Conference graph¶

In the mathematical area of graph theory, a conference graph is a strongly regular graph with parameters v, and It is the graph associated with a symmetric conference matrix, and consequently its order v must be 1 (modulo 4) and a sum of two squares.

Version: v1 · 2026-09-28 · History
Domain-specific #: 8640
Domain group: Formal Sciences
Origin domain: Mathematics
Subdomains: Graph Theory, Algebraic Graph Theory → Mathematics

Core Idea¶

Conference graph is treated here as the recurring graph theory identity summarized by this source-grounded definition: In the mathematical area of graph theory, a conference graph is a strongly regular graph with parameters v, and It is the graph associated with a symmetric conference matrix, and consequently its order v must be 1 (modulo 4) and a sum of two squares.

In the mathematical area of graph theory, a conference graph is a strongly regular graph with parameters v, and It is the graph associated with a symmetric conference matrix, and consequently its order v must be 1 (modulo 4) and a sum of two squares. Conference graphs are known to exist for all small values of v allowed by the restrictions, e.g., v = 5, 9, 13, 17, 25, 29, and (the Paley graphs) for all prime powers congruent to 1 (modulo 4). However, there are many values of v that are allowed, for which the existence of a conference graph is unknown.

The smallest value of v which has no Paley graph but does have a conference graph is v = 45, found in 1978. The next smallest, v = 65, was found over 4 decades later in 2021. As of now, the smallest open case is v = 85.

For Conference graph, the abstraction is narrower than the article's general subject matter: a positive case must preserve In the mathematical area of graph theory, a conference graph is a strongly regular graph with parameters v, and It is the graph associated with a symmetric conference matrix, and consequently its order v must be 1 (modulo 4) and a sum of two squares. Retaining only the name, a familiar example, or a downstream effect is insufficient. The specialist roles and tests remain anchored in graph theory, which is why this identity is domain-specific rather than prime.

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Perfectly Even Friend Maps

Imagine dots joined by lines, like a map of who is friends with whom, built so everything is perfectly even: every dot has the same number of friends, and pairs of dots share friends in a very regular way. A conference graph is a special kind of such perfectly even friend-map. It can only be made with certain numbers of dots, like 5, 9 or 13.

Perfectly Even Friend Networks

A graph is a set of dots with lines between some of them. In a strongly regular graph, every dot has the same number of lines, any two connected dots share the same number of common neighbors, and any two unconnected dots also share the same number of common neighbors. A Conference graph is a special strongly regular graph that comes from a special kind of number table called a conference matrix. Because of that, the number of dots has to follow strict rules: it must leave a remainder of 1 when divided by 4 and be the sum of two square numbers. Some allowed sizes are known to work, like 5, 9, and 13, but for some allowed sizes, nobody yet knows if one exists.

Strongly Regular Conference-Matrix Graph

In graph theory, a conference graph is a strongly regular graph — a graph where every vertex has the same number of neighbors, and every pair of vertices has a fixed number of common neighbors depending only on whether the pair is adjacent — with a particular set of parameters determined by its number of vertices v. It corresponds to a symmetric conference matrix, and because of that, v must be 1 more than a multiple of 4 and a sum of two squares. Conference graphs exist for all small allowed values like 5, 9, 13, 17, 25, and 29, and the Paley graphs give one for every prime power that is 1 more than a multiple of 4. But for many allowed values, it's unknown whether one exists. The smallest size with a conference graph but no Paley graph is 45, found in 1978; the next, 65, was found in 2021, and 85 is currently the smallest open case.

A Conference graph is a strongly regular graph with a specific parameter set determined by its order v, and it is the graph associated with a symmetric conference matrix. That correspondence forces two necessary conditions on the order: v ≡ 1 (mod 4) and v is a sum of two squares. Existence is known for all small admissible orders, for example v = 5, 9, 13, 17, 25, 29, and for every prime power congruent to 1 mod 4 via the Paley graphs. The conditions are necessary but not known to be sufficient: many admissible v have unresolved existence. The smallest order with a conference graph but no Paley graph is v = 45, found in 1978; the next, v = 65, was found in 2021; the smallest open case is v = 85. The identity requires both strong regularity with the conference parameters and the link to a symmetric conference matrix.

Structural Signature¶

Sig role-phrases:

Defining carrier — Conference graphs are known to exist for all small values of v allowed by the restrictions, e.g., v = 5, 9, 13, 17, 25, 29, and (the Paley graphs) for all prime powers congruent to 1 (modulo 4).
Constitutive relation — However, there are many values of v that are allowed, for which the existence of a conference graph is unknown.
Operating condition — The smallest value of v which has no Paley graph but does have a conference graph is v = 45, found in 1978.
Recognition evidence — The next smallest, v = 65, was found over 4 decades later in 2021.
Admissible variation — The eigenvalues of a conference graph need not be integers, unlike those of other strongly regular graphs.
Characteristic consequence — If the graph is connected, the eigenvalues are k with multiplicity 1, and two other eigenvalues,.
Failure boundary — The complement of a conference graph is always a conference graph with the same parameters, and in many cases is self-complementary, such as for all the Paley graphs.

