Incidence matrices of projective planes and of some regular bipartite graphs of girth 6 with few vertices
Let q be a prime power and r=0,1...,q−3. Using the Latin squares obtained by multiplying each entry of the addition table of the Galois field of order q by an element distinct from zero, we obtain the incidence matrices of projective planes and the incidence matrices of (q−r)-regular bipartite graph...
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| Tipo de recurso: | artículo |
| Fecha de publicación: | 2008 |
| País: | España |
| Institución: | Universitat Politècnica de Catalunya (UPC) |
| Repositorio: | UPCommons. Portal del coneixement obert de la UPC |
| Idioma: | inglés |
| OAI Identifier: | oai:upcommons.upc.edu:2117/9495 |
| Acceso en línea: | https://hdl.handle.net/2117/9495 https://dx.doi.org/10.1137/070688225 |
| Access Level: | acceso abierto |
| Palabra clave: | Magic squares Projective planes Cages Bipartite graphs Quadrats màgics Geometria projectiva Grafs, Teoria de Àrees temàtiques de la UPC::Matemàtiques i estadística::Matemàtica discreta::Teoria de grafs |
| Sumario: | Let q be a prime power and r=0,1...,q−3. Using the Latin squares obtained by multiplying each entry of the addition table of the Galois field of order q by an element distinct from zero, we obtain the incidence matrices of projective planes and the incidence matrices of (q−r)-regular bipartite graphs of girth 6 and $q^2$−rq−1 vertices in each partite set. Moreover, in this work two Latin squares of order q−1 with entries belonging to {0,1,..., q}, not necessarily the same, are defined to be quasi row-disjoint if and only if the cartesian product of any two rows contains at most one pair (χ,χ) with χ≠0. Using these quasi row-disjoint Latin squares we find (q−1)-regular bipartite graphs of girth 6 with $q^2$−q−2 vertices in each partite set. Some of these graphs have the smallest number of vertices known so far among the regular graphs with girth 6. |
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