Spherical Two-Distance Sets

• Spherical two-distance sets are finite collections of unit vectors on a sphere with exactly two distinct inner products, important in extremal geometry and combinatorics.
• They are characterized by their Gram matrix and Seidel matrix formulations, connecting these sets to strongly regular graphs, equiangular lines, and tight frame theory.
• Advanced techniques such as semidefinite programming and spectral methods yield sharp bounds and exact cardinalities, with implications for spherical designs and high-dimensional partitioning.

A spherical two-distance set is a finite set $S=\{x_1,\dots,x_N\}\subset \mathbb R^n$ of unit vectors such that the inner products of distinct vectors take only two values. Equivalently, for $S\subset S^{d-1}$, the Euclidean distances between distinct points take only two values because $\|x-y\|^2=2-2\langle x,y\rangle$. The subject lies at the intersection of extremal geometry, algebraic combinatorics, harmonic analysis, and frame theory. Its central problems concern maximal cardinality, structural classification, and graph-theoretic realization; modern treatments connect spherical two-distance sets to strongly regular graphs, spherical designs, equiangular lines, semidefinite programming, and Seidel-matrix spectral methods (Barg et al., 2014, Glazyrin et al., 2016, Chen et al., 31 Aug 2025).

1. Definition, basic notation, and equivalent formulations

A standard formulation takes $X=\{v_1,\dots,v_n\}\subset \mathbb S^{d-1}$ with

$$\langle v_i,v_j\rangle\in\{a,b\},\qquad -1\le a<b<1,\quad i\neq j.$$

The parameters $a$ and $b$ are the two possible inner products between distinct unit vectors. In geometric terms, the larger inner product corresponds to the smaller angle, and the smaller inner product to the larger angle (Chen et al., 31 Aug 2025).

The Gram-matrix viewpoint is fundamental. If $G=(\langle x_i,x_j\rangle)_{ij}$, then for a two-distance set one may write

$G=I+a\Phi_1+b\Phi_2,$

where $\Phi_1,\Phi_2$ are the adjacency matrices of two complementary graphs on the same vertex set. Thus the geometry is encoded by a graph recording which pairs realize one inner product and which realize the other (Barg et al., 2014).

The special case $S\subset S^{d-1}$0 is the equiangular case. In that regime the two-distance problem collapses to the theory of equiangular lines, and several arguments used in the general setting become exceptional. The non-equiangular regime $S\subset S^{d-1}$1 is therefore structurally distinct (Barg et al., 2014).

Several notational conventions occur in the literature. The maximum size of a spherical two-distance set in $S\subset S^{d-1}$2 is denoted by $S\subset S^{d-1}$3 in work on exact cardinalities (Yu, 2016, Barg et al., 2012). For fixed angles $S\subset S^{d-1}$4, the maximum size of a spherical $S\subset S^{d-1}$5-code in $S\subset S^{d-1}$6 is denoted by

$S\subset S^{d-1}$7

which is the natural fixed-angle asymptotic variant (Jiang et al., 2020).

2. Absolute bounds, parameter-dependent bounds, and exact cardinalities

The classical absolute upper bound for spherical $S\subset S^{d-1}$8-distance sets, due to Delsarte–Goethals–Seidel, is

$S\subset S^{d-1}$9

Specialized to $\|x-y\|^2=2-2\langle x,y\rangle$0, this gives

$\|x-y\|^2=2-2\langle x,y\rangle$1

which is the standard absolute bound for spherical two-distance sets (Hegedüs et al., 2020). A Gerzon-type polynomial argument recovers the same value when the two distinct inner products sum to zero; for $\|x-y\|^2=2-2\langle x,y\rangle$2 this again yields $\|x-y\|^2=2-2\langle x,y\rangle$3 (Datta et al., 2021).

