6-Regular Strongly Regular Signed Graphs

• 6-regular strongly regular signed graphs are defined on graphs where each vertex has degree 6 and edges carry ±1 signs, uniting several algebraic frameworks.
• They include multiple formal approaches such as FSRSG (focused on signed 2-path counts), two-eigenvalue STE models, and quasi-balanced weighing matrices linked to classical SRGs.
• Their complete classification by net-degree and explicit spectral constructions offer insights for developing infinite families and constructing Ramanujan graphs.

Searching arXiv for recent and foundational papers on 6-regular strongly regular signed graphs and related signed strongly regular graph frameworks. In the literature on signed graphs, 6-regular strongly regular signed graphs form a technically rich but definition-dependent class. A signed graph is a graph whose edges carry signs in $\{+1,-1\}$, and 6-regularity means that every vertex has total degree $6$. The phrase “strongly regular signed graph” is used in at least three closely related ways: as a finite strongly regular signed graph (FSRSG), as a signed graph with only two adjacency eigenvalues (STE), and as a Stanić-type strongly regular signed graph determined by uniform signed $2$-walk counts; a further matrix-theoretic formulation uses quasi-balanced weighing matrices whose absolute values are adjacency matrices of ordinary strongly regular graphs (Ramezani, 2019, Kharaghani et al., 2022, Yu et al., 1 Aug 2025).

1. Signed-graph framework and 6-regularity

A signed graph is a pair $\Sigma=(G,\sigma)$ or $\dot G=(G,\sigma)$, where $G$ is a simple graph and $\sigma:E(G)\to\{+1,-1\}$ assigns a sign to each edge. Its signed adjacency matrix is the matrix $A=(a^\sigma_{ij})$ with

$$a^\sigma_{ij}= \begin{cases} \sigma(v_iv_j), & v_i\sim v_j,\ 0, & \text{otherwise.} \end{cases}$$

The ordinary adjacency matrix of the ground graph is the absolute-value matrix $|A|$ (Ramezani, 2019, Yu et al., 1 Aug 2025).

For a 6-regular signed graph, the underlying graph has degree $6$0 at every vertex. In the net-regular framework one further distinguishes

$6$1

so that $6$2, and defines the net-degree by

$6$3

A signed graph is $6$4-net-regular if $6$5 for all vertices. For degree $6$6, the possible nonnegative cases treated explicitly are $6$7, $6$8, and $6$9, corresponding respectively to $2$0, $2$1, and $2$2 (Yu et al., 1 Aug 2025).

The spectrum of a signed graph is the multiset of eigenvalues of its signed adjacency matrix, written in the form

$2$3

This spectral viewpoint is central in the two-eigenvalue theory and remains structurally important in the other frameworks (Ramezani, 2019).

2. Three notions of signed strong regularity

One major framework is the FSRSG formalism. A signed regular graph $2$4 is a finite strongly regular signed graph $2$5 when its underlying graph is $2$6-regular on $2$7 vertices, the difference $2$8 between positive and negative $2$9-paths joining adjacent vertices is a constant $\Sigma=(G,\sigma)$0, and the same difference for nonadjacent vertices is a constant $\Sigma=(G,\sigma)$1. In this language, a signed graph with only two distinct adjacency eigenvalues is an STE, and a basic structural theorem states that a signed graph is a STE if and only if it is a FSRSG with $\Sigma=(G,\sigma)$2 (Ramezani, 2019).

A second framework, due to Stanić and used in the 2025 classification of degree $\Sigma=(G,\sigma)$3, defines a strongly regular signed graph by requiring that every entry of $\Sigma=(G,\sigma)$4 depends only on whether the corresponding pair of vertices is equal, positively adjacent, negatively adjacent, or nonadjacent. Thus there exist integers $\Sigma=(G,\sigma)$5 such that

$\Sigma=(G,\sigma)$6

Equivalently,

$\Sigma=(G,\sigma)$7

This is the signed analogue of the standard strongly regular graph relation (Yu et al., 1 Aug 2025).

