Block-Regular Steiner Triple Systems

• Block-regular Steiner triple systems are defined by uniform block organization manifesting in strongly regular block graphs and incidence matrices.
• Spectral methods link nowhere-zero integer eigenvectors to minimum flow problems and regularity bounds, offering practical L∞-norm measures based on order congruences.
• Incidence graph constructions and combinatorial decompositions, such as transversal subdesigns and flower partitions, yield precise classifications including STS(21) cases.

Block-regular Steiner triple systems are not introduced as a single formal class in several recent arXiv treatments; instead, the literature makes block-side regularity precise through block graphs, incidence graphs, completely regular codes, nowhere-zero flows, block-avoiding sequencings, expansion properties, and tightly constrained subdesign decompositions. A Steiner triple system of order $v$, written $\operatorname{STS}(v)$, is a pair $(V,\mathcal B)$ in which $\mathcal B$ is a collection of 3-subsets of $V$, called blocks or triples, such that every pair of distinct points lies in exactly one block. In this sense, the topic concerns how the blocks of an STS organize themselves into regular graph-theoretic, algebraic, and structural patterns (Bespalov et al., 2023, Araujo-Pardo et al., 2019, Erskine et al., 2022, Guan et al., 2019, Blázsik et al., 2019).

1. Basic objects and interpretations of block-side regularity

For a classical Steiner triple system, the primary block-side objects are the incidence matrix and the block graph. If $S$ is an STS, its incidence matrix $W_S$ has rows indexed by points and columns by blocks. The block graph $T_S$ has the blocks as vertices, with two distinct blocks adjacent when they intersect nontrivially. The cited spectral treatment also works with $q$-ary Steiner triple systems, where blocks are 3-subspaces of $\mathbb F_q^n$ and points are 1-subspaces; the classical case is $\operatorname{STS}(v)$0. In that framework, the block graph is strongly regular, and it can be viewed as an induced subgraph of the distance-2 graph of the corresponding Grassmann graph $\operatorname{STS}(v)$1 (Bespalov et al., 2023).

A second regularity viewpoint comes from the point-block incidence graph. For a Steiner system $\operatorname{STS}(v)$2, the point-block incidence graph is bipartite, one part consisting of points and the other of blocks. In this graph, every block has degree $\operatorname{STS}(v)$3, and every point has degree

$\operatorname{STS}(v)$4

For Steiner triple systems, this becomes the biregular pair $\operatorname{STS}(v)$5. A plausible implication is that one natural meaning of “block-regular” is the uniformity of the block side in the incidence graph: each block has the same size, and the block part is degree-homogeneous (Araujo-Pardo et al., 2019).

A third viewpoint is structural rather than algebraic. Work on $\operatorname{STS}(v)$6 containing transversal subdesigns $\operatorname{STS}(v)$7 shows that the block set can decompose into a transversal component together with three highly constrained “petals,” one of which is a genuine sub-STS and two of which are almost-sub-STS. This suggests that block-regularity can also refer to rigid subdesign organization, not only to graph eigenstructure or incidence parameters (Guan et al., 2019).

2. Spectral formulation via block graphs

The most explicit algebraic treatment is formulated in terms of nowhere-zero integer eigenvectors of block graphs. A vector is a nowhere-zero integer vector if all of its entries are nonzero integers, and the quantity studied is the minimum possible $\operatorname{STS}(v)$8-norm among such vectors in a specified eigenspace. For a distance-regular graph $\operatorname{STS}(v)$9 and an eigenvalue indexed by $(V,\mathcal B)$0, the paper denotes the corresponding minimum by

$(V,\mathcal B)$1

with the source notation using the eigenspaces $(V,\mathcal B)$2 and $(V,\mathcal B)$3 (Bespalov et al., 2023).

For Steiner triple systems, the second eigenspace of the block graph is exactly the nullspace of the incidence matrix. Proposition 4 states that for a $(V,\mathcal B)$4-ary STS $(V,\mathcal B)$5, a nonzero vector $(V,\mathcal B)$6 satisfies

$(V,\mathcal B)$7

if and only if $(V,\mathcal B)$8 is a $(V,\mathcal B)$9-eigenvector of the block graph. The same proposition identifies flows with these eigenvectors: a vector $\mathcal B$0 is a nowhere-zero $\mathcal B$1-flow for $\mathcal B$2 if and only if $\mathcal B$3 is a nowhere-zero integer $\mathcal B$4-eigenvector of $\mathcal B$5. The paper derives this from the spectral relation between a biregular incidence graph and its two halved graphs. This is the central exact equivalence linking block-graph spectral theory to minimum nowhere-zero flows (Bespalov et al., 2023).

