{"title":"Turán's theorem, from edges to eigenvalues","authors":"Leo Torres","abstract":"Turán's theorem is the founding result of extremal graph theory: among all graphs on $n$ vertices with no clique on $r+1$ vertices, the balanced complete multipartite graph has the most edges. We present the theorem through five classical proofs and follow one of them, the variational argument of Motzkin and Straus, upward into spectral graph theory, where Nikiforov's spectral strengthening returns and proves the original theorem in two lines. All mathematics herein is known; the contribution is the form of its presentation.","keywords":[],"year":2026,"slug":"turan","version":1,"url_hash":"a53f35091073","license":"cc-by-4.0","doi":"10.5281/zenodo.22199342","published_at":"2026-08-31T07:54:06.184311+00:00","subject":"Mathematics","storage_type":"archive","canonical_url":"https://scroll.press/2026/turan","version_url":"https://scroll.press/2026/turan/v1","paper_url":"https://scroll.press/2026/turan/paper","paper_version_url":"https://scroll.press/2026/turan/v1/paper","html_url":"https://scroll.press/2026/turan/paper","html_sha256":"a53f35091073dae25f5047825ac09b0e64a634d76e62eb884bf4212f4d93f7de","html_bytes":null,"cite_as":{"bibtex":"@misc{turan2026,\n  author = {Torres, Leo},\n  title = {Turán's theorem, from edges to eigenvalues},\n  year = {2026},\n  publisher = {Scroll Press},\n  url = {https://scroll.press/2026/turan},\n  version = {1},\n  doi = {10.5281/zenodo.22199342}\n}","csl_json":{"type":"article","id":"a53f35091073","title":"Turán's theorem, from edges to eigenvalues","author":[{"family":"Torres","given":"Leo"}],"publisher":"Scroll Press","URL":"https://scroll.press/2026/turan","version":"1","issued":{"date-parts":[[2026]]},"DOI":"10.5281/zenodo.22199342","license":"cc-by-4.0"}}}