{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"institution":[{"id":[{"id":"https:\/\/ror.org\/03mb6wj31","id-type":"ROR","asserted-by":"publisher"},{"id":"https:\/\/www.isni.org\/000000041937028X","id-type":"ISNI","asserted-by":"publisher"},{"id":"https:\/\/www.wikidata.org\/entity\/Q1640731","id-type":"wikidata","asserted-by":"publisher"}],"name":"Universitat Polit\u00e8cnica de Catalunya","acronym":["UPC"]}],"indexed":{"date-parts":[[2026,2,6]],"date-time":"2026-02-06T17:37:55Z","timestamp":1770399475719,"version":"3.49.0"},"reference-count":0,"publisher":"Universitat Polit\u00e8cnica de Catalunya","license":[{"content-version":"vor","delay-in-days":0,"URL":"http:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"abstract":"<jats:p>In this thesis we use analytic combinatorics to deal with two related problems: graph enumeration and random graphs from constrained classes of graphs. We are interested in drawing a general picture of some graph families by determining, first, how many elements are there of a given possible size (graph enumeration), and secondly, what is the typical behaviour of an element of fixed size chosen uniformly at random, when the size tends to infinity (random graphs). The problems  concern graphs subject to global conditions, such as being planar and\/or with restrictions on the degrees of the vertices. \r\n\r\nIn Chapter 2 we analyse random planar graphs with minimum degree two and three. Using techniques from analytic combinatorics and the concepts of core and kernel of a graph, we obtain precise asymptotic estimates and analyse relevant parameters for random graphs, such as the number of edges or the size of the core, where we obtain Gaussian limit laws. More challenging is the extremal parameter equal to the size of the largest tree attached to the core. In this case we obtain a logarithmic estimate for the expected value together with a concentration result. \r\n\r\nIn Chapter 3 we study the number of subgraphs isomorphic to a fixed graph in subcritical classes of graphs. We obtain Gaussian limit laws with linear expectation and variance when the fixed graph is 2-connected. The main tool is the analysis of infinite systems of equations by Drmota, Gittenberger and Morgenbesser, using the theory of compact operators. Computing the exact constants for the first estimates of the moments is in general out of reach. For the class of series-parallel graphs we are able to compute them in some particular interesting cases. \r\n\r\nIn Chapter 4 we enumerate (arbitrary) graphs where the degree of every vertex belongs to a fixed subset of the natural numbers. In this case the associated generating functions are divergent and our analysis uses instead the so-called configuration model. We obtain precise asymptotic estimates for the number of graphs with given number of vertices and edges and subject to the degree restriction. Our results generalize widely previous special cases, such as d-regular graphs or graphs with minimum degree at least d.<\/jats:p>\n                <jats:p>En aquesta tesi utilitzem l'anal\u00edtica combinat\u00f2ria per treballar amb dos problemes relacionats: enumeraci\u00f3 de grafs i grafs aleatoris de classes de grafs amb restriccions. En particular ens interessa esbossar un dibuix general de determinades fam\u00edlies de grafs determinant, en primer lloc, quants grafs hi ha de cada mida possible (enumeraci\u00f3 de grafs), i, en segon lloc, quin \u00e9s el comportament t\u00edpic d'un element de mida fixa triat a l'atzar uniformement, quan aquesta mida tendeix a infinit (grafs aleatoris). Els problemes en qu\u00e8 treballem tracten amb grafs que satisfan condicions globals, com ara \u00e9sser planars, o b\u00e9 tenir restriccions en el grau dels v\u00e8rtexs. En el Cap\u00edtol 2 analitzem grafs planar aleatoris amb grau m\u00ednim dos i tres. Mitjan\u00e7ant t\u00e8cniques de combinat\u00f2ria anal\u00edtica i els conceptes de nucli i kernel d'un graf, obtenim estimacions asimpt\u00f2tiques precises i analitzem par\u00e0metres rellevants de grafs aleatoris, com ara el nombre d'arestes o la mida del nucli, on obtenim lleis l\u00edmit gaussianes. Tamb\u00e9 treballem amb un par\u00e0metre que suposa un repte m\u00e9s important: el par\u00e0metre extremal que es correspon amb la mida de l'arbre m\u00e9s gran que penja del nucli. En aquest cas obtenim una estimaci\u00f3 logar\u00edtmica per al seu valor esperat, juntament amb un resultat sobre la seva concentraci\u00f3. En el Cap\u00edtol 3 estudiem el nombre de subgrafs isomorfs a un graf fix en classes de grafs subcr\u00edtiques. Quan el graf fix \u00e9s biconnex, obtenim lleis l\u00edmit gaussianes amb esperan\u00e7a i vari\u00e0ncia lineals. L'eina principal \u00e9s l'an\u00e0lisi de sistemes infinits d'equacions donada per Drmota, Gittenberger i Morgenbesser, que utilitza la teoria d'operadors compactes. El c\u00e0lcul de les constants exactes de la primera estimaci\u00f3 dels moments en general es troba fora del nostre abast. Per a la classe de grafs s\u00e8rie-paral\u00b7lels podem calcular les constants en alguns casos particulars interessants. En el Cap\u00edtol 4 enumerem grafs (arbitraris) el grau de cada v\u00e8rtex dels quals pertany a un subconjunt fix dels nombres naturals. En aquest cas les funcions generatrius associades s\u00f3n divergents i la nostra an\u00e0lisi utilitza l'anomenat model de configuraci\u00f3. El nostre resultat consisteix a obtenir estimacions asimpt\u00f2tiques precises per al nombre de grafs amb un nombre de v\u00e8rtexs i arestes donat, amb la restricci\u00f3 dels graus. Aquest resultat generalitza \u00e0mpliament casos particulars existents, com ara grafs d-regulars, o grafs amb grau m\u00ednim com a m\u00ednim d.<\/jats:p>","DOI":"10.5821\/dissertation-2117-107956","type":"dissertation","created":{"date-parts":[[2023,10,14]],"date-time":"2023-10-14T01:28:29Z","timestamp":1697246909000},"approved":{"date-parts":[[2017,3,27]]},"source":"Crossref","is-referenced-by-count":0,"title":["Graph enumeration and random graphs"],"prefix":"10.5821","author":[{"sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Lander","family":"Ramos Garrido","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"3865","container-title":[],"original-title":[],"deposited":{"date-parts":[[2026,2,6]],"date-time":"2026-02-06T06:33:44Z","timestamp":1770359624000},"score":1,"resource":{"primary":{"URL":"https:\/\/hdl.handle.net\/2117\/107956"}},"subtitle":[],"editor":[{"given":"Marcos","family":"Noy Serrano","sequence":"first","affiliation":[],"role":[{"role":"editor","vocabulary":"crossref"}]}],"short-title":[],"issued":{"date-parts":[[null]]},"references-count":0,"URL":"https:\/\/doi.org\/10.5821\/dissertation-2117-107956","relation":{},"subject":[]}}