A third framework is matrix-theoretic. A weighing matrix $\Sigma=(G,\sigma)$8 is a $\Sigma=(G,\sigma)$9-matrix of order $\dot G=(G,\sigma)$0 with $\dot G=(G,\sigma)$1. A quasi-balanced weighing matrix $\dot G=(G,\sigma)$2 signs a strongly regular graph when $\dot G=(G,\sigma)$3 is the adjacency matrix of an SRG$\dot G=(G,\sigma)$4. In that setting, a 6-regular signed strongly regular graph is represented by a matrix $\dot G=(G,\sigma)$5 with $\dot G=(G,\sigma)$6 in the real case or $\dot G=(G,\sigma)$7 in the complex Butson case (Kharaghani et al., 2022).

These notions overlap but are not identical. A common source of confusion is to treat them as a single universal definition. The literature instead gives parallel formalisms, each emphasizing a different invariant: signed $\dot G=(G,\sigma)$8-path counts, low adjacency-eigenvalue multiplicity, or weighing-matrix/association-scheme structure.

3. The spectral theory of 6-regular STEs

In the FSRSG/STE framework, the signed adjacency matrix of a STE satisfies

$\dot G=(G,\sigma)$9

For $G$0, the type-$G$1 family has

$G$2

so that

$G$3

These are exactly the symmetric weighing matrices of weight $G$4, and the spectrum has the form

$G$5

with equal multiplicities (Ramezani, 2019).

This family has an especially strong signed-regularity interpretation. Since $G$6, the signed count of length-$G$7 walks from $G$8 to $G$9 is $\sigma:E(G)\to\{+1,-1\}$0 whenever $\sigma:E(G)\to\{+1,-1\}$1. Hence, for every unordered pair of distinct vertices, the net signed number of $\sigma:E(G)\to\{+1,-1\}$2-paths vanishes. In FSRSG notation, these graphs are $\sigma:E(G)\to\{+1,-1\}$3 (Ramezani, 2019).

The 2019 paper also provides an explicit 6-regular STE of a different spectral type: the signed line graph $\sigma:E(G)\to\{+1,-1\}$4 is 6-regular with spectrum

$\sigma:E(G)\to\{+1,-1\}$5

Thus the 6-regular case is not spectrally rigid: both the type-$\sigma:E(G)\to\{+1,-1\}$6 family with eigenvalues $\sigma:E(G)\to\{+1,-1\}$7 and a type-$\sigma:E(G)\to\{+1,-1\}$8 example with eigenvalues $\sigma:E(G)\to\{+1,-1\}$9 occur (Ramezani, 2019).

The existence theory is infinite. If $A=(a^\sigma_{ij})$0 is a signed $A=(a^\sigma_{ij})$1-regular graph with spectrum

$A=(a^\sigma_{ij})$2

then the recursive construction

$A=(a^\sigma_{ij})$3

produces a signed $A=(a^\sigma_{ij})$4-regular graph with spectrum

$A=(a^\sigma_{ij})$5

Using an infinite family of 4-regular signed graphs with spectrum $A=(a^\sigma_{ij})$6, one obtains, in particular, infinitely many connected signed 6-regular graphs with distinct eigenvalues $A=(a^\sigma_{ij})$7 (Ramezani, 2019).

The same paper places these graphs in a Ramanujan context. For $A=(a^\sigma_{ij})$8, the maximum eigenvalue in the type-$A=(a^\sigma_{ij})$9 family is $$a^\sigma_{ij}= \begin{cases} \sigma(v_iv_j), & v_i\sim v_j,\ 0, & \text{otherwise.} \end{cases}$$0, while the unsigned Ramanujan bound is $$a^\sigma_{ij}= \begin{cases} \sigma(v_iv_j), & v_i\sim v_j,\ 0, & \text{otherwise.} \end{cases}$$1. The paper states that the resulting signed families yield infinite families of $$a^\sigma_{ij}= \begin{cases} \sigma(v_iv_j), & v_i\sim v_j,\ 0, & \text{otherwise.} \end{cases}$$2-regular Ramanujan graphs via lift constructions, and the 6-regular case is one instance of that general mechanism (Ramezani, 2019).