Sharper estimates appear once additional structure is imposed. If a spherical $\|x-y\|^2=2-2\langle x,y\rangle$4-distance set has strength $\|x-y\|^2=2-2\langle x,y\rangle$5, then the classical absolute bound improves to

$\|x-y\|^2=2-2\langle x,y\rangle$6

This is the familiar Neumaier-type improvement for spherical $\|x-y\|^2=2-2\langle x,y\rangle$7-designs and is closely tied to strongly regular graphs and Krein-parameter conditions (Nozaki et al., 2010).

A different refinement depends explicitly on the two inner products. Using a Seidel matrix

$\|x-y\|^2=2-2\langle x,y\rangle$8

one obtains upper bounds that split according to the sign of $\|x-y\|^2=2-2\langle x,y\rangle$9: $X=\{v_1,\dots,v_n\}\subset \mathbb S^{d-1}$0 and

$X=\{v_1,\dots,v_n\}\subset \mathbb S^{d-1}$1

The second bound coincides with the relative bound for equiangular lines in one higher dimension, reflecting a bijective correspondence between spherical two-distance sets with $X=\{v_1,\dots,v_n\}\subset \mathbb S^{d-1}$2 and equiangular line systems in $X=\{v_1,\dots,v_n\}\subset \mathbb S^{d-1}$3 with common angle

$X=\{v_1,\dots,v_n\}\subset \mathbb S^{d-1}$4

When equality holds, the Seidel matrix has exactly two distinct eigenvalues, corresponding to an equiangular tight frame (Chen et al., 31 Aug 2025).

For large configurations there is also a rigidity theorem of Larman–Rogers–Seidel type: if $X=\{v_1,\dots,v_n\}\subset \mathbb S^{d-1}$5, then the inner products satisfy

$X=\{v_1,\dots,v_n\}\subset \mathbb S^{d-1}$6

for some integer

$X=\{v_1,\dots,v_n\}\subset \mathbb S^{d-1}$7

with

$X=\{v_1,\dots,v_n\}\subset \mathbb S^{d-1}$8

This reduction is a key ingredient in semidefinite programming approaches (Barg et al., 2012).

Lower bounds come from explicit constructions. The standard example is given by the midpoints of edges of a regular simplex, yielding

$X=\{v_1,\dots,v_n\}\subset \mathbb S^{d-1}$9

A substantial body of work shows that this lower bound is often exact. Semidefinite programming proves

$$\langle v_i,v_j\rangle\in\{a,b\},\qquad -1\le a<b<1,\quad i\neq j.$$0

and

$$\langle v_i,v_j\rangle\in\{a,b\},\qquad -1\le a<b<1,\quad i\neq j.$$1

while also recording the special values $$\langle v_i,v_j\rangle\in\{a,b\},\qquad -1\le a<b<1,\quad i\neq j.$$2, $$\langle v_i,v_j\rangle\in\{a,b\},\qquad -1\le a<b<1,\quad i\neq j.$$3, $$\langle v_i,v_j\rangle\in\{a,b\},\qquad -1\le a<b<1,\quad i\neq j.$$4, $$\langle v_i,v_j\rangle\in\{a,b\},\qquad -1\le a<b<1,\quad i\neq j.$$5, $$\langle v_i,v_j\rangle\in\{a,b\},\qquad -1\le a<b<1,\quad i\neq j.$$6, and $$\langle v_i,v_j\rangle\in\{a,b\},\qquad -1\le a<b<1,\quad i\neq j.$$7 (Barg et al., 2012). A later reduction to equiangular lines extends the exact formula to

$$\langle v_i,v_j\rangle\in\{a,b\},\qquad -1\le a<b<1,\quad i\neq j.$$8

for

$$\langle v_i,v_j\rangle\in\{a,b\},\qquad -1\le a<b<1,\quad i\neq j.$$9

except for

$a$0

which are exactly the values

$a$1

In a complementary formulation, the maximum size is proved to be $a$2 for all $a$3, except possibly when $a$4; these are precisely the dimensions tied to the open problem of tight spherical $a$5-designs (Yu, 2016, Glazyrin et al., 2016).

