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Girth-regular and edge-girth-regular graphs : master's thesis
Zavrtanik Drglin, Ajda (Author), Jajcay, Robert (Mentor) More about this mentor... This link opens in a new window, Potočnik, Primož (Co-mentor)

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Abstract
In this work we discuss girth-regular and edge-girth-regular graphs. The signature of a vertex u in a graph is a k-tuple of integers, ordered from the smallest to the largest, where each integer represents the number of girth cycles that contain an edge, incident with u. We say that a graph is girth-regular, if every vertex has the same signature. If every edge is contained in the same number of girth cycles, the graph is edge-girth-regular. We present the known results about girth-regular and edge-girth-regular graphs, classify cubic graphs of both types up to girth 5, look at tetravalent edge-girth-regular graphs and present some constructions of infinite families of such graphs. We then present some new results on tetravalent edge-girth-regular graphs and the classification of tetravalent edge-girth-regular Cayley graphs of Abelian groups.

Language:English
Keywords:graph, girth, girth-regular, edge-girth-regular
Work type:Master's thesis/paper (mb22)
Organization:FMF - Faculty of Mathematics and Physics
Year:2019
UDC:519.1
COBISS.SI-ID:18739289 This link opens in a new window
Views:413
Downloads:156
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Secondary language

Language:Slovenian
Title:Ožinsko-regularni in povezavno-ožinsko-regularni grafi
Abstract:
Magistrska naloga obravnava ožinsko-regularne in povezavno-ožinsko-regularne grafe. Podpis vozlišča u v grafu je k-terica celih števil, urejenih po velikosti od najmanjšega do največjega, kjer vsako število predstavlja število ožinskih ciklov, v katerih je vsebovana posamezna povezava, incidenčna z u. Pravimo, da je graf ožinsko-regularen (oz. tipa GR), če imajo vsa vozlišča v grafu enak podpis. Če velja, da je vsaka povezava v grafu vsebovana v enakem številu ožinskih ciklov, pravimo, da je graf povezavno-ožinsko-regularen (oz. tipa EGR). V delu predstavimo že znane rezultate o grafih tipa GR in EGR, posebej natančno pregledamo kubične grafe obeh tipov in tetravalentne grafe tipa EGR ter nekaj konstrukcij neskončnih družin takih grafov. Nato predstavimo nekaj novih rezultatov o grafih tipa EGR in klasifikacijo vseh tetravalentnih Cayleyevih grafov Abelovih grup – kaj mora veljati, da je tak graf lahko tipa EGR, ter v koliko ožinskih ciklih se potemtakem lahko nahaja vsaka povezava tega grafa.

Keywords:graf, ožina, ožinsko-regularen, povezavno-ožinsko-regularen

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