Quadratic Embedding Constant

• Quadratic Embedding Constant (QEC) is a numerical invariant defined by a constrained quadratic maximization on a graph’s distance matrix, indicating whether a graph’s metric admits a quadratic embedding.
• It connects variational definitions and spectral formulations, linking eigenvalue bounds and explicit formulas for families like path graphs, cycles, and strongly regular graphs.
• QEC offers practical insights through transformation laws under graph operations, enabling classification of graphs and identification of structural obstructions using clear algebraic and geometric criteria.

The Quadratic Embedding Constant (QEC), usually denoted $\mathrm{QEC}(G)$, is a numerical invariant of a connected graph $G$ defined from its distance matrix by a constrained quadratic maximization. It occupies a central position in the study of graph metrics of negative type, quadratic embeddings into Hilbert space, and distance spectra. In finite settings it is the maximal value of the distance quadratic form on unit vectors orthogonal to the all-ones vector; in general it is formulated on finitely supported mean-zero unit functions. The sign of $\mathrm{QEC}(G)$ determines whether the graph metric admits a quadratic embedding, while explicit formulas and transformation laws are now known for paths, strongly regular graphs, joins, star products, fan graphs, and corona graphs (Młotkowski, 2022, Obata, 2 Jan 2025).

1. Variational definition and geometric meaning

Let $G=(V,E)$ be a connected graph with graph distance $d(x,y)$. For finite $V$, with distance matrix $D=[d(x,y)]_{x,y\in V}$, the quadratic embedding constant is

$$\mathrm{QEC}(G)=\max\{f^\top Df:\mathbf{1}^\top f=0,\ \|f\|_2=1\}.$$

For infinite graphs it is defined as the supremum of

$\sum_{x,y\in V} d(x,y)f(x)f(y)$

over finitely supported real functions $f$ satisfying $G$0 and $G$1 (Młotkowski, 2022).

This variational formulation identifies QEC as the extremal value of the distance quadratic form on the codimension-two constraint set given by unit norm and orthogonality to constants. In the finite case, compactness guarantees that the maximum is attained (Baskoro et al., 2019).

The geometric significance is given by Schoenberg’s criterion. The distance matrix is conditionally negative definite if and only if

$G$2

and this is equivalent to $G$3. Hence a connected graph belongs to the QE class precisely when there exists a map $G$4 into a Hilbert space such that

$G$5

or equivalently, precisely when $G$6 (Młotkowski, 2022, Baskoro et al., 2019).

A complementary analytic characterization relates QEC to the kernel

$G$7

The condition $G$8 is equivalent to positivity of this matrix for all $G$9 (Baskoro et al., 2019). This places QEC simultaneously in Euclidean distance geometry, kernel positivity, and graph metric theory.

2. Spectral formulations and matrix characterizations

Because the distance matrix is real symmetric, QEC admits a spectral interpretation. If $\mathrm{QEC}(G)$0 are the eigenvalues of the distance matrix, then

$\mathrm{QEC}(G)$1

for finite connected graphs (Młotkowski, 2022, Baskoro et al., 2019). The strict upper bound reflects Perron–Frobenius positivity of the top distance eigenvector, which is not orthogonal to $\mathrm{QEC}(G)$2.

When the graph is transmission regular, so that the distance matrix has constant row sum and $\mathrm{QEC}(G)$3 is a distance eigenvector, the variational problem reduces exactly to the second-largest distance eigenvalue: $\mathrm{QEC}(G)$4 This identity is used explicitly for strongly regular graphs and other transmission-regular families (Obata, 2 Jan 2025).

A Lagrange multiplier formulation converts the constrained optimization into a stationary system. Writing

$\mathrm{QEC}(G)$5

stationarity yields

$\mathrm{QEC}(G)$6

and the maximizing multiplier $\mathrm{QEC}(G)$7 is precisely $\mathrm{QEC}(G)$8 (Baskoro et al., 2019). This formulation underlies many exact computations.

For graphs of diameter at most $\mathrm{QEC}(G)$9, the distance matrix simplifies to

$G=(V,E)$0

where $G=(V,E)$1 is the adjacency matrix. In the regular case this gives

$G=(V,E)$2

so QEC is controlled directly by the smallest adjacency eigenvalue (Lou et al., 2020). This reduction is fundamental in the strongly regular case.

3. Path graphs as the foundational explicit case

Path graphs provide the canonical model for the theory. For

$G=(V,E)$3

the distance matrix is $G=(V,E)$4 (Młotkowski, 2022).

