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.

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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.

Scope of Application¶

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.
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.

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.

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,.

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.

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.

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