3. Tight frames, spherical designs, and strongly regular graphs

A central structural theorem concerns two-distance sets that are also tight frames. A finite unit-norm set $a$6 is a tight frame if

$a$7

and for a unit-norm tight frame one has

$a$8

Equivalently, the Gram matrix has spectrum

$a$9

A two-distance tight frame is precisely a spherical two-distance set that is also a finite unit-norm tight frame (Barg et al., 2014).

The main theorem in this setting states that if $b$0 is a non-equiangular two-distance finite unit-norm tight frame in $b$1 and $b$2, then $b$3 is either a spherical two-distance $b$4-design or a shifted $b$5-design, and in either case arises as a spherical embedding of a strongly regular graph. Conversely, every strongly regular graph gives rise, through the standard Delsarte–Goethals–Seidel spherical embedding, to two-distance tight frames. Together with Waldron’s earlier result on the equiangular case, this completely characterizes finite two-distance tight frames (Barg et al., 2014).

A strongly regular graph has parameters

$b$6

meaning $b$7-regularity on $b$8 vertices, $b$9 common neighbors for adjacent vertices, and $G=(\langle x_i,x_j\rangle)_{ij}$0 common neighbors for nonadjacent vertices. If $G=(\langle x_i,x_j\rangle)_{ij}$1 is its adjacency matrix, then its spectrum consists of the trivial eigenvalue $G=(\langle x_i,x_j\rangle)_{ij}$2 and two nontrivial eigenvalues

$G=(\langle x_i,x_j\rangle)_{ij}$3

with multiplicities

$G=(\langle x_i,x_j\rangle)_{ij}$4

Projecting the standard basis of $G=(\langle x_i,x_j\rangle)_{ij}$5 onto one of the nontrivial eigenspaces and normalizing yields the spherical embeddings $G=(\langle x_i,x_j\rangle)_{ij}$6 and $G=(\langle x_i,x_j\rangle)_{ij}$7, each a spherical two-distance set (Barg et al., 2014).

The intermediate classification of spherical two-distance $G=(\langle x_i,x_j\rangle)_{ij}$8-designs is especially sharp:

Any spherical two-distance $G=(\langle x_i,x_j\rangle)_{ij}$9-design is either $G=I+a\Phi_1+b\Phi_2,$0, $G=I+a\Phi_1+b\Phi_2,$1, or a regular simplex.

Here a spherical $G=I+a\Phi_1+b\Phi_2,$2-design is equivalently a unit-norm tight frame with centroid at the origin, characterized by

$G=I+a\Phi_1+b\Phi_2,$3

If the centroid does not vanish, the configuration must be similar to an $G=I+a\Phi_1+b\Phi_2,$4-dimensional two-distance $G=I+a\Phi_1+b\Phi_2,$5-design lying in a subsphere of radius

$G=I+a\Phi_1+b\Phi_2,$6

the “shifted $G=I+a\Phi_1+b\Phi_2,$7-design” case (Barg et al., 2014).

4. Graph encodings and spectral representation theory

The graph-theoretic encoding of spherical two-distance sets extends well beyond the tight-frame case. One direction starts from geometry: given a two-distance set, declare two vertices adjacent when the corresponding pair realizes one of the two distances or inner products. The other direction starts from a graph and asks for a two-distance realization of its vertices. This bridge underlies several spectral characterizations (Barg et al., 2014, Noman et al., 2022).

A general result states that every graph $G=I+a\Phi_1+b\Phi_2,$8 can be embedded in a Euclidean space as a two-distance set. One may then define the Euclidean representation number $G=I+a\Phi_1+b\Phi_2,$9, the spherical representation number $\Phi_1,\Phi_2$0, and the J-spherical representation number $\Phi_1,\Phi_2$1 (Musin, 2016). In the Cayley–Menger framework, if $\Phi_1,\Phi_2$2 is the multiplicity of a distinguished root $\Phi_1,\Phi_2$3 of the discriminating polynomial, then

$\Phi_1,\Phi_2$4

Sphericality is controlled by the circumradius invariant $\Phi_1,\Phi_2$5: $\Phi_1,\Phi_2$6 For J-spherical representations, one has

$\Phi_1,\Phi_2$7

Every graph except a complete graph admits a J-spherical representation, and it is unique up to isometry (Musin, 2016).

