Conference graph

Unsolved problem in mathematics

Does there exist a conference graph for every number of vertices

v>1

where

v\equiv 1{\bmod {4}}

and

v

is an odd sum of two squares?

More unsolved problems in mathematics

In the mathematical area of graph theory, a conference graph is a strongly regular graph with parameters v, k = (v − 1)/2, λ = (v − 5)/4, and μ = (v − 1)/4. 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.^[1]

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.^[2] The next smallest, v = 65, was found over 4 decades later in 2021.^[3]^[4] As of now, the smallest open case is v = 85.^[4]

The eigenvalues of a conference graph need not be integers, unlike those of other strongly regular graphs. If the graph is connected, the eigenvalues are k with multiplicity 1, and two other eigenvalues,

{\frac {-1\pm {\sqrt {v}}}{2}},

each with multiplicity (v − 1)/2.

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.

References

^ Brouwer, A.E.; Cohen, A.M.; Neumaier, A. (1989). Distance Regular Graphs (PDF). Berlin, New York: Springer-Verlag. ISBN 978-3-540-50619-5.
^ Mathon, Rudolf (1978). "Symmetric Conference Matrices of Order pq² + 1". Canadian Journal of Mathematics. 30 (2): 321–331. doi:10.4153/CJM-1978-029-1.
^ Gritsenko, Oleg (2021). "On strongly regular graph with parameters (65; 32; 15; 16)". arXiv:2102.05432 [math.CO].
^ ^a ^b Brouwer, Andries E.; Van Maldeghem, Hendrik (2022). "8.2 Conference matrices and conference graphs". Strongly regular graphs (PDF). New Mathematical Monographs. Vol. 41. American Mathematical Society. pp. 189–190. ISBN 978-1-316-51203-6.

External links

(sequence A057653 in the OEIS), odd numbers that are the sum of 2 squares
(sequence A085759 in the OEIS), prime powers of the form 4n+1.

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[bcn89-1] Brouwer, A.E.; Cohen, A.M.; Neumaier, A. (1989). Distance Regular Graphs (PDF). Berlin, New York: Springer-Verlag. ISBN 978-3-540-50619-5.

[2] Mathon, Rudolf (1978). "Symmetric Conference Matrices of Order pq² + 1". Canadian Journal of Mathematics. 30 (2): 321–331. doi:10.4153/CJM-1978-029-1.

[3] Gritsenko, Oleg (2021). "On strongly regular graph with parameters (65; 32; 15; 16)". arXiv:2102.05432 [math.CO].

[bv22-4] Brouwer, Andries E.; Van Maldeghem, Hendrik (2022). "8.2 Conference matrices and conference graphs". Strongly regular graphs (PDF). New Mathematical Monographs. Vol. 41. American Mathematical Society. pp. 189–190. ISBN 978-1-316-51203-6.

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