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| Coxeter graph | |
|---|---|
| Name | Coxeter graph |
| Vertices | 28 |
| Edges | 42 |
| Automorphism group | PSL(2,7) |
| Named after | H. S. M. Coxeter |
Coxeter graph is a 3-regular undirected graph with 28 vertices and 42 edges notable for its girth 7, cubic symmetry, and role as a counterexample in several combinatorial problems. It arises in the study of finite simple groups, algebraic graph theory, and incidence geometry, and it connects to constructions by influential figures such as H. S. M. Coxeter, Élie Cartan, William Burnside, Felix Klein, and Johann Carl Friedrich Gauss.
The Coxeter graph is defined as a cubic graph on 28 vertices exhibiting girth 7 and automorphism group isomorphic to PSL(2,7), which is closely related to the groups studied by Évariste Galois, Camille Jordan, and Ferdinand Frobenius. Its diameter and radius equal 4, and its chromatic number is 3, attributes that link it to investigations by George David Birkhoff, Issai Schur, and Richard Dedekind into coloring and permutation groups. The graph is non-Hamiltonian, a property compared and contrasted with examples by William Tutte and constructions in the work of Paul Erdős and Alfréd Rényi. Eigenvalue multiplicities and integrality considerations connect to themes explored by Issai Schur and Alfred Young.
Standard constructions of the Coxeter graph use coset graphs of PSL(2,7), voltage assignments over covering graphs studied by Félix Le Vavasseur and methods analogous to those in W. T. Tutte's covering graph theory. One can obtain it from quotients of the Heawood graph linked to Thomas Kirkman and Arthur Cayley via operations reminiscent of work by Augustin-Louis Cauchy and Johann G. Dirichlet on permutation representations. Its full automorphism group equals PSL(2,7)],] a simple group central in the classification program led by William Burnside and later by Daniel Gorenstein and Michael Aschbacher, and it exhibits vertex-transitive but not arc-transitive features discussed by Marston Morse and John Conway.
The spectrum of the Coxeter graph features eigenvalues with algebraic multiplicities reflecting representations of PSL(2,7) and connections to character theory developed by Frobenius and Isaac Schur. Its adjacency matrix has spectral radius constrained by bounds proven in results of Hoffman and Noga Alon, and its eigenvalue distribution ties into Ramanujan-type phenomena studied by Srinivasa Ramanujan and contemporary work by Peter Sarnak. The characteristic polynomial and interlacing properties connect to linear algebraic techniques from John von Neumann and Hermann Weyl, and the graph's spectral gap informs expansion properties explored by Noga Alon and László Lovász.
The Coxeter graph relates to the Heawood graph associated with the Fano plane and configurations considered by Gino Fano and Émile Borel. It appears in families alongside the Petersen graph studied by Félix Klein and Philip Hall and the Tutte–Coxeter graphs connected to projective geometries of Galois fields investigated by Évariste Galois and Richard Dedekind. Embeddings and coverings link it to polyhedral constructions examined by H. S. M. Coxeter and to Klein quartic geometry tied to Felix Klein and modular curves in the work of Bernhard Riemann and Alexander Grothendieck.
The Coxeter graph serves as an extremal example in studies by Paul Erdős on girth versus chromatic number and in constructions employed in graph minors theory developed by Neil Robertson and Paul Seymour. It is used in coding theory contexts reminiscent of developments by Claude Shannon and Richard Hamming, and in model examples for symmetry breaking and representation theory treated by Isaac Schur and Emmy Noether. The graph provides test cases in computational algebra systems influenced by David Cox and John H. Conway and features in enumerative problems addressed by George Pólya and Gian-Carlo Rota.
Named after H. S. M. Coxeter, who popularized symmetry and regular polytopes alongside studies by Eugène Ehrhart and L. E. J. Brouwer, the Coxeter graph entered the literature through investigations in the mid-20th century related to cubic graphs from contributors such as W. T. Tutte and F. Harary. Its role in linking group theoretic ideas from Évariste Galois and classification results from William Burnside situates it historically among central developments in algebra and combinatorics documented by Saunders Mac Lane and Nicholas Bourbaki.
Category:Regular graphs