Share to:

Cayley's formula

In mathematics, Cayley's formula is a result in graph theory named after Arthur Cayley. It states that for every positive integer $n$ , the number of trees on $n$ labeled vertices is $n^{n-2}$ .

The formula equivalently counts the number of spanning trees of a complete graph with labeled vertices (sequence A000272 in the OEIS).

Proof

Many proofs of Cayley's tree formula are known.^[1] One classical proof of the formula uses Kirchhoff's matrix tree theorem, a formula for the number of spanning trees in an arbitrary graph involving the determinant of a matrix. Prüfer sequences yield a bijective proof of Cayley's formula. Another bijective proof, by André Joyal, finds a one-to-one transformation between n-node trees with two distinguished nodes and maximal directed pseudoforests. A proof by double counting due to Jim Pitman counts in two different ways the number of different sequences of directed edges that can be added to an empty graph on n vertices to form from it a rooted tree; see Double counting (proof technique) § Counting trees.

History

The formula was first discovered by Carl Wilhelm Borchardt in 1860, and proved via a determinant.^[2] In a short 1889 note, Cayley extended the formula in several directions, by taking into account the degrees of the vertices.^[3] Although he referred to Borchardt's original paper, the name "Cayley's formula" became standard in the field.

Other properties

Cayley's formula immediately gives the number of labelled rooted forests on n vertices, namely $(n + 1) n - 1$ . Each labelled rooted forest can be turned into a labelled tree with one extra vertex, by adding a vertex with label $n + 1$ and connecting it to all roots of the trees in the forest.

There is a close connection with rooted forests and parking functions, since the number of parking functions on n cars is also $(n + 1) n - 1$ . A bijection between rooted forests and parking functions was given by M. P. Schützenberger in 1968.^[4]

Generalizations

The following generalizes Cayley's formula to labelled forests: Let T_n,k be the number of labelled forests on n vertices with k connected components, such that vertices 1, 2, ..., k all belong to different connected components. Then $T n, k = k n n - k - 1$ .^[5]

References

^ Aigner, Martin; Ziegler, Günter M. (1998). Proofs from THE BOOK. Springer-Verlag. pp. 141–146.
^ Borchardt, C. W. (1860). "Über eine Interpolationsformel für eine Art Symmetrischer Functionen und über Deren Anwendung". Math. Abh. der Akademie der Wissenschaften zu Berlin: 1–20.
^ Cayley, A. (1889). "A theorem on trees". Quart. J. Pure Appl. Math. 23: 376–378.
^ Schützenberger, M. P. (1968). "On an enumeration problem". Journal of Combinatorial Theory. 4: 219–221. MR 0218257.
^ Takács, Lajos (March 1990). "On Cayley's formula for counting forests". Journal of Combinatorial Theory, Series A. 53 (2): 321–323. doi:10.1016/0097-3165(90)90064-4.

Read more information:

Benda bersejarah

Untuk kegunaan lain, lihat Benda bersejarah (disambiguasi). Centaur bertarung dengan Lapith Benda bersejarah adalah benda dari zaman kuno, khususnya peradaban Laut Tengah: Zaman klasik Yunani dan Romawi, Mesir Kuno dan budaya Timur Dekat Kuno lainnya. Artefak-artefak dari periode-periode sebelumnya seperti Mesolitikum dan peradaban lain dari Asia dan tempat lain juga disebut dengan istilah tersebut. Fenomena memberikan nilai tinggi terhadap artefak-artefak kuno ditemukan di budaya lainnya, terut…

Peter Lorre

Peter Lorre (26 Juni 1904-23 Maret 1964), lahir László Loewenstein merupakan seorang aktor berkebangsaan Amerika Serikat-Austria. Dia dilahirkan di Ruzomberok, Austria-Hungaria. Dia berkarier di dunia film sejak tahun 1932 hingga wafat tahun 1964 di Los Angeles. Filmografi Die Verschwundene Frau M (1931) Bomben auf Monte Carlo Die Koffer des Herrn O.F. Fünf von der Jazzband F.P.1 antwortet nicht Der Weisse Dämon Stupéfiants Schuss im Morgengrauen Was Frauen träumen Unsichtbare Gegner Les R…

Industrial metal

Halaman ini berisi artikel tentang genre musik. Untuk penggunaan metal(logam) dalam industri, lihat Pengolahan logam. Industrial metalMinistry di Hellfest tahun 2017. Dari kiri ke kanan: Al Jourgensen, Jason Christopher dan Cesar Soto. Keyboardis John Bechdel ada di latar belakang.Sumber aliranIndustrial rockheavy metalSumber kebudayaanPertengahan-1980an; Britania Raya, Amerika Serikat, Jerman, dan SwissBentuk turunanNu metalTopik lainnya Metal alternatif avant-garde metal Neue Deutsche Härte N…