The first eigenspace is described through incidence maps from points to blocks. For $\mathcal B$6, the set of real-valued nonzero vectors on points whose coordinates sum to zero, the cited work shows that $\mathcal B$7 is precisely the set of all $\mathcal B$8-eigenvectors of the block graph. For $\mathcal B$9, any $V$0-eigenvector restricts to a $V$1-eigenvector on the blocks of $V$2, and every $V$3-eigenvector extends uniquely to a $V$4-eigenvector. This places the first eigenspace of the block graph inside the spectral theory of Johnson and Grassmann graphs (Bespalov et al., 2023).

3. Norm minimization, flows, and completely regular codes

The first-eigenvalue minimization problem behaves differently for block graphs of Steiner triple systems than for Johnson graphs. For Johnson graphs, the cited work obtains several infinite-series results and solves the case $V$5 for sufficiently large $V$6. For Steiner triple system block graphs, the result is markedly uniform: if $V$7 is a Steiner triple system of order $V$8, $V$9, then

$S$0

and

$S$1

The proof is constructive. For $S$2, the construction begins with a point vector with entries $S$3, and for $S$4, it uses an alternating assignment on an auxiliary graph built from triples through two fixed points. In both cases, the induced block eigenvectors have small $S$5-norm (Bespalov et al., 2023).

For the second eigenspace, the main flow theorem concerns Assmus–Mattson systems. Any Assmus–Mattson Steiner triple system of order $S$6 admits a zero-sum $S$7-flow, equivalently

$S$8

The construction starts with a Steiner triple system $S$9 of order $W_S$0, applies the Assmus–Mattson construction to obtain a new STS $W_S$1 of order $W_S$2, and then defines a bounded integer vector on the blocks of $W_S$3 so that all pointwise sums cancel. The resulting vector is nowhere-zero and all values are bounded by $W_S$4, hence it yields a zero-sum $W_S$5-flow. The paper emphasizes that this conclusion holds regardless of the flow properties of the original STS (Bespalov et al., 2023).

The same paper links small-norm eigenvectors to completely regular codes in block graphs. A completely regular code with covering radius $W_S$6 produces a two-valued eigenvector, and minimum $W_S$7-norm $W_S$8 occurs exactly when such a code exists. For block graphs of Steiner triple systems, a $W_S$9, second-eigenvalue completely regular code is exactly a 1-design of blocks. The paper lists several constructions: blocks through a fixed point, Steiner subsystems of order $T_S$0, unions of such objects, 1-subdesigns, and small Steiner subsystems as $T_S$1 codes. In the projective or Hamming STS, whose block graph is a Grassmann graph $T_S$2, these codes coincide with Cameron–Liebler line classes. For STS of order 13 or 15, all completely regular codes with $T_S$3 in the first eigenspace are exactly the ones from the standard constructions listed in the paper (Bespalov et al., 2023).

4. Incidence graphs and bipartite biregular cages

The incidence-graph formulation gives a different and very concrete regularity theory. A bipartite biregular $T_S$4-graph is a bipartite graph of even girth $T_S$5 whose degree set is $T_S$6, with all vertices in the same partite set having the same degree. An $T_S$7-bipartite biregular cage is such a graph of minimum possible order, and a bipartite biregular Moore cage is one that attains the adjusted Moore lower bound (Araujo-Pardo et al., 2019).

For Steiner systems, the correspondence is exact. Theorem 8 states that the point-block incidence graph of a Steiner system $T_S$8, $T_S$9, is a bipartite biregular $q$0-Moore cage, where

$q$1

The girth is $q$2: bipartiteness forces even cycles, a 4-cycle would imply that two distinct blocks share the same two points, and three points not all contained in one block generate a 6-cycle via the three blocks containing the three pairs of points. The total number of vertices is

$q$3

which matches the Moore-type lower bound for girth $q$4 (Araujo-Pardo et al., 2019).