Earlier work established that the sequence $G=(V,E)$5 is strictly increasing and converges to $G=(V,E)$6, and used this path scale to formulate classification problems for finite connected graphs (Baskoro et al., 2019). A prior implicit description also expressed $G=(V,E)$7 as the negative of the maximal $G=(V,E)$8 for which a matrix built from $G=(V,E)$9 is positive definite, and from this obtained

$d(x,y)$0

by passage to the limit (Młotkowski et al., 2018).

The decisive closed formula is

$d(x,y)$1

together with

$d(x,y)$2

for the infinite one-sided and two-sided paths (Młotkowski, 2022). Since these values are negative, all finite and infinite path graphs are of QE class.

The derivation proceeds through a positive-definiteness threshold for the matrix family

$d(x,y)$3

For path graphs,

$d(x,y)$4

and determinant identities reduce this to a recurrence for polynomials $d(x,y)$5: $d(x,y)$6

$d(x,y)$7

A key factorization is

$d(x,y)$8

which isolates the positivity threshold and yields the explicit QEC formula (Młotkowski, 2022).

The relation with the distance spectrum is sharp. If $d(x,y)$9 denotes the second-largest distance eigenvalue, then

$V$0

for even $V$1, whereas

$V$2

for odd $V$3 (Młotkowski, 2022).

The maximizing vector can also be written explicitly: $V$4 It satisfies

$V$5

and

$V$6

so its normalized version attains $V$7 (Młotkowski, 2022).

4. Exact formulas for representative graph families

A substantial part of the modern theory consists of closed QEC formulas for concrete graph classes. Representative examples are as follows.

Graph family	QEC formula or criterion
Complete graph $V$8	$V$9
Odd cycle $D=[d(x,y)]_{x,y\in V}$0	$D=[d(x,y)]_{x,y\in V}$1
Even cycle $D=[d(x,y)]_{x,y\in V}$2	$D=[d(x,y)]_{x,y\in V}$3
Complete bipartite graph $D=[d(x,y)]_{x,y\in V}$4	$D=[d(x,y)]_{x,y\in V}$5
Two-clique graph $D=[d(x,y)]_{x,y\in V}$6	$D=[d(x,y)]_{x,y\in V}$7
Star product $D=[d(x,y)]_{x,y\in V}$8	$D=[d(x,y)]_{x,y\in V}$9
Strongly regular graph $$\mathrm{QEC}(G)=\max\{f^\top Df:\mathbf{1}^\top f=0,\ \|f\|_2=1\}.$$0, $$\mathrm{QEC}(G)=\max\{f^\top Df:\mathbf{1}^\top f=0,\ \|f\|_2=1\}.$$1	$$\mathrm{QEC}(G)=\max\{f^\top Df:\mathbf{1}^\top f=0,\ \|f\|_2=1\}.$$2 with $$\mathrm{QEC}(G)=\max\{f^\top Df:\mathbf{1}^\top f=0,\ \|f\|_2=1\}.$$3

These formulas come from several strands of the literature: complete graphs and cycles from early classification work, complete bipartite and join formulas from the study of graph joins, two-clique and star-product formulas from clique-graph methods, and the strongly regular formula from the distance-spectral analysis of diameter-$$\mathrm{QEC}(G)=\max\{f^\top Df:\mathbf{1}^\top f=0,\ \|f\|_2=1\}.$$4 regular graphs (Baskoro et al., 2019, Lou et al., 2020, Baskoro et al., 2023, Obata, 2 Jan 2025).

For strongly regular graphs the result is especially rigid. If $$\mathrm{QEC}(G)=\max\{f^\top Df:\mathbf{1}^\top f=0,\ \|f\|_2=1\}.$$5 with $$\mathrm{QEC}(G)=\max\{f^\top Df:\mathbf{1}^\top f=0,\ \|f\|_2=1\}.$$6, then

$$\mathrm{QEC}(G)=\max\{f^\top Df:\mathbf{1}^\top f=0,\ \|f\|_2=1\}.$$7

and

$$\mathrm{QEC}(G)=\max\{f^\top Df:\mathbf{1}^\top f=0,\ \|f\|_2=1\}.$$8

Moreover, among strongly regular graphs with $$\mathrm{QEC}(G)=\max\{f^\top Df:\mathbf{1}^\top f=0,\ \|f\|_2=1\}.$$9, one theorem states that

$\sum_{x,y\in V} d(x,y)f(x)f(y)$0

except for

$\sum_{x,y\in V} d(x,y)f(x)f(y)$1

(Obata, 2 Jan 2025).

