Smith graph

For the nomogram technique in radio frequency engineering, see Smith chart. For the Biggs–Smith distance-regular graph, see Biggs–Smith graph.

In the mathematical field of graph theory, a Smith graph is either of two kinds of graph.

  • It is a graph whose adjacency matrix has largest eigenvalue at most 2,[1] or has spectral radius 2[2] or at most 2.[3] The graphs with spectral radius 2 form two infinite families and three sporadic examples; if we ask for spectral radius at most 2 then there are two additional infinite families and three more sporadic examples. The infinite families with spectral radius less than 2 are the paths and the paths with one extra edge attached to the vertex next to an endpoint; the infinite families with spectral radius exactly 2 are the cycles and the paths with an extra edge attached to each of the vertices next to an endpoint.

These are also the simply laced affine (and finite, if the spectral radius may be less than 2) Dynkin diagrams.

References

  1. ^ John H. Smith (June 2–14, 1969). "Some properties of the spectrum of a graph". In Richard Guy (ed.). Combinatorial Structures and Their Applications. Proceedings of the Calgary International Conference on Combinatorial Structures and Their Applications. University of Calgary, Calgary, Alberta, Canada: Gordon and Breach. pp. 403–406.
  2. ^ Radosavljević, Z.; Mihailović, B.; Rašajski, M. (2008). "Decomposition of Smith graphs in maximal reflexive cacti". Discrete Mathematics. 308 (2–3): 355–366. doi:10.1016/j.disc.2006.11.049.
  3. ^ Cvetković, Dragoš (2017). "Spectral Theory of Smith Graphs". Bulletin (Académie Serbe des Sciences et des Arts. Classe des Sciences Mathématiques et Naturelles. Sciences Mathématiques) (42): 19–40. JSTOR 26359061.
  4. ^ Cameron, P. J.; Goethals, J.-M.; Seidel, J. J. (1978). "Strongly regular graphs having strongly regular subconstituents". Journal of Algebra. 55 (2): 257–280. doi:10.1016/0021-8693(78)90220-X. MR 0523457.


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