Net-Regular Strongly Regular Signed Graphs

• Net-regular strongly regular signed graphs are signed graphs with a regular underlying structure and uniform net-degrees, ensuring the same difference between positive and negative edge counts at every vertex.
• They are analyzed using Stanić's strong regularity conditions and quasi-balanced weighing matrices, which lead to explicit classifications in low degrees such as 5 and 6.
• The research interconnects combinatorial, matrix-theoretic, and spectral approaches, demonstrating how local sign constraints and association schemes yield both finite and infinite families of structured graphs.

Searching arXiv for relevant papers on net-regular strongly regular signed graphs and quasi-balanced weighing matrices. A net-regular strongly regular signed graph is a signed graph in which the underlying graph is regular, the signed adjacency matrix satisfies a strong-regularity condition formulated through uniform signed 2-walk counts, and every vertex has the same net-degree $d^+(v)-d^-(v)$. In the contemporary literature, this topic lies at the intersection of signed graph theory, strongly regular graph theory, weighing matrices, and association schemes. Two strands are especially relevant. One strand studies strongly regular signed graphs in the sense of Stanić and classifies net-regular examples of small degree, notably degrees $5$ and $6$ (Yu et al., 31 Jul 2025, Yu et al., 1 Aug 2025). The other strand studies quasi-balanced weighing matrices whose support matrices are adjacency matrices of strongly regular graphs; in that setting, signed strongly regular graphs arise as signings of strongly regular graphs, and in many cases the resulting structure is equivalent to an association scheme (Kharaghani et al., 2022). A related spectral perspective comes from signed regular graphs with exactly two distinct eigenvalues, where signed strong regularity appears through the notion of FSRSG and the special case $p=0$ (Ramezani, 2019).

1. Definitions and formal framework

A signed graph is a pair

$\dot{G}=(G,\sigma),$

where $G$ is a simple graph and

$\sigma:E(G)\to \{+1,-1\}.$

If $v_i\sim v_j$, then the signed adjacency matrix satisfies

$a_{ij}^\sigma=\sigma(v_iv_j),$

and otherwise $a_{ij}^\sigma=0$ (Yu et al., 1 Aug 2025). The matrix $5$0 is the signed adjacency matrix, and its eigenvalues are the eigenvalues of the signed graph (Yu et al., 1 Aug 2025).

For a vertex $5$1, the degree in the underlying graph is $5$2, while $5$3 and $5$4 denote the numbers of positive and negative incident edges. A signed graph is $5$5-regular if every vertex has degree $5$6, and it is $5$7-net-regular if every vertex has the same net-degree

$5$8

Hence, if the graph is both $5$9-regular and $6$0-net-regular, then

$6$1

for every vertex (Yu et al., 1 Aug 2025). This uniformity condition is the defining feature of net-regularity.

The strongly regular signed graph condition used in the recent degree-$6$2 and degree-$6$3 classification papers is the definition of Stanić. A signed graph $6$4 is strongly regular if it is neither homogeneous complete nor edgeless, and there exist integers

$6$5

such that the entries of $6$6 depend only on whether two vertices are equal, positively adjacent, negatively adjacent, or nonadjacent. Explicitly,

$6$7

Equivalently, the defining matrix identity is

$6$8

which the papers rewrite as

$6$9

using $p=0$0 (Yu et al., 31 Jul 2025, Yu et al., 1 Aug 2025).

When $p=0$1 is $p=0$2-net-regular, the all-ones vector $p=0$3 is an eigenvector of $p=0$4, $p=0$5, and $p=0$6, yielding the key parameter constraint

$p=0$7

This equation is one of the principal numerical tools in the classification theory of net-regular strongly regular signed graphs (Yu et al., 31 Jul 2025, Yu et al., 1 Aug 2025).

2. Structural classes and local constraints

The modern classification theory relies heavily on the five classes $p=0$8 recalled from Koledin–Stanić. These classes partition inhomogeneous strongly regular signed graphs according to the relations among $p=0$9, $\dot{G}=(G,\sigma),$0, and $\dot{G}=(G,\sigma),$1. In brief, $\dot{G}=(G,\sigma),$2 and $\dot{G}=(G,\sigma),$3 have $\dot{G}=(G,\sigma),$4, $\dot{G}=(G,\sigma),$5, $\dot{G}=(G,\sigma),$6, and $\dot{G}=(G,\sigma),$7 have $\dot{G}=(G,\sigma),$8, and the cases are further separated by whether $\dot{G}=(G,\sigma),$9 or $G$0 (Yu et al., 31 Jul 2025, Yu et al., 1 Aug 2025).