For fan graphs $\sum_{x,y\in V} d(x,y)f(x)f(y)$2, the QEC is expressed through the minimal zero of a polynomial built from Chebyshev polynomials: $\sum_{x,y\in V} d(x,y)f(x)f(y)$3 where $\sum_{x,y\in V} d(x,y)f(x)f(y)$4 is the minimal zero of a polynomial $\sum_{x,y\in V} d(x,y)f(x)f(y)$5 related to Chebyshev polynomials of the second kind. For even $\sum_{x,y\in V} d(x,y)f(x)f(y)$6,

$\sum_{x,y\in V} d(x,y)f(x)f(y)$7

while for odd $\sum_{x,y\in V} d(x,y)f(x)f(y)$8 explicit two-sided bounds are available (Młotkowski et al., 2024, Młotkowski et al., 2024).

5. Behavior under graph operations

QEC interacts in a controlled way with several graph operations. For finite star products

$\sum_{x,y\in V} d(x,y)f(x)f(y)$9

if each factor has negative QEC, then

$f$0

where $f$1 (Młotkowski et al., 2018). If all factors are of QE class and one factor has QEC $f$2, then the star product has QEC $f$3 (Młotkowski et al., 2018).

For joins of regular graphs, if $f$4 is $f$5-regular on $f$6 vertices, then

$f$7

which yields explicit values for complete bipartite graphs, wheel graphs, friendship graphs, complete split graphs, and joins involving strongly regular graphs (Lou et al., 2020). More recent work extends join formulas to joins with arbitrary regular graphs and complete multipartite graphs, and derives Cartesian-product formulas such as

$f$8

for non-QE graphs $f$9, together with analogous formulas for $G$00 (Choudhury et al., 14 Aug 2025).

The corona operation produces a more analytic transformation law. Under spectral assumptions on $G$01,

$G$02

where $G$03 is built from the main eigenvalues of the adjacency matrix of $G$04 (Ferdi et al., 19 Dec 2025). A later refinement gives

$G$05

with $G$06 and $G$07 determined by singular spectral contributions of $G$08 (Ferdi et al., 16 Mar 2026). When $G$09 is $G$10-regular on $G$11 vertices,

$G$12

and the same work proves

$G$13

for regular $G$14 (Ferdi et al., 16 Mar 2026).

These formulas show that QEC is not merely an isolated invariant; it is functorial enough to support exact transfer laws under several nontrivial graph constructions.

6. Structural obstructions, clique geometry, and classification

A major structural threshold is $G$15. If a connected graph satisfies

$G$16

then it cannot contain an induced diamond $G$17 or an induced claw $G$18, because both of those graphs have QEC exactly $G$19. The same line of argument gives further forbidden induced subgraphs: $G$20 since induced $G$21 forces QEC at least $G$22 and induced $G$23 forces QEC at least $G$24 (Baskoro et al., 2019).

A sharper clique-theoretic description is available. If $G$25 denotes the clique graph whose vertices are the maximal cliques of $G$26, then for graphs with $G$27: $G$28 adjacent maximal cliques intersect in exactly one vertex, and no three distinct maximal cliques have nonempty triple intersection (Baskoro et al., 2023). In that precise sense, such graphs admit a cactus-like decomposition into maximal cliques glued along cut vertices.

This structure supports classification along the increasing path scale

$G$29

If

$G$30

then

$G$31

(Baskoro et al., 2023). The intervals near the bottom of the scale are already classified: below $G$32 only complete graphs occur, and between $G$33 and $G$34 the graphs are exactly $G$35 with $G$36 together with $G$37 (Baskoro et al., 2019).

Beyond clique obstructions, theta graphs provide a different source of non-QE behavior. For bipartite graphs, Tanaka’s criterion links quadratic embeddability to the absence of Tanaka quintuples; for theta graphs the existence of Tanaka quintuples and modified Tanaka quintuples yields large families of non-QE examples. In particular, non-QE theta graphs are primary non-QE graphs, and there exists a non-QE theta graph on each number of vertices $G$38 (Młotkowski et al., 2024).

Taken together, these results present QEC as both a spectral invariant and a structural classifier. Its sign governs quadratic embeddability, its exact values encode delicate spectral data of the distance matrix, and threshold ranges such as $G$39 force stringent global geometry on the graph.