In the special case $q$5, the paper completely solves the existence problem for $q$6-bipartite biregular cages. Since an $q$7 exists if and only if $q$8 and $q$9, and since in an STS one has

$\mathbb F_q^n$0

Steiner triple systems immediately yield the Moore-cage cases $\mathbb F_q^n$1. The full classification is: $\mathbb F_q^n$2

$\mathbb F_q^n$3

For the exceptional congruence class $\mathbb F_q^n$4, the construction begins with an $\mathbb F_q^n$5, whose incidence graph is a $\mathbb F_q^n$6-Moore cage, and then deletes one point together with all blocks incident with that point. The resulting graph has all remaining blocks of degree $\mathbb F_q^n$7, all remaining points of degree $\mathbb F_q^n$8, no 4-cycles, girth $\mathbb F_q^n$9, and the exact minimum order stated above (Araujo-Pardo et al., 2019).

5. Sequencing, spreading, and expansion as regularity-adjacent notions

A different strand of the literature studies regularity through the avoidance or propagation of blocks rather than through eigenvectors. An $\operatorname{STS}(v)$00-good sequencing of an $\operatorname{STS}(v)$01 is a permutation of the points such that no $\operatorname{STS}(v)$02 consecutive points contain a block. The cited work proves that every $\operatorname{STS}(v)$03 with $\operatorname{STS}(v)$04 has a 3-good sequencing, every $\operatorname{STS}(v)$05 with $\operatorname{STS}(v)$06 has a 4-good sequencing, and every 3-chromatic $\operatorname{STS}(v)$07 with $\operatorname{STS}(v)$08 has a 5-good sequencing. It also gives a general coloring criterion: if an $\operatorname{STS}(v)$09 admits a coloring in which every color class except possibly one has at least

$\operatorname{STS}(v)$10

points, then the system admits an $\operatorname{STS}(v)$11-good sequencing. The paper explicitly relates sequencing to colorings and independent sets, and states that this connects naturally to block-regularity in the sense that a sequencing is more “regular” when it can be arranged to avoid blocks over longer intervals (Erskine et al., 2022).

The expansion literature uses linear triple systems and their shadow graphs. For a linear triple system $\operatorname{STS}(v)$12, the shadow graph $\operatorname{STS}(v)$13 has exactly those pairs that are covered by triples of $\operatorname{STS}(v)$14. The closure operator $\operatorname{STS}(v)$15 is defined from the neighborhood process in the shadow graph, and a linear triple system is spreading if $\operatorname{STS}(v)$16 for every nontrivial subset $\operatorname{STS}(v)$17. It is weakly spreading if the same holds whenever $\operatorname{STS}(v)$18 is the vertex set of a subfamily of more than one triple. The paper states the implication chain

$\operatorname{STS}(v)$19

and, specifically for Steiner triple systems, that an STS is subsystem-free, that is, spreading, if and only if it is strongly connected (Blázsik et al., 2019).

The same work introduces an $\operatorname{STS}(v)$20-expander STS: for every nontrivial $\operatorname{STS}(v)$21 with $\operatorname{STS}(v)$22,

$\operatorname{STS}(v)$23

Its main expander theorem states that for odd prime $\operatorname{STS}(v)$24 there exists an $\operatorname{STS}(v)$25 such that

$\operatorname{STS}(v)$26

for every $\operatorname{STS}(v)$27 with $\operatorname{STS}(v)$28. A corollary states that for every sufficiently large $\operatorname{STS}(v)$29, there exists an STS of order $\operatorname{STS}(v)$30 satisfying the same inequality, and hence for every $\operatorname{STS}(v)$31 one can find an STS $\operatorname{STS}(v)$32 with

$\operatorname{STS}(v)$33

that is “almost $\operatorname{STS}(v)$34-expander.” These notions are not identified in the source as block-regularity proper, but they are regularity-adjacent constraints on how blocks interact through the shadow graph (Blázsik et al., 2019).

6. Transversal subdesigns, flowers, and the STS(21) classification

The most detailed finite structural classification in the supplied literature concerns Steiner triple systems of order $\operatorname{STS}(v)$35 that contain a transversal subdesign $\operatorname{STS}(v)$36. A transversal design $\operatorname{STS}(v)$37 consists of a point set partitioned into three groups $\operatorname{STS}(v)$38, each of size $\operatorname{STS}(v)$39, together with triples meeting each group in exactly one point, such that every pair of points from different groups lies in exactly one block. The paper proves that if an $\operatorname{STS}(v)$40 contains a sub-$\operatorname{STS}(v)$41, then its block set decomposes as

$\operatorname{STS}(v)$42

where the supports of $\operatorname{STS}(v)$43 are $\operatorname{STS}(v)$44, $\operatorname{STS}(v)$45, $\operatorname{STS}(v)$46 with $\operatorname{STS}(v)$47, two of these are almost-sub-STS with missing triple $\operatorname{STS}(v)$48, and the remaining one is a genuine sub-STS. This is the paper’s main structural decomposition (Guan et al., 2019).

