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oai:recercat.cat:2117/1848692026-02-09T06:46:18Zcom_2072_1033col_2072_452950

00925njm 22002777a 4500 dc Abajo Casado, Encarnación author Balbuena Martínez, Maria Camino Teófila author Bendala García, Manuel Francisco author Marcote Ordax, Francisco Javier author 2019-10-01 © 2019. This manuscript version is made available under the CC-BY-NC-ND 4.0 license http://creativecommons.org/licenses/by-nc-nd/4.0/ A -graph is a -regular graph with girth and a -cage is a -graph with the fewest possible number of vertices . Constructing -cages and determining the order are both very hard problems. For this reason, an intensive line of research is devoted to constructing smaller -graphs than previously known ones, providing in this way new upper bounds to each time such a graph is constructed. The paper focuses on girth , where cages are known only for degrees . We construct -graphs using and extending techniques of amalgamation into the incidence graphs of elliptic semiplanes of type introduced and exposed by Funk (2009). The order of these graphs provides better upper bounds on than those known so far, for values of such that either or. Peer Reviewed Postprint (author's final draft) Àrees temàtiques de la UPC::Matemàtiques i estadística::Matemàtica discreta::Combinatòria Àrees temàtiques de la UPC::Matemàtiques i estadística::Equacions diferencials i integrals Combinatorial analysis Differential equations, Elliptic Small regular graphs Cage Girth Amalgam Elliptic semiplane Combinacions (Matemàtica) Equacions diferencials el·líptiques Classificació AMS::05 Combinatorics::05E Algebraic combinatorics Classificació AMS::35 Partial differential equations::35J Partial differential equations of elliptic type Improving bounds on the order of regular graphs of girth 5