What It Is Not¶

Not the whole field of graph theory. The node requires the specific identity stated by In the mathematical area of graph theory, a conference graph is a strongly regular graph with parameters v, and It is the graph associated with a symmetric conference matrix, and consequently its order v must be 1 (modulo 4) and a sum of two squares.
Not an over-broad reading. The eigenvalues of a conference graph need not be integers, unlike those of other strongly regular graphs.
Not an over-broad reading. However, there are many values of v that are allowed, for which the existence of a conference graph is unknown.
Not an over-broad reading. Conference graphs are known to exist for all small values of v allowed by the restrictions, e.g., v = 5, 9, 13, 17, 25, 29, and (the Paley graphs) for all prime powers congruent to 1 (modulo 4).
Not automatically Conference Matrix. Retrieval proximity does not establish equivalence; the two identities must be compared by carrier, operation, and failure boundary.

Scope of Application¶

Conference graph applies literally inside graph theory wherever the source-defined carrier and relation can be established. Its documented habitats include:

Documented setting. Conference graphs are known to exist for all small values of v allowed by the restrictions, e.g., v = 5, 9, 13, 17, 25, 29, and (the Paley graphs) for all prime powers congruent to 1 (modulo 4).
Documented setting. However, there are many values of v that are allowed, for which the existence of a conference graph is unknown.
Documented setting. The smallest value of v which has no Paley graph but does have a conference graph is v = 45, found in 1978.
Documented setting. The next smallest, v = 65, was found over 4 decades later in 2021.
Documented setting. The eigenvalues of a conference graph need not be integers, unlike those of other strongly regular graphs.
Documented setting. If the graph is connected, the eigenvalues are k with multiplicity 1, and two other eigenvalues,.

Outside graph theory, the name should be retained only when these same operational conditions survive; otherwise the comparison belongs to the broader parent Classification or should be marked as analogy.

Clarity¶

A clear use of Conference graph names the carrier, the operative relation, and the conditions under which the source treats the identity as present. The minimal definition is In the mathematical area of graph theory, a conference graph is a strongly regular graph with parameters v, and It is the graph associated with a symmetric conference matrix, and consequently its order v must be 1 (modulo 4) and a sum of two squares. The strongest recognition evidence in the frozen account is: The next smallest, v = 65, was found over 4 decades later in 2021. A report should distinguish that evidence from a proxy, consequence, or common implementation. It should also state the qualification The eigenvalues of a conference graph need not be integers, unlike those of other strongly regular graphs. so that a reader can reproduce the classification rather than infer it from topical resemblance.

Manages Complexity¶

Conference graph compresses multiple graph theory details into a stable diagnostic relation. The source shows both the central mechanism—however, there are many values of v that are allowed, for which the existence of a conference graph is unknown.—and the practical consequence—if the graph is connected, the eigenvalues are k with multiplicity 1, and two other eigenvalues,. This compression makes cases comparable while leaving parameters, conventions, exceptions, and evidential quality explicit. It is lossy by design: local history and implementation details may be omitted only when they do not alter the defining relation.

Abstract Reasoning¶

Type the carrier. Identify the graph theory entities to which the claim applies.
State the relation. Use the source-grounded identity: In the mathematical area of graph theory, a conference graph is a strongly regular graph with parameters v, and It is the graph associated with a symmetric conference matrix, and consequently its order v must be 1 (modulo 4) and a sum of two squares.
Check operation and conditions. The smallest value of v which has no Paley graph but does have a conference graph is v = 45, found in 1978.
Demand recognition evidence. The next smallest, v = 65, was found over 4 decades later in 2021.
Test variation. Change an implementation or setting while preserving the eigenvalues of a conference graph need not be integers, unlike those of other strongly regular graphs.
Run the collapse test. Remove the defining operation; if the label still seems equally apt, only a topic or correlate was retained.
Reduce cautiously. When the specialist conditions cannot be carried, route the residual comparison to Classification.

Knowledge Transfer¶

Within the home domain. Knowledge about Conference graph transfers literally when a new case preserves the same carrier type, relation, and recognition test. Conference graphs are known to exist for all small values of v allowed by the restrictions, e.g., v = 5, 9, 13, 17, 25, 29, and (the Paley graphs) for all prime powers congruent to 1 (modulo 4). However, there are many values of v that are allowed, for which the existence of a conference graph is unknown.

Beyond the home domain. No canonical parent is asserted for Conference graph. An outside case receives the specialist name only when the same typed roles and rejection conditions can be filled literally; otherwise the comparison remains an analogy pending later graph densification.