An eigenvalue-based reformulation uses the projected Gram matrix. If $\Phi_1,\Phi_2$8 is the adjacency matrix of $\Phi_1,\Phi_2$9 and $S\subset S^{d-1}$00 has orthonormal columns spanning the orthogonal complement of the all-ones vector, then for a normalized two-distance representation one obtains

$S\subset S^{d-1}$01

This leads to exact formulas for $S\subset S^{d-1}$02 and $S\subset S^{d-1}$03 in terms of extremal eigenvalues of $S\subset S^{d-1}$04, and to the clean J-spherical formula

$S\subset S^{d-1}$05

where $S\subset S^{d-1}$06 is the largest eigenvalue of the complement graph (Alfakih, 2018).

For graphs on $S\subset S^{d-1}$07 vertices, spherical representability in $S\subset S^{d-1}$08 admits an especially precise spectral criterion. Let $S\subset S^{d-1}$09 have eigenvalues $S\subset S^{d-1}$10, and let

$S\subset S^{d-1}$11

be the orthogonal projection onto $S\subset S^{d-1}$12. Then $S\subset S^{d-1}$13 has a spherical representation in $S\subset S^{d-1}$14 if and only if the maximum eigenvalue of $S\subset S^{d-1}$15 equals $S\subset S^{d-1}$16, and the multiplicity of $S\subset S^{d-1}$17 in $S\subset S^{d-1}$18 equals its multiplicity in $S\subset S^{d-1}$19, excluding $S\subset S^{d-1}$20 if $S\subset S^{d-1}$21. In this formulation the distance ratio is determined by

$S\subset S^{d-1}$22

The same framework also determines the lowest-dimensional spherical realization in terms of the multiplicity of $S\subset S^{d-1}$23 (Noman et al., 2022).

These results make precise a common theme in the subject: the existence of a spherical two-distance realization is often equivalent to the positive semidefiniteness and rank behavior of a small collection of graph-derived matrices, while extremal or highly symmetric configurations correspond to rigid spectral multiplicity patterns.

5. Extremal constructions and Borsuk-type counterexamples

Strongly regular graphs provide concrete high-dimensional examples with striking extremal behavior. A general Euclidean representation starts from a strongly regular graph $S\subset S^{d-1}$24 with parameters $S\subset S^{d-1}$25, chooses one eigenspace, and produces vectors $S\subset S^{d-1}$26 satisfying

$S\subset S^{d-1}$27

Thus adjacency and nonadjacency translate directly into the two inner products (Bondarenko, 2013).

The graph $S\subset S^{d-1}$28 with parameters

$S\subset S^{d-1}$29

yields a two-distance set of $S\subset S^{d-1}$30 points on the unit sphere $S\subset S^{d-1}$31. In this case the positive eigenspace has dimension $S\subset S^{d-1}$32, and the two inner products are

$S\subset S^{d-1}$33

Because the diameter is realized by nonadjacent vertices, any partition into smaller-diameter parts corresponds to a partition of the graph into cliques. The paper proves that $S\subset S^{d-1}$34 has no $S\subset S^{d-1}$35-clique, hence

$S\subset S^{d-1}$36

so any such partition needs at least

$S\subset S^{d-1}$37

parts. Equivalently, the configuration cannot be partitioned into $S\subset S^{d-1}$38 parts of smaller diameter, proving

$S\subset S^{d-1}$39

This answers Larman’s question negatively for two-distance sets and reduces the smallest dimension in which Borsuk’s conjecture is known to be false in the two-distance spherical setting (Bondarenko, 2013).

