorithmic linear algebra.

A squared distance matrix is a symmetric hollow matrix whose entries encode squared pairwise distances. In graph-theoretic usage, for a connected graph $G$ with graph distance $d_{ij}$, the squared distance matrix $\Delta(G)$ has diagonal entries $0$ and off-diagonal entries $d_{ij}^2$; equivalently, $\Delta(G)=D(G)\circ D(G)$ when $D(G)$ is the distance matrix. In Euclidean distance matrix theory, a matrix $D=(d_{ij})$ is an Euclidean distance matrix precisely when $d_{ij}=\|p^i-p^j\|^2$ for some point configuration, so the standard EDM is already a squared distance matrix. The same formal object therefore appears in graph spectra, Euclidean embedding, inverse problems, algorithmic linear algebra, and several optimization frameworks (Das et al., 2020, Alfakih, 2019, Bapat, 2018).

1. Definitions and ambient settings

For a connected graph $G$ on vertices $d_{ij}$0, the squared distance matrix is defined entrywise by

$d_{ij}$1

where $d_{ij}$2 is the length of a shortest path. Several papers also write $d_{ij}$3, the Hadamard square of the distance matrix $d_{ij}$4 (Das et al., 2020, Howell et al., 2022). In weighted trees with scalar edge weights, the same definition is used with $d_{ij}$5 equal to the sum of edge weights on the unique $d_{ij}$6-path, and in trees with matrix weights the construction becomes block-valued, with $d_{ij}$7 as an $d_{ij}$8 block (Bapat, 2018, Mahato et al., 2022).

For finite point sets in metric spaces, the squared distance matrix is likewise $d_{ij}$9. In Euclidean geometry this coincides with the classical Euclidean distance matrix, defined by $\Delta(G)$0 for points $\Delta(G)$1 in Euclidean space (Alfakih, 2019). The infinite-size variant considered on a countable set $\Delta(G)$2 in a Riemannian manifold keeps the same entrywise definition $\Delta(G)$3 (Plakhotnikov, 13 Sep 2025).

This common definition hides substantial variation in structure. In some settings the matrix is governed by combinatorial block patterns, as in complete multipartite or starlike block graphs; in others it is constrained by realizability in Euclidean space, sphericality, or positivity of associated Gram-type matrices. A plausible implication is that the main theory of squared distance matrices is not a single theorem family but an interface between metric geometry and matrix analysis.

2. Euclidean realizability and spherical constraints

A real symmetric zero-diagonal matrix $\Delta(G)$4 is an Euclidean distance matrix if there exist points $\Delta(G)$5 such that

$\Delta(G)$6

A central criterion is the Gram-matrix test: for any vector $\Delta(G)$7 with $\Delta(G)$8,

$\Delta(G)$9

and $0$0 is an EDM iff $0$1. With $0$2, this becomes $0$3, where $0$4. For spherical EDMs, the vector $0$5 satisfying $0$6 is decisive: if the embedding dimension is $0$7, then sphericality is equivalent to $0$8, equivalently $0$9, and the radius is

$d_{ij}^2$0

Unit sphericality is therefore characterized by $d_{ij}^2$1 (Alfakih, 2019).

The one-entry perturbation problem for unit spherical EDMs gives a refined local classification. If one fixes all entries of a unit spherical EDM $d_{ij}^2$2 except the symmetric pair $d_{ij}^2$3, then the admissible set

$d_{ij}^2$4

can take only three forms: a nontrivial interval, exactly two points one of which is $d_{ij}^2$5, or the singleton $d_{ij}^2$6. The analysis uses Gram matrices, Gale transforms, and a Cayley–Menger reformulation. In the Cayley–Menger approach,

$d_{ij}^2$7

and the paper shows that $d_{ij}^2$8 is unit spherical iff $d_{ij}^2$9 is nonspherical (Alfakih, 2019).

These results separate Euclidean realizability from mere hollow symmetry. A squared distance matrix can be symmetric and zero-diagonal without being an EDM, whereas unit spherical EDMs satisfy stringent rank, positivity, and perturbative constraints.

3. Spectral theory for finite graph classes

For graph-based squared distance matrices, inertia and energy are prominent invariants. If $\Delta(G)=D(G)\circ D(G)$0 is Hermitian, its inertia is

$\Delta(G)=D(G)\circ D(G)$1

the counts of positive, negative, and zero eigenvalues (Howell et al., 2022). In complete multipartite graphs $\Delta(G)=D(G)\circ D(G)$2, the distance structure is especially rigid: vertices in the same part are at distance $\Delta(G)=D(G)\circ D(G)$3, vertices in different parts are at distance $\Delta(G)=D(G)\circ D(G)$4, so the diagonal blocks of $\Delta(G)=D(G)\circ D(G)$5 are $\Delta(G)=D(G)\circ D(G)$6 and the off-diagonal blocks are $\Delta(G)=D(G)\circ D(G)$7. From this block form one obtains the characteristic polynomial

$\Delta(G)=D(G)\circ D(G)$8

If every part has size at least $\Delta(G)=D(G)\circ D(G)$9, then

$D(G)$0

If singleton parts are present, with $D(G)$1, then

$D(G)$2

For fixed $D(G)$3 and $D(G)$4, both the squared distance energy and the spectral radius are maximal at the complete split graph $D(G)$5 and minimal at the Turán graph $D(G)$6 (Das et al., 2020).

Trees exhibit another rigid pattern. If $D(G)$7 is a tree with $D(G)$8 leaves and $D(G)$9 degree-2 vertices, then the Bapat–Sivasubramanian theorem gives [ i_