Examples¶

Canonical¶

The complement of a conference graph is always a conference graph with the same parameters, and in many cases is self-complementary, such as for all the Paley graphs. This case is canonical because it supplies a concrete carrier and lets the defining relation be checked rather than merely named.

Mapped back: carrier → the entities in the documented case; operation → In the mathematical area of graph theory, a conference graph is a strongly regular graph with parameters v, and It is the graph associated with a symmetric conference matrix, and consequently its order v must be 1 (modulo 4) and a sum of two squares; recognition evidence → The next smallest, v = 65, was found over 4 decades later in 2021

Applied / In Practice¶

Conference graphs are known to exist for all small values of v allowed by the restrictions, e.g., v = 5, 9, 13, 17, 25, 29, and (the Paley graphs) for all prime powers congruent to 1 (modulo 4). The applied case shows how the identity is used under a second setting or qualification while keeping the same operative relation.

Mapped back: changed setting → the applied context; invariant → In the mathematical area of graph theory, a conference graph is a strongly regular graph with parameters v, and It is the graph associated with a symmetric conference matrix, and consequently its order v must be 1 (modulo 4) and a sum of two squares; boundary → the case exits the class when the eigenvalues of a conference graph need not be integers, unlike those of other strongly regular graphs

Structural Tensions¶

T1 — Stable identity versus admissible variation. The eigenvalues of a conference graph need not be integers, unlike those of other strongly regular graphs. The tension matters because emphasizing only one side either dissolves the identity or overstates what the evidence and domain conventions warrant.

Diagnostic: Which changes preserve the defining relation, and which replace it?

T2 — Recognition versus proxy. However, there are many values of v that are allowed, for which the existence of a conference graph is unknown. The tension matters because emphasizing only one side either dissolves the identity or overstates what the evidence and domain conventions warrant.

Diagnostic: Does the cited evidence establish the identity or only a correlated sign?

T3 — Definition versus implementation. Conference graphs are known to exist for all small values of v allowed by the restrictions, e.g., v = 5, 9, 13, 17, 25, 29, and (the Paley graphs) for all prime powers congruent to 1 (modulo 4). The tension matters because emphasizing only one side either dissolves the identity or overstates what the evidence and domain conventions warrant.

Diagnostic: Is the observed implementation constitutive, optional, or merely common?

T4 — Scope versus overextension. The smallest value of v which has no Paley graph but does have a conference graph is v = 45, found in 1978. The tension matters because emphasizing only one side either dissolves the identity or overstates what the evidence and domain conventions warrant.

Diagnostic: Can every claimed application fill the same typed roles without metaphor?

T5 — Transfer versus domain accent. Conference graphs are known to exist for all small values of v allowed by the restrictions, e.g., v = 5, 9, 13, 17, 25, 29, and (the Paley graphs) for all prime powers congruent to 1 (modulo 4). The tension matters because emphasizing only one side either dissolves the identity or overstates what the evidence and domain conventions warrant.

Diagnostic: Does the receiving case instantiate Conference graph literally, co-instantiate Classification, or only resemble it?

T6 — Autonomy versus reduction. However, there are many values of v that are allowed, for which the existence of a conference graph is unknown. The tension matters because emphasizing only one side either dissolves the identity or overstates what the evidence and domain conventions warrant.

Diagnostic: What does Conference graph distinguish that the broader parent Classification leaves together?

Structural–Framed Character¶

Conference graph is mixed or framed-leaning. Its structural side is the repeatable organization summarized by In the mathematical area of graph theory, a conference graph is a strongly regular graph with parameters v, and It is the graph associated with a symmetric conference matrix, and consequently its order v must be 1 (modulo 4) and a sum of two squares. Its framed side is the graph theory vocabulary that fixes the carrier, evidence, exceptions, and admissible transformations.

Evaluative weight: the identity can be stated descriptively even when applications carry practical stakes. Human-practice dependence: the source-grounded carrier determines whether the relation exists independently or is constituted by a practice. Institutional origin: disciplinary conventions stabilize the name and test. Vocabulary portability: The smallest value of v which has no Paley graph but does have a conference graph is v = 45, found in 1978. Import versus recognition: literal transfer requires the same mechanism; shape alone is analogy.

Its portable skeleton is Classification. Its character: a recurring specialist identity whose thin organization can be abstracted, while its operational meaning remains domain-bound.

