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Wide diameter of Cartesian graph bundles
Iztok Banič, Janez Žerovnik, 2010, published scientific conference contribution

Abstract: Fault tolerance and transmission delay of networks are important concepts in network design. The notions are strongly related to connectivity and diameter of a graph, and have been studied by many authors. Wide diameter of a graph combines studying connectivity with the diameter of a graph. Diameter with width ▫$k$▫ of a graph ▫$G$▫, ▫$k$▫-diameter, is defined as the minimum integer ▫$d$▫ for which there exist at least ▫$k$▫ internally disjoint paths of length at most ▫$d$▫ between any two distinct vertices in ▫$G$▫. Denote by ▫${mathscr D}_c(G)$▫ the ▫$c$▫-diameter of ▫$G$▫ and ▫$kappa(G)$▫ the connectivity of ▫$G$▫. In the context of computer networks, wide diameters of Cartesian graph products have been recently studied by many authors. Cartesian graph bundles is a class of graphs that is a generalization of the Cartesian graph products. Let ▫$G$▫ be a Cartesian graph bundle with fiber ▫$F$▫ over base ▫$B$▫, ▫$0 < a le kappa(F)$▫, and ▫$0 < b le kappa(B)$▫. We prove that ▫${mathscr D}_{a+b}(G) le {mathscr D}_a(F) + {mathscr D}_b(B) + 1$▫. Moreover, if ▫$G$▫ is a graph bundle with fiber ▫$F ne K_2$▫ over base ▫$B ne K_2$▫, then ▫${mathscr D}_{a+b}(G) le {mathscr D}_a(F) + {mathscr D}_b(B)$▫. The bounds are tight.
Keywords: mathematics, graph theory, Cartesian graph products, Cartesian graph bundles, wide diameter
Published: 07.06.2012; Views: 1135; Downloads: 59
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Petar Filipov, 2012, undergraduate thesis

Abstract: The Subprime crisis that occurred in 2007-2008, was the most disturbing and severe crisis since The Great Depression. The global stock market indexes and housing prices noted a sharp drop. Big corporations, conglomerates, defaulted, were acquired and bailed out, since they could not meet their financing obligations. Substantial number of people lost their jobs worldwide and a lot of sectors in the economy were struggling. Within this Bachelor seminar paper, it is presented how the credit rating agencies influenced the credit crunch. The main issue discussed in this seminar work is the role of the rating agencies in one of the greatest crushes in history. They (the agencies), actually, participated in creating the structured financial products, which during the housing bubble were increasingly trading. These structured financial products were very complex and highly risky and maybe, the most crucial cause of the crisis. First, in the seminar paper it is described how the agencies work and their origins. As in the last century, Standard and Poor’s and Moody’s, emerged to great power in the financial system with assigning ratings to issuers of debt. This position in the financial world that the agencies held, has brought many academicians to doubting the possible moral hazard problems within the credit ratings business. Furthermore, it is explained how the financial turmoil emerged and the trends that preceded the credit crunch. Questions regarding the role of the agencies in the securitization process are discussed. For instance, were the rating agencies biased; were the ratings on the structured financial products inflated and if so, what were the incentives for it? Several theories and empirical evidence has been put forward concerning the importance of the ratings in the securitization process. It is concluded in this paper, that the ratings assigned to the financial derivatives were inflated and the agencies noted increased returns from it, which presents the incentives of the agencies during the housing bubble. These empirical findings also indicate of moral hazard problem within the ratings system.
Keywords: Credit rating agencies, Financial crisis, Structured financial products, Securitization, Inflated ratings, Moral hazard
Published: 29.11.2012; Views: 1009; Downloads: 70
.pdf Full text (749,27 KB)

Distance-balanced graphs
Janja Jerebic, Sandi Klavžar, Douglas F. Rall, 2005

Abstract: V članku so vpeljani razdaljno uravnoteženi grafi kot grafi, v katerih ima vsaka povezava ▫$uv$▫ naslednjo lastnost: število točk, ki so bližje ▫$u$▫ kot ▫$v$▫, je enako kot število točk, ki so bližje ▫$v$▫ kot ▫$u$▫. Dobljene so osnovne lastnosti teh grafov. Novi koncept je povezan z grafovskimi simetrijami, študirane so tudi lokalne operacije na grafih glede na razdaljno uravnoteženost. Karakterizirani so razdaljno uravnoteženi kartezični in leksikografski produkti grafov. Postavljenih je več odprtih problemov.
Keywords: matematika, teorija grafov, razdalja, razdaljno uravnoteženi grafi, produkti grafov, povezanost, mathematics, graph theory, graph distance, distance-balanced graphs, graph products, connectivity
Published: 10.07.2015; Views: 567; Downloads: 58
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Fault-diameter of Cartesian product of graphs and Cartesian graph bundles
Iztok Banič, Janez Žerovnik, 2006

