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# Paulus Graphs

The Paulus graphs are the 15 strongly regular graphs on 25 nodes with parameters and the 10 strongly regular graphs on 26 nodes with parameters (26, 10, 3, 4).

They are implemented in the Wolfram Language as GraphData["Paulus", n, i].

The -Paulus graph is isomorphic to the 25-Paley graph.

The 25-node Paulus graphs are cospectral, as are the 26-node Paulus graphs, so none of these is determined by spectrum.

The -Paulus graph has the largest possible graph automorphism group order of all 26-node Paulus graphs (namely 120), and is sometimes known as the Paulus-Rozenfeld-Thompson (or PRT) graph and denoted (Gyürki et al. 2020).

The Paulus graphs are pancyclic.

The -, -, and -Paulus graphs have the apparently rather unusual property of being both integral graphs and identity graphs.

Chang Graphs, Cospectral Graphs, Determined by Spectrum, Paley Graph, Paulus-Rozenfeld-Thompson Graph, Strongly Regular Graph

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## References

Brouwer, A. E. "Paulus Graphs." http://www.win.tue.nl/~aeb/drg/graphs/Paulus.html.Brouwer, A. E.; Cohen, A. M.; and Neumaier, A. Distance-Regular Graphs. New York: Springer-Verlag, p. 37, 1989.DistanceRegular.org. "Paulus graphs () (14 graphs, 7 pairs)." [Excludes the 25-Paley graph.] https://www.distanceregular.org/graphs/paulus25.html.DistanceRegular.org. "Paulus graphs () (10 graphs)." https://www.distanceregular.org/graphs/paulus26.html.Gyürki, Š.; Klin, M.; and Ziv-Av, M. "The Paulus-Rozenfeld-Thompson Graph on 26 Vertices Revisited and Related Combinatorial Structures." In Isomorphisms, Symmetry and Computations in Algebraic Graph Theory: Pilsen, Czech Republic, October 3-7, 2016 (Ed. G. A. Jones, I. Ponomarenko, and J. Širáň). Cham, Switzerland: Springer Nature, pp. 73-154, 2020.Paulus, A. J. L. "Conference Matrices and Graphs of Order 26." Technische Hogeschool Eindhoven. Report WSK 73/06, Eindhoven, 1973.

Paulus Graphs

## Cite this as:

Weisstein, Eric W. "Paulus Graphs." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/PaulusGraphs.html