Two lemmas are repeatedly used in the degree-$G$1 and degree-$G$2 classifications. First, negation preserves the SRSG class: if $G$3, then $G$4, with $G$5 and $G$6 interchanged (Yu et al., 31 Jul 2025, Yu et al., 1 Aug 2025). This permits reduction to nonnegative net-degree. Second, for a connected non-complete net-regular SRSG in $G$7, the number of negative walks of length $G$8 between any two vertices is even; if there are $G$9 such walks, then they arise in $\sigma:E(G)\to \{+1,-1\}.$0 matched positive/negative common-neighbor pairs (Yu et al., 31 Jul 2025, Yu et al., 1 Aug 2025). This parity phenomenon imposes strong restrictions on allowable local sign patterns.

Triangle structure is another central organizing principle. The degree-$\sigma:E(G)\to \{+1,-1\}.$1 and degree-$\sigma:E(G)\to \{+1,-1\}.$2 analyses distinguish balanced triangles, unbalanced triangles of the first type, and unbalanced triangles of the second type, and repeatedly derive contradictions from the coexistence of these triangle types with fixed values of $\sigma:E(G)\to \{+1,-1\}.$3, $\sigma:E(G)\to \{+1,-1\}.$4, and the parameters $\sigma:E(G)\to \{+1,-1\}.$5 (Yu et al., 31 Jul 2025, Yu et al., 1 Aug 2025). A plausible implication is that local signed triangle data serve as an effective finite surrogate for more global signed walk constraints in low-degree classification problems.

3. Net-regularity through quasi-balanced signings of strongly regular graphs

A second major framework comes from quasi-balanced weighing matrices. A weighing matrix of order $\sigma:E(G)\to \{+1,-1\}.$6 and weight $\sigma:E(G)\to \{+1,-1\}.$7 is a $\sigma:E(G)\to \{+1,-1\}.$8-matrix $\sigma:E(G)\to \{+1,-1\}.$9 satisfying

$v_i\sim v_j$0

The matrix is quasi-balanced when its absolute value matrix $v_i\sim v_j$1, defined by $v_i\sim v_j$2, satisfies

$v_i\sim v_j$3

and the product $v_i\sim v_j$4 has at most two off-diagonal entries (Kharaghani et al., 2022). If $v_i\sim v_j$5 has exactly one off-diagonal entry, then $v_i\sim v_j$6 is an ordinary balanced weighing matrix, so quasi-balanced matrices generalize balanced ones (Kharaghani et al., 2022).

The key case for signed graph theory occurs when $v_i\sim v_j$7 is the adjacency matrix $v_i\sim v_j$8 of a strongly regular graph with parameters $v_i\sim v_j$9, i.e.

$a_{ij}^\sigma=\sigma(v_iv_j),$0

Then $a_{ij}^\sigma=\sigma(v_iv_j),$1 is said to sign the graph (Kharaghani et al., 2022). In this language, a graph is signable if some of the nonzero entries of $a_{ij}^\sigma=\sigma(v_iv_j),$2 can be changed to $a_{ij}^\sigma=\sigma(v_iv_j),$3 so that the resulting matrix is a quasi-balanced weighing matrix (Kharaghani et al., 2022). This ties the signed condition

$a_{ij}^\sigma=\sigma(v_iv_j),$4

to the combinatorial regularity of the underlying strongly regular graph.

For signings over $a_{ij}^\sigma=\sigma(v_iv_j),$5, the paper defines the multisets

$a_{ij}^\sigma=\sigma(v_iv_j),$6

If each element of $a_{ij}^\sigma=\sigma(v_iv_j),$7 appears exactly $a_{ij}^\sigma=\sigma(v_iv_j),$8 times in $a_{ij}^\sigma=\sigma(v_iv_j),$9 when $a_{ij}^\sigma=0$0, and exactly $a_{ij}^\sigma=0$1 times when $a_{ij}^\sigma=0$2, then the signing is called srg-balanced (Kharaghani et al., 2022). The paper proves that this is equivalent to a matrix identity which reduces to

$a_{ij}^\sigma=0$3

so an srg-balanced signing is automatically a Butson weighing matrix (Kharaghani et al., 2022). Moreover, if a signing over the $a_{ij}^\sigma=0$4-th roots of unity exists for a prime $a_{ij}^\sigma=0$5, then $a_{ij}^\sigma=0$6 must divide both $a_{ij}^\sigma=0$7 and $a_{ij}^\sigma=0$8, and the signing is srg-balanced (Kharaghani et al., 2022). This is a strong arithmetic restriction on net-regular or balanced signed strongly regular graphs in the root-of-unity setting.