Ryo Kiyuna

Ryo KiyunaRyo Kiyuna pada 2018Informasi pribadiLahir12 Juli 1990 (umur 33)Okinawa, Jepang[1] OlahragaNegaraJepangOlahragaKarate Ryo Kiyuna (lahir 12 Juli 1990)[2] adalah karateka Jepang. Ia merupakan peraih medali emas tiga kali pada Kejuaraan Karate Dunia dan dua kali medali emas pada tim kata putra, selain Arata Kinjo dan Takuya Uemura. Ia merupakan empat kali juara Asia berturut - turut. Ia dijadwalkan akan mewakili Jepang pada Olimpiade Musim Panas 2020 yang diselenggara…

Lee Deok-hwa

Dalam nama Korean ini, nama keluarganya adalah Lee. Lee Deok-hwaLee Deok-hwa pada tahun 2019Lahir8 Mei 1952 (umur 71)Seoul, Korea SelatanAlmamaterUniversitas DonggukPekerjaanAktorTahun aktif1972-sekarangSuami/istriKim Bo-ok [ko]Anak2Nama KoreaHangul이덕화 Hanja李德華 Alih AksaraI Deok-hwaMcCune–ReischauerI Dŏkhwa Lee Deok-hwa (lahir 8 Mei 1952) adalah pemeran Korea Selatan. Karir Lee Deok-hwa mempelajari Teater dan Film di Universitas Dongguk, dan memulai debut ak…

Daftar Wali Kota Bandung

Daftar ini belum tentu lengkap. Anda dapat membantu Wikipedia dengan mengembangkannya. Wali Kota BandungPetahanaBambang Tirtoyuliono(Penjabat)sejak 20 September 2023Pemerintah Kota BandungKediamanPendopo Kota Bandung, Balonggede, Regol, Bandung 40251Masa jabatan5 tahun dan dapat dipilih kembali untuk satu kali masa jabatanDibentuk1 Juli 1917; 106 tahun lalu (1917-07-01)Pejabat pertamaBertus Coops (Hindia Belanda, 1917)Ating Atma di Nata (Republik Indonesia, 1945)WakilWakil Wali Kota Ba…

Skandal Perjamuan Natal

Skandal Perjamuan Natal Ilustrasi edisi Inggris pertamaPengarangAgatha ChristiePerancang sampulTidak diketahuiNegaraBritania RayaBahasaInggrisGenreNovel kejahatanPenerbitCollins Crime ClubTanggal terbit24 Oktober 1960Jenis mediaCetak (sampul keras & sampul kertas)Halaman256 halaman (edisi pertama, sampul keras)Didahului olehKucing di Tengah Burung Dara Diikuti olehDouble Sin and Other Stories Skandal Perjamuan Natal atau The Adventure of the Christmas Pudd…

Mata (siklon)

Gambar Badai Isabel yang terlihat dari Stasiun Luar Angkasa Internasional menunjukkan mata yang jelas di pusat badai. Mata adalah daerah yang sebagian besar cuacanya tenang di pusat siklon tropis yang kuat. Mata badai adalah area yang kira-kira bundar, biasanya berdiameter 30–65 kilometer (19–40 mi). Dikelilingi oleh dinding mata, sebuah cincin badai petir yang menjulang tinggi di mana cuaca paling parah dan angin kencang terjadi. Tekanan barometrik terendah topan terjadi di mata dan bi…

الدوري البولندي الممتاز 1950

يفتقر محتوى هذه المقالة إلى الاستشهاد بمصادر. فضلاً، ساهم في تطوير هذه المقالة من خلال إضافة مصادر موثوق بها. أي معلومات غير موثقة يمكن التشكيك بها وإزالتها. (يونيو 2019) الدوري البولندي الممتاز 1950 تفاصيل الموسم الدوري البولندي الممتاز النسخة 24 البلد بولندا المنظم ات…

Calcio Padova 1986-1987

Voce principale: Calcio Padova. Calcio PadovaStagione 1986-1987Una formazione del Padova 1986-87 Sport calcio Squadra Padova Allenatore Adriano Buffoni Presidente Marino Puggina Serie C12º posto (promozione in serie B) Coppa Italia Serie CSemifinale Maggiori presenzeCampionato: Donati (34) Miglior marcatoreCampionato: Mariani (9) 1985-86 1987-88 Si invita a seguire il modello di voce Questa pagina raccoglie i dati riguardanti l'Associazione Calcio Padova nelle competizioni ufficiali della …

Gabriele Cioffi

Italian footballer (born 1975) Gabriele Cioffi Cioffi in 2023Personal informationDate of birth (1975-09-07) 7 September 1975 (age 48)Place of birth Florence, ItalyHeight 1.96 m (6 ft 5 in)Position(s) DefenderYouth career SesteseSenior career*Years Team Apps (Gls)1992–1996 Sestese 77 (2)1996–1997 Marsala 12 (2)1996–1997 Poggibonsi 10 (0)1997–1999 Spezia 55 (1)1999–2001 Arezzo 19 (0)2001–2002 Taranto 4 (0)2002–2005 Novara 72 (7)2005–2006 Mantova 52 (5)2006–200…