For $\operatorname{STS}(v)$49, the paper introduces the notion of a flower: a partition of the 21 points into $\operatorname{STS}(v)$50 with sizes $\operatorname{STS}(v)$51, where the system has one sub-$\operatorname{STS}(v)$52 and two almost-sub-$\operatorname{STS}(v)$53 supported on $\operatorname{STS}(v)$54, $\operatorname{STS}(v)$55, $\operatorname{STS}(v)$56, all with missing triple $\operatorname{STS}(v)$57. It then proves that an $\operatorname{STS}(v)$58 has such a flower if and only if it has a sub-$\operatorname{STS}(v)$59 with groups $\operatorname{STS}(v)$60. Thus, in order 21, flowers and transversal subdesigns are equivalent descriptions of the same structural phenomenon (Guan et al., 2019).

The interaction of multiple flowers is highly rigid. If an $\operatorname{STS}(v)$61 has two different flowers with stems $\operatorname{STS}(v)$62 and $\operatorname{STS}(v)$63, then $\operatorname{STS}(v)$64, and the system contains a sub-$\operatorname{STS}(v)$65 on a support of the form $\operatorname{STS}(v)$66. The classification then splits into two cases. If both stems are blocks, the system has exactly 7 sub-$\operatorname{STS}(v)$67 and exactly 7 sub-$\operatorname{STS}(v)$68; if at most one stem is a block, then it has exactly 3 sub-$\operatorname{STS}(v)$69, and either 3 or 1 sub-$\operatorname{STS}(v)$70. The seven-support case is organized by the seven lines of the Fano plane on the seven parts $\operatorname{STS}(v)$71 (Guan et al., 2019).

The computational classification is exhaustive. There are exactly $\operatorname{STS}(v)$72 isomorphism classes of $\operatorname{STS}(v)$73 containing a transversal subdesign $\operatorname{STS}(v)$74. Among them, $\operatorname{STS}(v)$75 have exactly one sub-$\operatorname{STS}(v)$76, $\operatorname{STS}(v)$77 have exactly three, and $\operatorname{STS}(v)$78 have seven. The authors also checked resolvability and found $\operatorname{STS}(v)$79 resolvable isomorphism classes. These data do not define block-regularity as a named class, but they do show that controlled subdesign structure can be classified very finely and can force strong regular overlap patterns (Guan et al., 2019).

7. Synthesis and scope of the concept

Taken together, these papers support a precise but plural understanding of block-regular Steiner triple systems. In one sense, block-regularity is spectral: the block graph is strongly regular, its first and second eigenspaces are described through incidence maps, and minimum nowhere-zero integer eigenvectors encode both small-norm block-graph structure and minimum nowhere-zero flows. In another sense, block-regularity is incidence-theoretic: the point-block incidence graph of an STS is a $\operatorname{STS}(v)$80-biregular graph and, when the Steiner congruence conditions are met, a Moore cage. In a third sense, block-regularity is structural: block-avoiding sequencings, spreading and expansion in shadow graphs, and flower decompositions with transversal subdesigns all impose strong regularity conditions on how blocks can cluster, propagate, or overlap (Bespalov et al., 2023, Araujo-Pardo et al., 2019, Erskine et al., 2022, Blázsik et al., 2019, Guan et al., 2019).

Several of the cited papers explicitly note that they do not introduce “block-regular” as a separate formal term. This suggests that the topic is best treated as an umbrella for a family of regularity phenomena centered on the block side of an STS rather than as a single universally standardized definition. Within that umbrella, the sharpest exact results presently assembled in the supplied literature are the equivalence

$\operatorname{STS}(v)$81

the resulting interpretation of minimum nowhere-zero flows as second-eigenvalue minimization in the block graph, the universal first-eigenvalue bounds $\operatorname{STS}(v)$82 or $\operatorname{STS}(v)$83 according to $\operatorname{STS}(v)$84, the zero-sum $\operatorname{STS}(v)$85-flow theorem for Assmus–Mattson systems of order at least $\operatorname{STS}(v)$86, the Moore-cage correspondence for incidence graphs, and the complete classification of the $\operatorname{STS}(v)$87 isomorphism classes of $\operatorname{STS}(v)$88 containing $\operatorname{STS}(v)$89 (Bespalov et al., 2023, Araujo-Pardo et al., 2019, Guan et al., 2019).