4. Complete classification of connected 6-regular net-regular SRSGs

Within Stanić’s definition, the classification problem for connected 6-regular net-regular strongly regular signed graphs is solved completely. The main result is that there are exactly three, six, and four connected 6-regular strongly regular signed graphs with net-degree $$a^\sigma_{ij}= \begin{cases} \sigma(v_iv_j), & v_i\sim v_j,\ 0, & \text{otherwise.} \end{cases}$$3, $$a^\sigma_{ij}= \begin{cases} \sigma(v_iv_j), & v_i\sim v_j,\ 0, & \text{otherwise.} \end{cases}$$4, and $$a^\sigma_{ij}= \begin{cases} \sigma(v_iv_j), & v_i\sim v_j,\ 0, & \text{otherwise.} \end{cases}$$5, respectively (Yu et al., 1 Aug 2025).

Net-degree $$a^\sigma_{ij}= \begin{cases} \sigma(v_iv_j), & v_i\sim v_j,\ 0, & \text{otherwise.} \end{cases}$$6	Count	Parameter sets $$a^\sigma_{ij}= \begin{cases} \sigma(v_iv_j), & v_i\sim v_j,\ 0, & \text{otherwise.} \end{cases}$$7
$$a^\sigma_{ij}= \begin{cases} \sigma(v_iv_j), & v_i\sim v_j,\ 0, & \text{otherwise.} \end{cases}$$8	3	$$a^\sigma_{ij}= \begin{cases} \sigma(v_iv_j), & v_i\sim v_j,\ 0, & \text{otherwise.} \end{cases}$$9; $|A|$0; $|A|$1
$|A|$2	6	$|A|$3; $|A|$4; $|A|$5; $|A|$6; $|A|$7; $|A|$8
$|A|$9	4	$6$00; $6$01

For net-degree $6$02, the specialization of the basic parameter relation is

$6$03

The negative subgraph is 1-regular, hence a perfect matching, and the analysis eliminates all but three parameter sets, giving the graphs $6$04, $6$05, and $6$06 (Yu et al., 1 Aug 2025).

For net-degree $6$07, where $6$08 is 4-regular and $6$09 is 2-regular, the parameter equation becomes

$6$10

The classification is more intricate and uses parity of negative $6$11-walks, local sign constraints, and exhaustive tests against the complete lists of 6-regular graphs on $6$12, $6$13, and $6$14 vertices. The outcome is exactly six graphs: $6$15, $6$16, $6$17, $6$18, $6$19, and $6$20 (Yu et al., 1 Aug 2025).

For net-degree $6$21, where both $6$22 and $6$23 are 3-regular, the parameter relation becomes

$6$24

Only two structures survive up to sign: $6$25 and $6$26, together with their negations. In the net-degree $6$27 case, negation preserves the net-degree and swaps $6$28 and $6$29; this yields the four signed graphs counted in the theorem (Yu et al., 1 Aug 2025).

5. Representative structures and explicit 6-regular examples

The classified net-regular examples are built on highly symmetric underlying graphs. For $6$30 with parameters $6$31, the underlying graph is $6$32. For $6$33 with parameters $6$34, the positive subgraph is a disjoint union of two copies of $6$35 and the negative edges form a perfect matching between them. For $6$36 with parameters $6$37, the graph is obtained by a detailed local analysis starting from a negative edge and enforcing the required signed $6$38-walk counts (Yu et al., 1 Aug 2025).

Among the net-degree $6$39 examples, $6$40 with parameters $6$41 is realized on $6$42, while $6$43 with parameters $6$44 sits on a specific 9-vertex 6-regular graph $6$45. The graph $6$46 with parameters $6$47 has underlying graph $6$48, the unique strongly regular graph with parameters $6$49. The graph $6$50 with parameters $6$51 is characterized by the fact that $6$52 is a union of $6$53’s and $6$54 is a union of $6$55’s. The 8-vertex examples $6$56 and $6$57 both sit on the unique 6-regular graph on 8 vertices (Yu et al., 1 Aug 2025).