Tinia Valles

Valles on Mars Tinia VallesThe Tinia Valles, as seen by HiRISE (click for full size image to view the dark slope streaks)Coordinates4°42′S 149°00′W / 4.7°S 149°W / -4.7; -149 The Tinia Valles are a set of channels in an ancient valley in the Memnonia quadrangle of Mars, located at 4.7° south latitude and 149° west longitude. They are 18.7 km long and were named after a classical river in Italy.[1] The associated valley has many dark slope streaks on…

Halcyon RB80

Non-depth-compensated passive addition semi-closed circuit rebreather The Halcyon RB80 is a non-depth-compensated passive addition semi-closed circuit rebreather of similar external dimensions to a standard AL80 scuba cylinder (11-litre, 207-bar aluminium cylinder, 185 mm diameter and about 660 mm long). It was originally developed by Reinhard Buchaly (RB) in 1996 for the cave exploration dives conducted by the European Karst Plain Project (EKPP).[1] About 1/10 of the respired …

ERV3

ERV3-1 المعرفات الأسماء المستعارة ERV3-1, ERV-R, ERV3, ERVR, HERV-R, HERVR, envR, endogenous retrovirus group 3 member 1, endogenous retrovirus group 3 member 1, envelope معرفات خارجية الوراثة المندلية البشرية عبر الإنترنت 131170 HomoloGene: 128310 GeneCards: 2086 علم الوجود الجيني الوظيفة الجزيئية • وظيفة جزيئة المكونات الخلوية • غلاف الفيروس• virion• ح…

Carlos Lopez-Cantera

19th Lieutenant Governor of Florida Carlos Lopez-Cantera19th Lieutenant Governor of FloridaIn officeFebruary 3, 2014 – January 7, 2019GovernorRick ScottPreceded byJennifer CarrollSucceeded byJeanette NuñezProperty Appraiser of Miami-Dade CountyIn officeJanuary 1, 2013[1] – February 3, 2014Preceded byPedro GarciaSucceeded byPedro GarciaMember of the Florida House of Representativesfrom the 113th districtIn officeNovember 2, 2004[2] – N…

Minamifurano, Hokkaido

Minamifurano 南富良野町KotaprajaBalai Kota Minamifurano BenderaEmblemLokasi Minamifurano di Hokkaido (Subprefektur Kamikawa)MinamifuranoLokasi di JepangKoordinat: 43°10′N 142°34′E / 43.167°N 142.567°E / 43.167; 142.567Koordinat: 43°10′N 142°34′E / 43.167°N 142.567°E / 43.167; 142.567NegaraJepangWilayahHokkaidoPrefektur Hokkaido (Subprefektur Kamikawa)DistrikSorachiPemerintahan • WalikotaHideki TakahashiLuas…

Bosnian diaspora

People of Bosnian heritage who live outside Bosnia and Herzegovina Not to be confused with Bosniak diaspora.This article needs additional citations for verification. Please help improve this article by adding citations to reliable sources. Unsourced material may be challenged and removed.Find sources: Bosnian diaspora – news · newspapers · books · scholar · JSTOR (October 2010) (Learn how and when to remove this message)You can help expand this article wi…

Languaculture

This article has multiple issues. Please help improve it or discuss these issues on the talk page. (Learn how and when to remove these template messages) This article relies excessively on references to primary sources. Please improve this article by adding secondary or tertiary sources. Find sources: Languaculture – news · newspapers · books · scholar · JSTOR (June 2011) (Learn how and when to remove this message) The topic of this article may not meet W…

Strada statale 48 bis delle Dolomiti

Questa voce o sezione sull'argomento strade d'Italia non cita le fonti necessarie o quelle presenti sono insufficienti. Puoi migliorare questa voce aggiungendo citazioni da fonti attendibili secondo le linee guida sull'uso delle fonti. Strada statale 48 bisdelle DolomitiLocalizzazioneStato Italia Regioni Veneto Trentino-Alto Adige DatiClassificazioneStrada statale InizioMisurina FineCarbonin Lunghezza8,800 km Provvedimento di istituzioneLegge 17 maggio 1928, n. 1094 GestoreVe…

2016年美國總統選舉

2016年美國總統選舉 ← 2012 2016年11月8日 2020 → 538個選舉人團席位獲勝需270票民意調查投票率55.7%[1][2] ▲ 0.8 % 获提名人唐納·川普希拉莉·克林頓政党共和黨民主党家鄉州紐約州紐約州竞选搭档迈克·彭斯蒂姆·凱恩选举人票 304[3][4][註 1] 227[5] 胜出州/省 30 + 緬-2 20 + DC 民選得票 62,984,828[6] 65,853,514[6] 得…

Kembali kehalaman sebelumnya