1. An application of the Sakai's theorem to the characterization of H*-algebrasBorut Zalar, 1995, original scientific article Abstract: The well-known Sakai's theorem, which states that every derivation acting on a von Neumann algebra is inner, is used to obtain a new elegant proof of the Saworotnow's characterization theorem for associative ▫$H^\ast$▫-algebras via two-sided ▫$H^\ast$▫-algebras. This proof completely avoids structure theory.
Keywords: mathematics, functional analysis, ▫$H^\ast$▫-algebra, involution, automorphism, derivation, centralizer, von Neumann algebra
Published in DKUM: 14.06.2017; Views: 903; Downloads: 164
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2. Tree-like isometric subgraphs of hypercubesBoštjan Brešar, Wilfried Imrich, Sandi Klavžar, 2003, original scientific article Abstract: Tree-like isometric subgraphs of hypercubes, or tree-like partial cubes as we call them, are a generalization of median graphs. Just as median graphs they capture numerous properties of trees, but may contain larger classes of graphs that may be easier to recognize than the class of median graphs. We investigate the structure of tree-like partial cubes, characterize them, and provide examples of similarities with trees and median graphs. For instance, we show that the cube graph of tree-like partial cube is dismantlable. This in particular implies that every tree-like partial cube ▫$G$▫ contains a cube that is invariant under every automorphism of ▫$G$▫. We also show that weak retractions preserve tree-like partial cubes, which in turn implies that every contraction of a tree-like partial cube fixes a cube. The paper ends with several Frucht-type results and a list of open problems.
Keywords: mathematics, graph theory, Isometric embeddings, partial cubes, expansion procedures, trees, median graphs, graph automorphisms, automorphism groups, dismantlable graphs
Published in DKUM: 31.03.2017; Views: 1709; Downloads: 390
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3. The distinguishing chromatic number of Cartesian products of two complete graphsJanja Jerebic, Sandi Klavžar, 2010, published scientific conference contribution Abstract: Označitev grafa ▫$G$▫ je razlikovalna, če jo ohranja le trivialni avtomorfizem grafa ▫$G$▫. Razlikovalno kromatično število grafa ▫$G$▫ je najmanjše naravno število, za katero obstaja razlikovalna označitev grafa, ki je hkrati tudi dobro barvanje. Za vse ▫$k$▫ in ▫$n$▫ je določeno razlikovalno kromatično število kartezičnih produktov ▫$K_kBox K_n$▫. V večini primerov je enako kromatičnemu številu, kar med drugim odgovori na vprašanje Choia, Hartkeja and Kaula, ali obstajajo še kakšni drugi grafi, za katere velja enakost.
Keywords: teorija grafov, razlikovalno kromatično število, grafovski avtomorfizem, kartezični produkt grafov, graph theory, distinguishing chromatic number, graph automorphism, Cartesian product of graphs
Published in DKUM: 10.07.2015; Views: 1152; Downloads: 96
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4. Distinguishing infite graphsWilfried Imrich, Sandi Klavžar, Vladimir Ivanovič Trofimov, 2007, original scientific article Abstract: Razlikovalno število, ▫$D(G)$▫, grafa ▫$G$▫, je najmanjše kardinalno število ▫$aleph$▫, tako da ▫$G$▫ premore označitev z ▫$aleph$▫ oznakami, ki jo ohranja samo trivialni avtomorfizem. Dokažemo, da je razlikovalno število števnega slučajnega grafa enako dva in da imajo drevesom podobni grafi z ne več kot kontinuum vozlišči razlikovalno število enako dva. Določimo tudi razlikovalno število za mnoge razrede neskončnih kartezičnih produktov. Na primer, ▫$D(Q_n) = 2$▫, kjer je ▫$Q_n$▫ neskončna hiperkocka dimenzije ▫$n$▫.