For net-degree $6$58, $6$59 and $6$60 are the surviving cases. In $6$61 with parameters $6$62, both $6$63 and $6$64 decompose into disjoint unions of copies of $6$65. The construction partitions the vertex set into four negative $6$66’s and links them with positive $6$67’s in a structured pattern (Yu et al., 1 Aug 2025).

A different explicit 6-regular example appears in the quasi-balanced weighing-matrix framework. The paper on quasi-balanced weighing matrices constructs a strictly quasi-balanced signing over $6$68 of the SRG$6$69, giving a Butson weighing matrix $6$70 with $6$71. The paper also states that no srg-balanced signing exists for this graph over $6$72, and its search finds none over $6$73 (Kharaghani et al., 2022).

These examples should not be conflated. The 16-vertex graph $6$74 in the Stanić framework and the complex signing of SRG$6$75 in the quasi-balanced framework both have degree $6$76, but the papers present them as instances of different signed-regularity theories.

6. Related small complete signed graphs and a common misconception

A recurrent misconception is to identify signed complete graphs on six vertices with 6-regular signed graphs. This is incorrect: every signed $6$77 has underlying degree $6$78, not $6$79. The paper on signed complete graphs on six vertices is therefore adjacent to, but not part of, the classification of 6-regular strongly regular signed graphs (Sehrawat et al., 2018).

That paper proves that there are exactly 16 different signatures on $6$80 up to switching isomorphism. It works with switching, balance, minimal signatures, and negative-cycle invariants, and it classifies the 16 switching classes by canonical negative-edge patterns such as paths, cycles, a triangle, a 5-cycle, a 6-cycle, and two disjoint triangles. The classes are distinguished by the counts of negative 3-, 4-, and 5-cycles (Sehrawat et al., 2018).

Although the paper does not compute spectra, it identifies highly symmetric classes such as $6$81 and $6$82, the latter being the union of two disjoint negative triangles. This is relevant context because it shows how switching-equivalence and cycle-sign distributions organize small signed complete graphs, but these objects remain 5-regular and therefore do not answer the 6-regular problem directly (Sehrawat et al., 2018).

The distinction between switching and isomorphism is also important. In the $6$83 classification, switching-isomorphism is the basic equivalence relation. In the 2025 classification of net-regular SRSGs, the classification is up to isomorphism of signed graphs, and sign negation $6$84 is treated explicitly because it preserves $6$85 and swaps the parameters $6$86 and $6$87 (Sehrawat et al., 2018, Yu et al., 1 Aug 2025).

7. Significance, interaction of frameworks, and natural directions

Taken together, the three main lines of work show that the degree-$6$88 case is unusually well developed. In the FSRSG/STE theory, 6-regular signed graphs occur both as explicit low-order examples and as infinite families with spectrum $6$89, satisfying $6$90 and serving as inputs for the construction of 6-regular Ramanujan graphs (Ramezani, 2019). In the Stanić net-regular theory, the connected 6-regular case is completely classified, with exact counts for net-degree $6$91, $6$92, and $6$93 (Yu et al., 1 Aug 2025). In the quasi-balanced weighing-matrix theory, 6-regular signed SRGs are linked to 8-class symmetric association schemes and to concrete constructions such as the strictly quasi-balanced signing of SRG$6$94 (Kharaghani et al., 2022).

This suggests that “6-regular strongly regular signed graphs” is best viewed not as a single class but as an intersection zone between several mature formalisms. One axis emphasizes two-eigenvalue signed adjacency matrices; another emphasizes the quadratic relation for signed $6$95-walks; a third emphasizes signings of ordinary strongly regular graphs by weighing matrices and the associated Bose–Mesner algebra.

A plausible implication is that the degree-$6$96 case functions as a benchmark for comparing these frameworks. It contains infinite spectral families, complete finite classifications in the net-regular setting, and explicit links to association schemes and Ramanujan constructions. The natural next questions, as indicated by the degree-by-degree program in the net-regular literature and the existence problems in the quasi-balanced literature, concern higher valencies, structural comparison of the competing definitions, and the extent to which classical strongly regular graphs admit signings that remain strongly regular in one or more signed senses (Kharaghani et al., 2022, Yu et al., 1 Aug 2025).