{"title":"关于Corank两边二部图和简单扩展欧几里得图","authors":"Marcin Gąsiorek, D. Simson, Katarzyna Zając","doi":"10.1109/SYNASC.2014.17","DOIUrl":null,"url":null,"abstract":"We continue the Coxeter spectral study of finite connected loop-free edge-bipartite graphs Δ, with m+2 ≥ 3 vertices (a class of signed graphs), started in [SIAM J. Discrete Math., 27(2013), 827-854] by means of the complex Coxeter spectrum specc<sub>Δ</sub> ⊆ C and presented in our talks given in SYNASC12 and SYNASC13. Here, we study non-negative edge-bipartite graphs of corank two, in the sense that the symmetric Gram matrix G<sub>Δ</sub> ∈ M<sub>m+2</sub>(Z) of Δ is positive semi-definite of rank m ≥ 1. Extending each of the simply laced Euclidean diagrams A<sub>m</sub>, m ≥ 1, D<sub>m</sub>, m ≥ 4, Ẽ<sub>6</sub>, Ẽ<sub>7</sub>, Ẽ<sub>8</sub> by one vertex, we construct a family of loop-free corank two diagrams Ã<sub>m</sub><sup>2</sup>, D̃<sub>m</sub><sup>2</sup>, Ẽ<sub>6</sub><sup>2</sup>, Ẽ<sub>7</sub><sup>2</sup>, Ẽ<sub>8</sub><sup>2</sup> (called simply extended Euclidean diagrams) such that they classify all connected corank two loop-free edge-bipartite graphs Δ, with m + 2 ≥ 3 vertices, up to Z-congruence Δ ~z Δ'. Here Δ ~z Δ' means that G<sub>Δ'</sub>, = B<sup>tr</sup> ·G<sub>Δ</sub> ·B, for some B ∈ M<sub>m+2</sub>(Z) such that det B = ±1. We present algorithms that generate all such edge-bipartite graphs of a given size m + 2 ≥ 3, together with their Coxeter polynomials, and the reduced Coxeter numbers, using symbolic and numeric computer calculations in Python. Moreover, we prove that for any corank two connected loop-free edge-bipartite graph Δ, with m + 2 ≥ 3 vertices, there exists a simply extended Euclidean diagram D such that Δ ~z D.","PeriodicalId":150575,"journal":{"name":"2014 16th International Symposium on Symbolic and Numeric Algorithms for Scientific Computing","volume":null,"pages":null},"PeriodicalIF":0.0000,"publicationDate":"2014-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"12","resultStr":"{\"title\":\"On Corank Two Edge-Bipartite Graphs and Simply Extended Euclidean Diagrams\",\"authors\":\"Marcin Gąsiorek, D. Simson, Katarzyna Zając\",\"doi\":\"10.1109/SYNASC.2014.17\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"We continue the Coxeter spectral study of finite connected loop-free edge-bipartite graphs Δ, with m+2 ≥ 3 vertices (a class of signed graphs), started in [SIAM J. Discrete Math., 27(2013), 827-854] by means of the complex Coxeter spectrum specc<sub>Δ</sub> ⊆ C and presented in our talks given in SYNASC12 and SYNASC13. Here, we study non-negative edge-bipartite graphs of corank two, in the sense that the symmetric Gram matrix G<sub>Δ</sub> ∈ M<sub>m+2</sub>(Z) of Δ is positive semi-definite of rank m ≥ 1. Extending each of the simply laced Euclidean diagrams A<sub>m</sub>, m ≥ 1, D<sub>m</sub>, m ≥ 4, Ẽ<sub>6</sub>, Ẽ<sub>7</sub>, Ẽ<sub>8</sub> by one vertex, we construct a family of loop-free corank two diagrams Ã<sub>m</sub><sup>2</sup>, D̃<sub>m</sub><sup>2</sup>, Ẽ<sub>6</sub><sup>2</sup>, Ẽ<sub>7</sub><sup>2</sup>, Ẽ<sub>8</sub><sup>2</sup> (called simply extended Euclidean diagrams) such that they classify all connected corank two loop-free edge-bipartite graphs Δ, with m + 2 ≥ 3 vertices, up to Z-congruence Δ ~z Δ'. Here Δ ~z Δ' means that G<sub>Δ'</sub>, = B<sup>tr</sup> ·G<sub>Δ</sub> ·B, for some B ∈ M<sub>m+2</sub>(Z) such that det B = ±1. We present algorithms that generate all such edge-bipartite graphs of a given size m + 2 ≥ 3, together with their Coxeter polynomials, and the reduced Coxeter numbers, using symbolic and numeric computer calculations in Python. Moreover, we prove that for any corank two connected loop-free edge-bipartite graph Δ, with m + 2 ≥ 3 vertices, there exists a simply extended Euclidean diagram D such that Δ ~z D.\",\"PeriodicalId\":150575,\"journal\":{\"name\":\"2014 16th International Symposium on Symbolic and Numeric Algorithms for Scientific Computing\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2014-09-01\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"12\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"2014 16th International Symposium on Symbolic and Numeric Algorithms for Scientific Computing\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.1109/SYNASC.2014.17\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"2014 16th International Symposium on Symbolic and Numeric Algorithms for Scientific Computing","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1109/SYNASC.2014.17","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
On Corank Two Edge-Bipartite Graphs and Simply Extended Euclidean Diagrams
We continue the Coxeter spectral study of finite connected loop-free edge-bipartite graphs Δ, with m+2 ≥ 3 vertices (a class of signed graphs), started in [SIAM J. Discrete Math., 27(2013), 827-854] by means of the complex Coxeter spectrum speccΔ ⊆ C and presented in our talks given in SYNASC12 and SYNASC13. Here, we study non-negative edge-bipartite graphs of corank two, in the sense that the symmetric Gram matrix GΔ ∈ Mm+2(Z) of Δ is positive semi-definite of rank m ≥ 1. Extending each of the simply laced Euclidean diagrams Am, m ≥ 1, Dm, m ≥ 4, Ẽ6, Ẽ7, Ẽ8 by one vertex, we construct a family of loop-free corank two diagrams Ãm2, D̃m2, Ẽ62, Ẽ72, Ẽ82 (called simply extended Euclidean diagrams) such that they classify all connected corank two loop-free edge-bipartite graphs Δ, with m + 2 ≥ 3 vertices, up to Z-congruence Δ ~z Δ'. Here Δ ~z Δ' means that GΔ', = Btr ·GΔ ·B, for some B ∈ Mm+2(Z) such that det B = ±1. We present algorithms that generate all such edge-bipartite graphs of a given size m + 2 ≥ 3, together with their Coxeter polynomials, and the reduced Coxeter numbers, using symbolic and numeric computer calculations in Python. Moreover, we prove that for any corank two connected loop-free edge-bipartite graph Δ, with m + 2 ≥ 3 vertices, there exists a simply extended Euclidean diagram D such that Δ ~z D.