This quasi-balanced framework is not identical to the Stanić-type SRSG framework used in the degree-$a_{ij}^\sigma=0$9 and degree-$5$00 classifications, but the two theories overlap in their focus on regular signed support, uniform signed common-neighbor data, and rigid algebraic structure. This suggests that net-regularity can be viewed both as a local sign-balance constraint and as a manifestation of matrix-theoretic regularity in suitably structured signings of strongly regular graphs.

4. Association schemes and algebraic-combinatorial equivalence

One of the principal conceptual advances in the quasi-balanced theory is the identification of signed strongly regular graph signings with association schemes. If $5$01 is a quasi-balanced weighing matrix with

$5$02

where $5$03 is the adjacency matrix of a strongly regular graph, then seven adjacency matrices $5$04 are defined using block forms involving $5$05, $5$06, their transposes, and the matrix

$5$07

Theorem 4.1 states that if $5$08 is quasi-balanced and $5$09 is an SRG $5$10, then the matrices $5$11 form an association scheme (Kharaghani et al., 2022). The theorem also gives explicit first and second eigenmatrices in terms of the eigenvalues $5$12 of the SRG (Kharaghani et al., 2022).

The converse is equally significant. Theorem 4.2 proves that if an association scheme exists with the eigenmatrices from Theorem 4.1, then there exists a quasi-balanced weighing matrix $5$13 whose absolute value is an SRG with the corresponding parameters (Kharaghani et al., 2022). Thus the correspondence is two-way:

• quasi-balanced signed SRG $5$14 association scheme;
• association scheme with the specified eigenstructure $5$15 quasi-balanced signed SRG (Kharaghani et al., 2022).

Analogous two-way correspondences hold when $5$16 is the incidence matrix of a symmetric group divisible design or the adjacency matrix of a divisible design graph, under regularity conditions such as

$5$17

or

$5$18

(Kharaghani et al., 2022). Although these results go beyond strongly regular graphs proper, they reinforce the same point: quasi-balanced signed objects are often precisely the graph-theoretic shadows of association schemes.

For net-regular strongly regular signed graphs, this association-scheme viewpoint is important because it indicates that uniformity of net-degree and signed 2-walk counts is not merely a combinatorial curiosity; it is frequently the surface manifestation of a deeper Bose–Mesner-type structure. A plausible implication is that classification results for net-regular SRSGs may increasingly depend on association-scheme methods as the degree grows.

5. Classification in degree 5

The degree-$5$19 classification determines all connected net-regular strongly regular signed graphs with degree $5$20 (Yu et al., 31 Jul 2025). Since negation preserves the class, it suffices to consider positive net-degree. Because $5$21, the possible positive net-degrees are $5$22 and $5$23 (Yu et al., 31 Jul 2025).

For the $5$24-net-regular case,

$5$25

so the negative edges form a perfect matching and the order is even (Yu et al., 31 Jul 2025). The classification yields exactly five connected $5$26-regular and $5$27-net-regular strongly regular signed graphs: $5$28 (Yu et al., 31 Jul 2025). One complete case occurs in $5$29, namely $5$30, producing $5$31 with parameters $5$32 (Yu et al., 31 Jul 2025). In the non-complete cases, the analysis divides into the presence or absence of unbalanced triangles, uses the identity

$5$33

and eliminates candidate parameter sets through local configuration arguments, divisibility constraints, and inspection of underlying $5$34-regular graphs of small order (Yu et al., 31 Jul 2025).

For the $5$35-net-regular case,

$5$36

(Yu et al., 31 Jul 2025). The complete case yields exactly one graph, $5$37, with parameters $5$38 (Yu et al., 31 Jul 2025). Among non-complete graphs, a finite set of $5$39 pairs survives preliminary restrictions, but all except one are eliminated. The only remaining graph is

$5$40

with parameters $5$41 (Yu et al., 31 Jul 2025). The construction is particularly concrete: $5$42, $5$43, and the vertex set simultaneously decomposes into three positive $5$44-cliques and four negative $5$45-cliques (Yu et al., 31 Jul 2025).

The final degree-$5$46 count is therefore seven connected examples up to sign reversal: five with net-degree $5$47 and two with net-degree $5$48 (Yu et al., 31 Jul 2025).

Net-degree	Number of connected 5-regular SRSGs	Named graphs
$5$49	5	$5$50
$5$51	2	$5$52

6. Classification in degree 6 and broader spectral context

The degree-$5$53 classification determines all connected $5$54-regular and net-regular strongly regular signed graphs (Yu et al., 1 Aug 2025). By negation symmetry, only nonnegative net-degrees need be treated: $5$55 (Yu et al., 1 Aug 2025).