Structural Core vs. Domain Accent¶

What is skeletal. In the mathematical area of graph theory, a conference graph is a strongly regular graph with parameters v, and It is the graph associated with a symmetric conference matrix, and consequently its order v must be 1 (modulo 4) and a sum of two squares. The stable skeleton is the typed relation expressed in that definition and the entry's recognition and collapse tests. The source identifies these operative conditions: Conference graphs are known to exist for all small values of v allowed by the restrictions, e.g., v = 5, 9, 13, 17, 25, 29, and (the Paley graphs) for all prime powers congruent to 1 (modulo 4). However, there are many values of v that are allowed, for which the existence of a conference graph is unknown. It further constrains recognition and variation through: The smallest value of v which has no Paley graph but does have a conference graph is v = 45, found in 1978. The next smallest, v = 65, was found over 4 decades later in 2021.

What is domain-bound. graph theory supplies the operative entities, technical vocabulary, warrants, and exceptions that make Conference graph literal. Its documented scope includes the condition that Conference graphs are known to exist for all small values of v allowed by the restrictions, e.g., v = 5, 9, 13, 17, 25, 29, and (the Paley graphs) for all prime powers congruent to 1 (modulo 4). Another bounded application condition is that However, there are many values of v that are allowed, for which the existence of a conference graph is unknown. These are not decorative examples; they determine which carrier and evidence can fill the abstraction's roles.

Why no parent is asserted. Removing those specialist details does not currently yield one live catalog node that is a necessary genus for every instance. The entry is therefore approved as unparented rather than attached by topical resemblance. Its collapse evidence remains specific—The eigenvalues of a conference graph need not be integers, unlike those of other strongly regular graphs.—and future graph densification may discover a defensible relation only if it preserves that boundary.

This entry is a kind of Network.

Approved unparented node. No current live node supplies a defensible necessary genus or structural prerequisite for Conference graph. The reviewed identity is: In the mathematical area of graph theory, a conference graph is a strongly regular graph with parameters v, and It is the graph associated with a symmetric conference matrix, and consequently its order v must be 1 (modulo 4) and a sum of two squares. The accelerated suggestion was declined because topical or lexical similarity does not establish hierarchy; the node is admitted without a parent pending later graph densification.
Related reasoning operations. Evidence, representation, comparison, classification, transformation, or evaluation may participate in particular cases, but participation does not make any one of them a necessary parent of every instance.

Relationships to Other Abstractions¶

Current abstraction Conference graph Domain-specific

Parents (1) — more general patterns this builds on

Conference graph is a kind of Network Prime

A conference graph is a graph/network with a specified strongly regular parameter family; Graph is a declared alias of the live Network Prime.

Hierarchy path (1) — routes to 1 parentless root

Conference graph → Network → Reservoir-Flux Network → Conservation Laws → Invariance

Neighborhood in Abstraction Space¶

Conference graph sits in a sparse region of the domain-specific corpus (68^th percentile for distinctiveness): few abstractions share its structure, so a faithful description tends to retrieve it precisely.

Family — Number-Theoretic Properties & Tests (20 abstractions)

Nearest neighbors

Computed from structural-signature embeddings · 2026-10-08

Not to Be Confused With¶

Classification. The parent omits the specialist differentia. Tell: Can the case establish In the mathematical area of graph theory, a conference graph is a strongly regular graph with parameters v, and It is the graph associated with a symmetric conference matrix, and consequently its order v must be 1 (modulo 4) and a sum of two squares?
Conference Matrix. A square real sign matrix with zero diagonal and mutually orthogonal rows of squared norm one less than its order, equivalently satisfying CCᵀ=(n−1)I, up to declared equivalence operations. Tell: Which entry's carrier, operation, and failure condition are satisfied?
Strongly regular graph. A regular graph with fixed numbers of common neighbors for every adjacent pair and for every nonadjacent pair, summarized by parameters (v,k,lambda,mu). Tell: Which entry's carrier, operation, and failure condition are satisfied?
Prime Graph. Represent a finite group by making the prime divisors of its order the vertices and joining distinct primes p and q exactly when the group contains an element of order pq. Tell: Which entry's carrier, operation, and failure condition are satisfied?
A measurement, proxy, or consequence. Those may provide evidence without being the identity. Tell: Would Conference graph remain present if the detector or downstream effect changed?
A metaphorical analogue. A similar shape outside graph theory lacks the specialist mechanism. Tell: Do the native roles transfer literally, or only the parent Classification?

References¶

Frozen Wikipedia discovery revision: https://en.wikipedia.org/wiki/Conference_graph (revision 1319329901).
Preserved source candidate: https://projecteuclid.org/journals/bulletin-of-the-american-mathematical-society-new-series/volume-24/issue-2/Review–A-E-Brouwer-A-M-Cohen-and-A/bams/1183656885.pdf
Preserved source candidate: https://homepages.cwi.nl/~aeb/math/srg/rk3/srgw.pdf

The frozen Wikipedia revision is discovery provenance. The retained source set was reviewed for identity, formal or operational relation, and scope. The encyclopedia's structural synthesis is bounded to those claims; a thin authority surface is recorded as a nonblocking source-strengthening repair rather than concealed.