Quotient-polynomial graphs
As a generalization of orbit-polynomial and distance-regular graphs, we introduce the concept of a quotient-polynomial graph. In these graphs every vertex u induces the same regular partition around u, where all vertices of each cell are equidistant from u. Some properties and characterizations of s...
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| Published in: | Linear algebra and its applications Vol. 488; pp. 363 - 376 |
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| Format: | Journal Article Publication |
| Language: | English |
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01.01.2016
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| ISSN: | 0024-3795, 1873-1856 |
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| Abstract | As a generalization of orbit-polynomial and distance-regular graphs, we introduce the concept of a quotient-polynomial graph. In these graphs every vertex u induces the same regular partition around u, where all vertices of each cell are equidistant from u. Some properties and characterizations of such graphs are studied. For instance, all quotient-polynomial graphs are walk-regular and distance-polynomial. Also, we show that every quotient-polynomial graph generates a (symmetric) association scheme. |
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| AbstractList | As a generalization of orbit-polynomial and distance-regular graphs, we introduce the concept of a quotient-polynomial graph. In these graphs every vertex u induces the same regular partition around u, where all vertices of each cell are equidistant from u. Some properties and characterizations of such graphs are studied. For instance, all quotient-polynomial graphs are walk-regular and distance-polynomial. Also, we show that every quotient-polynomial graph generates a (symmetric) association scheme. As a generalization of orbit-polynomial and distance-regular graphs, we introduce the concept of a quotient-polynomial graph. In these graphs every vertex uinduces the same regular partition around u, where all vertices of each cell are equidistant from u. Some properties and characterizations of such graphs are studied. For instance, all quotient-polynomial graphs are walk-regular and distance-polynomial. Also, we show that every quotient-polynomial graph generates a (symmetric) association scheme. Peer Reviewed |
| Author | Fiol, M.A. |
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| Cites_doi | 10.1016/0095-8956(82)90031-4 10.1006/jctb.1997.1778 10.1016/j.jcta.2010.10.005 10.1016/0024-3795(80)90180-9 10.1080/00029890.1963.11990038 10.1016/0024-3795(86)90235-1 10.1016/0012-365X(87)90023-9 10.1006/jctb.1996.0063 10.1023/A:1018628204156 |
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| Contributor | Universitat Politècnica de Catalunya. COMBGRAPH - Combinatòria, Teoria de Grafs i Aplicacions Universitat Politècnica de Catalunya. Departament de Matemàtiques |
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| Keywords | 05E30 Intersection numbers 05C50 Graph quotient Quotient-polynomial graph Walk-regular partition Orthogonal polynomials Distance-regular graph Eigenvalues Distance-faithful partition |
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| References | Beezer (br0020) 1986; 73 Beezer (br0030) 1987; 67 Biggs (br0010) 1974 Hoffman (br0140) 1963; 70 Dalfó, van Dam, Fiol, Garriga, Gorissen (br0060) 2011; 118 Godsil, McKay (br0130) 1980; 30 van Dam (br0070) 1999; 10 Delsarte (br0090) 1973; 10 Qiao, Du, Koolen (br0150) 2015 Godsil (br0120) 1993 Fiol, Garriga (br0100) 1997; 71 Brouwer, Cohen, Neumaier (br0040) 1989 Weichsel (br0160) 1982; 32 Fiol, Garriga, Yebra (br0110) 1996; 68 Dalfó (10.1016/j.laa.2015.09.053_br0060) 2011; 118 Godsil (10.1016/j.laa.2015.09.053_br0130) 1980; 30 Biggs (10.1016/j.laa.2015.09.053_br0010) 1974 Brouwer (10.1016/j.laa.2015.09.053_br0040) 1989 Beezer (10.1016/j.laa.2015.09.053_br0020) 1986; 73 Delsarte (10.1016/j.laa.2015.09.053_br0090) 1973; 10 Fiol (10.1016/j.laa.2015.09.053_br0110) 1996; 68 Fiol (10.1016/j.laa.2015.09.053_br0100) 1997; 71 Hoffman (10.1016/j.laa.2015.09.053_br0140) 1963; 70 van Dam (10.1016/j.laa.2015.09.053_br0070) 1999; 10 Godsil (10.1016/j.laa.2015.09.053_br0120) 1993 Weichsel (10.1016/j.laa.2015.09.053_br0160) 1982; 32 Qiao (10.1016/j.laa.2015.09.053_br0150) Beezer (10.1016/j.laa.2015.09.053_br0030) 1987; 67 |