Abstract: Cartesian graph bundles is a class of graphs that is a generalization of the Cartesian graph products. Let ▫$G$▫ be a ▫$k_G$▫-connected graph and ▫${mathcal{D}}_c(G)$▫ denote the diameter of ▫$G$▫ after deleting any of its ▫$c < k_G$▫ vertices. We prove that if ▫$G_1, G_2, dots, G_q$▫ are ▫$k_1$▫-connected, ▫$k_2$▫-connected,...,▫$k_q$▫-connected graphs and ▫$0 leq a_1 < k_1$▫, ▫$0 leq a_2 < k_2$▫,...,▫$0 leq a_q < k_q$▫ and ▫$a = a_1 + a_2 + dots + a_q + (q-1)$▫, then the fault diameter of ▫$G$▫, a Cartesian product of ▫$G_1$▫, ▫$G_2$▫,...,▫$G_q$▫, with ▫$a$▫ faulty nodes is ▫${mathcal{D}}_{a}(G) leq {mathcal{D}}_{a_1}(G_1)+{mathcal{D}}_{a_2}(G_2) + dots + {mathcal{D}}_{a_q}(G_q) + 1$▫. We also show that ▫${mathcal{D}}_{a+b+1}(G) leq {mathcal{D}}_a(F) + {mathcal{D}}_b(B) + 1$▫ if ▫$G$▫ is a graph bundle with fibre ▫$F$▫ over base ▫$B$▫, ▫$a leq k_F$▫, and ▫$b leq k_B$▫. As an auxiliary result we prove that connectivity of graph bundle ▫$G$▫ is at least ▫$k_F+k_B$▫.
Keywords: mathematics, graph theory, Cartesian graph bundles, Cartesian graph products, fault diameter, interconnection network
Published: 10.07.2015; Views: 360; Downloads: 15
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On the k-path vertex cover of some graph products
Marko Jakovac, Andrej Taranenko, 2013, original scientific article

Abstract: A subset S of vertices of a graph G is called a k-path vertex cover if every path of order k in G contains at least one vertex from S. Denote by ▫$psi_k$▫(G) the minimum cardinality of a k-path vertex cover in G. In this paper, improved lower and upper bounds for ▫$psi_k$▫ of the Cartesian and the strong product of paths are derived. It is shown that for ▫$psi_3$▫ those bounds are tight. For the lexicographic product bounds are presented for ▫$psi_k$▫, moreover ▫$psi_2$▫ and ▫$psi_3$▫ are exactly determined for the lexicographic product of two arbitrary graphs. As a consequence the independence and the dissociation number of the lexicographic product are given.
Keywords: matematika, teorija grafov, vozliščno pokritje, po poteh vozliščno pokritje, disociacijsko število, neodvisnostno število, grafovski produkti, mathematics, graph theory, vertex cover, path vertex cover, dissociation number, independence number, graph products
Published: 10.07.2015; Views: 479; Downloads: 10
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Characterizing subgraphs of Hamming graphs
Sandi Klavžar, Iztok Peterin, 2005, original scientific article

Abstract: Kartezični produkti polnih grafov so znani kot Hammingovi grafi. Z uporabo vložitev v kartezične produkte kvocientnih grafov so karakterizirani podgrafi, inducirani podgrafi in izometrični podgrafi Hammingovih grafov. Na primer, graf ▫$G$▫ je inducirani podgraf Hammingovega grafa natanko tedaj, ko obstaja označitev povezav grafa ▫$G$▫, ki zadošča naslednjima pogojema: (i) povezave trikotnika imajo isto oznako, (ii) za vsaki točki ▫$u$▫ in ▫$v$▫ na razdalji vsaj 2 obstajata dve taki oznaki, ki se pojavita na vsaki inducirani poti med ▫$u$▫ in ▫$v$▫.
Keywords: matematika, teorija grafov, Hammingovi grafi, inducirani podgrafi, izometrični podgrafi, kartezični produkt grafov, označevanje povezav, kvocientni grafi, mathematics, graph theory, Hamming graphs, induced subgraphs, isometric subgraphs, edge-labelings, Cartesian products, quotient graphs
Published: 10.07.2015; Views: 404; Downloads: 47
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