A second example comes from the strongly regular $S\subset S^{d-1}$40 graph with parameters

$S\subset S^{d-1}$41

Its Euclidean representation lies in $S\subset S^{d-1}$42, hence on $S\subset S^{d-1}$43, and has inner products

$S\subset S^{d-1}$44

Again diameter corresponds to nonadjacency, so the clique number controls Borsuk partitions. The paper shows that a clique cannot have more than $S\subset S^{d-1}$45 vertices, giving

$S\subset S^{d-1}$46

and therefore

$S\subset S^{d-1}$47

It also derives the nearby bounds

$S\subset S^{d-1}$48

by a standard stacking construction (Bondarenko, 2013).

These examples show that spherical two-distance sets are not merely highly symmetric packings; they can also force extremal partition behavior. A plausible implication is that the graph-theoretic rigidity responsible for exact cardinality results also underlies the failure of naive diameter-partition heuristics in high dimensions.

6. Fixed-angle asymptotics, signed graphs, and broader extensions

A modern asymptotic version fixes the two inner products and studies

$S\subset S^{d-1}$49

the largest size of a spherical $S\subset S^{d-1}$50-code in $S\subset S^{d-1}$51, for fixed

$S\subset S^{d-1}$52

The key parameters are

$S\subset S^{d-1}$53

The conjectural limit is expressed through a signed-graph invariant $S\subset S^{d-1}$54: $S\subset S^{d-1}$55 This conjecture is proved when $S\subset S^{d-1}$56, or when $S\subset S^{d-1}$57 under the stated $S\subset S^{d-1}$58-conditions. In particular, if $S\subset S^{d-1}$59 and $S\subset S^{d-1}$60, then

$S\subset S^{d-1}$61

if $S\subset S^{d-1}$62 and $S\subset S^{d-1}$63, then

$S\subset S^{d-1}$64

and if $S\subset S^{d-1}$65 with $S\subset S^{d-1}$66, then

$S\subset S^{d-1}$67

This is the first determination of $S\subset S^{d-1}$68 for nontrivial fixed values of $S\subset S^{d-1}$69 and $S\subset S^{d-1}$70 outside the equiangular setting (Jiang et al., 2020).

Two broader methodological extensions place spherical two-distance sets inside the general theory of few-distance spherical sets. First, the Petrov–Pohoata method, combined with Sylvester’s law of inertia and Gröbner bases, yields a short algebraic proof of the Delsarte–Goethals–Seidel bound for spherical $S\subset S^{d-1}$71-distance sets, and hence recovers $S\subset S^{d-1}$72 for the two-distance case (Hegedüs et al., 2020). Second, the polynomial method for spherical $S\subset S^{d-1}$73-distance sets with

$S\subset S^{d-1}$74

gives

$S\subset S^{d-1}$75

which for $S\subset S^{d-1}$76 again becomes the Gerzon bound $S\subset S^{d-1}$77 (Datta et al., 2021).

Related work on strength and antipodality shows how the two-distance problem interacts with association schemes, real ETFs, and Levenstein-equality packings. For spherical $S\subset S^{d-1}$78-distance sets of strength $S\subset S^{d-1}$79, the improved upper bound $S\subset S^{d-1}$80 can be read both as a geometric estimate and as a restriction on strongly regular graphs with suitable Krein-parameter behavior (Nozaki et al., 2010). In antipodal few-angle settings, the corresponding half-sets are equiangular or two-distance spherical embeddings of strongly regular graphs, which indicates that the structural role of two-distance geometry persists well beyond the literal $S\subset S^{d-1}$81 case (Xu et al., 2020).

Taken together, these developments show that spherical two-distance sets occupy a distinguished position among spherical codes. They are simultaneously constrained enough to admit exact classification results in several regimes, yet flexible enough to encode strongly regular graphs, tight frames, asymptotic signed-graph phenomena, and high-dimensional counterexamples in extremal geometry.