For net-degree $5$56,

$5$57

so $5$58 is a perfect matching (Yu et al., 1 Aug 2025). The key parameter equation becomes

$5$59

A structural lemma shows that such graphs contain no unbalanced triangle; then $5$60 is forced, because negatively adjacent vertices cannot share common neighbors (Yu et al., 1 Aug 2025). Exactly three connected $5$61-regular $5$62-net-regular SRSGs survive: $5$63 (Yu et al., 1 Aug 2025).

For net-degree $5$64,

$5$65

so $5$66 is a disjoint union of cycles (Yu et al., 1 Aug 2025). The governing identity is

$5$67

The analysis separates graphs containing an unbalanced triangle of the first type, an unbalanced triangle of the second type, or only balanced triangles (Yu et al., 1 Aug 2025). After extensive elimination, exactly six connected examples remain: $5$68 (Yu et al., 1 Aug 2025).

For net-degree $5$69,

$5$70

so both $5$71 and $5$72 are $5$73-regular (Yu et al., 1 Aug 2025). The relevant identity is

$5$74

The case $5$75 yields the unique graph $5$76 with parameters $5$77, while $5$78 yields the graph $5$79 with parameters $5$80 (Yu et al., 1 Aug 2025). The final count is exactly four connected $5$81-regular $5$82-net-regular SRSGs: $5$83 (Yu et al., 1 Aug 2025).

Altogether, there are

$5$84

connected $5$85-regular net-regular SRSGs (Yu et al., 1 Aug 2025).

Net-degree	Number of connected 6-regular SRSGs	Named graphs
$5$86	3	$5$87
$5$88	6	$5$89
$5$90	4	$5$91

A related but distinct spectral framework appears in the study of signed regular graphs with exactly two distinct eigenvalues. There a signed graph is called an FSRSG if it satisfies a signed analogue of strong regularity, and Theorem 1.1 states that a signed graph is a STE if and only if it is a FSRSG with $5$92 (Ramezani, 2019). For a STE, the adjacency matrix satisfies

$5$93

so the two eigenvalues are

$5$94

(Ramezani, 2019). The paper constructs infinite families, including connected signed $5$95-regular graphs with spectrum

$5$96

from semi-orthogonal weighing matrices (Ramezani, 2019), and more generally proves that for any $5$97 there are infinitely many connected signed $5$98-regular graphs with distinct eigenvalues $5$99 (Ramezani, 2019). Since STE is equivalent to FSRSG with $6$00, these families furnish explicit infinite families of strongly regular signed graphs in the signed sense (Ramezani, 2019).

7. Significance, scope, and common points of interpretation

The current theory shows that net-regular strongly regular signed graphs are highly rigid objects. In low degrees, the combination of fixed degree, fixed net-degree, strong regularity, and parity constraints on signed 2-walks leads to finite and explicit classifications (Yu et al., 31 Jul 2025, Yu et al., 1 Aug 2025). In the matrix-theoretic direction, quasi-balanced signings of strongly regular graphs are constrained by root-of-unity uniformity, divisibility conditions on the strongly regular graph parameters, and equivalence with association schemes (Kharaghani et al., 2022). In the spectral direction, the case of exactly two adjacency eigenvalues connects strongly regular signed graphs to weighing matrices, conference-type constructions, and infinite regular families (Ramezani, 2019).

A common misconception is to identify net-regularity with ordinary regularity of the underlying graph. The degree-$6$01 and degree-$6$02 papers make clear that net-regularity is a stronger signed condition, fixing the difference $6$03 at every vertex and thereby constraining the positive and negative subgraphs $6$04 and $6$05 separately (Yu et al., 31 Jul 2025, Yu et al., 1 Aug 2025). Another potential misconception is that all signed strongly regular graphs should arise from unsigned strongly regular graphs by arbitrary sign changes. The quasi-balanced framework indicates that meaningful signings are highly structured: when $6$06 is an SRG adjacency matrix, the relevant signings can force Butson-type conditions and association schemes (Kharaghani et al., 2022).

The broader significance of the subject is therefore twofold. First, it provides a refined signed analogue of strong regularity that captures uniform local interaction between positive and negative adjacencies. Second, it links signed graph theory to several algebraic-combinatorial frameworks—especially weighing matrices, Butson matrices, symmetric group divisible designs, divisible design graphs, and association schemes (Kharaghani et al., 2022). The low-degree classifications in degrees $6$07 and $6$08 demonstrate the strength of this framework in finite classification problems (Yu et al., 31 Jul 2025, Yu et al., 1 Aug 2025), while the two-eigenvalue constructions show that, beyond those low-degree cases, the theory also supports infinite spectral families (Ramezani, 2019).