| References_xml | – volume: 67 start-page: 139 year: 1987 end-page: 147 ident: br0030 article-title: Orbit polynomial graphs of prime order publication-title: Discrete Math. – year: 1974 ident: br0010 article-title: Algebraic Graph Theory – volume: 10 start-page: 69 year: 1999 end-page: 107 ident: br0070 article-title: Three-class association schemes publication-title: J. Algebraic Combin. – year: 1989 ident: br0040 article-title: Distance-Regular Graphs – volume: 118 start-page: 1094 year: 2011 end-page: 1113 ident: br0060 article-title: On almost distance-regular graphs publication-title: J. Combin. Theory Ser. A – volume: 10 year: 1973 ident: br0090 article-title: An algebraic approach to the association schemes of coding theory publication-title: Philips Res. Rep. Suppl. – volume: 70 start-page: 30 year: 1963 end-page: 36 ident: br0140 article-title: On the polynomial of a graph publication-title: Amer. Math. Monthly – volume: 30 start-page: 51 year: 1980 end-page: 61 ident: br0130 article-title: Feasibility conditions for the existence walk-regular graphs publication-title: Linear Algebra Appl. – volume: 73 start-page: 133 year: 1986 end-page: 146 ident: br0020 article-title: Trivalent orbit polynomial graphs publication-title: Linear Algebra Appl. – volume: 32 start-page: 156 year: 1982 end-page: 161 ident: br0160 article-title: On distance-regularity in graphs publication-title: J. Combin. Theory Ser. B – volume: 71 start-page: 162 year: 1997 end-page: 183 ident: br0100 article-title: From local adjacency polynomials to locally pseudo-distance-regular graphs publication-title: J. Combin. Theory Ser. B – year: 2015 ident: br0150 article-title: 2-walk-regular dihedrants from group-divisible designs – volume: 68 start-page: 179 year: 1996 end-page: 205 ident: br0110 article-title: Locally pseudo-distance-regular graphs publication-title: J. Combin. Theory Ser. B – year: 1993 ident: br0120 article-title: Algebraic Combinatorics – volume: 32 start-page: 156 year: 1982 ident: 10.1016/j.laa.2015.09.053_br0160 article-title: On distance-regularity in graphs publication-title: J. Combin. Theory Ser. B doi: 10.1016/0095-8956(82)90031-4 – volume: 71 start-page: 162 year: 1997 ident: 10.1016/j.laa.2015.09.053_br0100 article-title: From local adjacency polynomials to locally pseudo-distance-regular graphs publication-title: J. Combin. Theory Ser. B doi: 10.1006/jctb.1997.1778 – volume: 118 start-page: 1094 year: 2011 ident: 10.1016/j.laa.2015.09.053_br0060 article-title: On almost distance-regular graphs publication-title: J. Combin. Theory Ser. A doi: 10.1016/j.jcta.2010.10.005 – volume: 30 start-page: 51 year: 1980 ident: 10.1016/j.laa.2015.09.053_br0130 article-title: Feasibility conditions for the existence walk-regular graphs publication-title: Linear Algebra Appl. doi: 10.1016/0024-3795(80)90180-9 – year: 1989 ident: 10.1016/j.laa.2015.09.053_br0040 – volume: 70 start-page: 30 year: 1963 ident: 10.1016/j.laa.2015.09.053_br0140 article-title: On the polynomial of a graph publication-title: Amer. Math. Monthly doi: 10.1080/00029890.1963.11990038 – volume: 73 start-page: 133 year: 1986 ident: 10.1016/j.laa.2015.09.053_br0020 article-title: Trivalent orbit polynomial graphs publication-title: Linear Algebra Appl. doi: 10.1016/0024-3795(86)90235-1 – volume: 67 start-page: 139 year: 1987 ident: 10.1016/j.laa.2015.09.053_br0030 article-title: Orbit polynomial graphs of prime order publication-title: Discrete Math. doi: 10.1016/0012-365X(87)90023-9 – ident: 10.1016/j.laa.2015.09.053_br0150 – year: 1993 ident: 10.1016/j.laa.2015.09.053_br0120 – volume: 10 year: 1973 ident: 10.1016/j.laa.2015.09.053_br0090 article-title: An algebraic approach to the association schemes of coding theory publication-title: Philips Res. Rep. Suppl. – year: 1974 ident: 10.1016/j.laa.2015.09.053_br0010 – volume: 68 start-page: 179 year: 1996 ident: 10.1016/j.laa.2015.09.053_br0110 article-title: Locally pseudo-distance-regular graphs publication-title: J. Combin. Theory Ser. B doi: 10.1006/jctb.1996.0063 – volume: 10 start-page: 69 year: 1999 ident: 10.1016/j.laa.2015.09.053_br0070 article-title: Three-class association schemes publication-title: J. Algebraic Combin. doi: 10.1023/A:1018628204156 |
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| Snippet | As a generalization of orbit-polynomial and distance-regular graphs, we introduce the concept of a quotient-polynomial graph. In these graphs every vertex u... As a generalization of orbit-polynomial and distance-regular graphs, we introduce the concept of a quotient-polynomial graph. In these graphs every vertex... |
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| SubjectTerms | 05 Combinatorics 05C Graph theory 05E Algebraic combinatorics Classificació AMS Combinatorial analysis Combinatòria Distance-faithful partition Distance-regular graph Eigenvalues Grafs, Teoria de Graph quotient Graph theory Intersection numbers Matemàtica discreta Matemàtiques i estadística Orthogonal polynomials Quotient-polynomial graph Teoria de grafs Walk-regular partition Àrees temàtiques de la UPC |
| Title | Quotient-polynomial graphs |
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