Keywords: matematika, teorija grafov, razlikovalno število, avtomorfizem, neskončni grafi, slučajni graf, kartezični produkt grafov, kardinalna števila, ordinalna števila, mathematics, graph theory, distinguishing number, automorphism, infinite graphs, random graph, Cartesian product of graphs, ordinal numbers, cardinal numbers
Published in DKUM: 10.07.2015; Views: 1092; Downloads: 34
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5. Makhnëv, A. A. (RS-AOSUR); Nosov, V. V. (RS-AOSUR): Automorphisms of strongly regular Krein graphs without triangles. (Russian. Russian summary). - Algebra Logika 44 (2005), no. 3, 335--354, 384; translation in Algebra Logic 44 (2005), no. 3, 185--196Primož Potočnik, 2006, review, book review, critique Keywords: matematika, teorija grafov, avtomorfizem, Kreinov graf, Klebshov graf, ▫$n$▫-klika, mathematics, graph theory, automorphism, Krein graph, Klebsh graph, Higman-Sims graph, ▫$n$▫-clique, ▫$n$▫-coclique
Published in DKUM: 10.07.2015; Views: 826; Downloads: 39
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6. O'Reilly Regueiro, Eugenia (MEX-NAM-IM): Biplanes with flag-transitive automorphism groups of almost simple type, with alternating or sporadic socle. (English summary). - European J. Combin. 26 (2005), no. 5, 577--584.Primož Potočnik, 2006, review, book review, critique Keywords: matematika, teorija grup, grupa avtomorfizmov, mathematics, group theory, automorphism group
Published in DKUM: 10.07.2015; Views: 1045; Downloads: 38
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7. Cartesian powers of graphs can be distinguished by two labelsSandi Klavžar, Xuding Zhu, 2007, original scientific article Abstract: The distinguishing number ▫$D(G)$▫ of a graph ▫$G$▫ is the least integer ▫$d$▫ such that there is a ▫$d$▫-labeling of the vertices of ▫$G$▫ which is not preserved by any nontrivial automorphism. For a graph ▫$G$▫ let ▫$G^r$▫ be the ▫$r$▫-th power of ▫$G$▫ with respect to the Cartesian product. It is proved that ▫$D(G^r) = 2$▫ for any connected graph ▫$G$▫ with at least 3 vertices and for any ▫$r = 3$▫. This confirms and strengthens a conjecture of Albertson. Other graph products are also considered and a refinement of the Russell and Sundaram motion lemma is proved.
Keywords: matematika, teorija grafov, razlikovalno število, grafovski avtomorfizem, produkti grafov, mathematics, graph theory, distingushing number, graph automorphism, products of graphs
Published in DKUM: 10.07.2015; Views: 1260; Downloads: 88
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8. Distinguishing Cartesian powers of graphsWilfried Imrich, Sandi Klavžar, 2006, original scientific article Abstract: Razlikovalno število ▫$D(G)$▫ grafa je najmanjše celo število ▫$d$▫, za katero obstaja taka ▫$d$▫-označitev točk grafa ▫$G$▫, da je ne ohranja noben avtomorfizem grafa ▫$G$▫. Dokažemo, da je razlikovalno število kvadrata in višjih potenc povezanega grafa ▫$G ne K_2, K_3$▫, glede na kartezični produkt, vedno enako 2. Ta rezultat je močnejši od rezultatov Albertsona [Electron J Combin, 12 (2005), N17] za potence pra-grafov in tudi od rezultatov Klavžarja and Zhuja [European J. Combin, v tisku]. Bolj splošno, dokažemo tudi, da je ▫$(G Box H) = 2$▫, če sta ▫$G$▫ in ▫$H$▫ relativno tuja grafa in je ▫$|H| le |G| < 2^{|H|} - |H|$▫. Pod podobnimi pogoji veljajo sorodni rezultati tudi za potence grafov glede na krepki in direktni produkt grafov.
Keywords: matematika, teorija grafov, razlikovalno število, grafovski avtomorfizem, produkti grafov, mathematics, graph theory, distingushing number, graph automorphism, products of graphs
Published in DKUM: 10.07.2015; Views: 1234; Downloads: 97
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9. Distinguishing labellings of group action on vector spaces and graphsSandi Klavžar, Tsai-Lien Wong, Xuding Zhu, 2006, original scientific article Abstract: ▫$Gamma$▫ deluje na množico ▫$X$▫. ▫$k$▫-označitev ▫$X$▫ je preslikava ▫$c: to {1,2,...,k}$▫. Označitev ▫$c$▫ množice ▫$X$▫ je razlikovalna (glede na delovanje ▫$Gamma$▫), če za vsak ▫$g in Gamma$▫, ▫$g ne {mathrm{id}}_X$▫ obstaja element ▫$x in X$▫, tako da je ▫$c(x) ne c(g(x))$▫. Razlikovalno število, ▫$D_Gamma(X)$▫, delovanja ▫$Gamma$▫ na ▫$X$▫, je najmanjši ▫$k$▫, za katerega obstaja ▫$k$▫-označitev, ki je razlikovalna. V tem članku študiramo razlikovalno število linearne grupe ▫$GL_n(K)$▫ nad poljem ▫$K$▫, ki deluje na vektorski prostor ▫$K^n$▫ in razlikovalno število grupe avtomorfizmov Aut▫$(G)$▫ grafa ▫$G$▫, ki deluje na ▫$V(G)$▫. Slednje je poimenovano razlikovalno število grafa ▫$G$▫ in označeno z ▫$D(G)$▫. V članku so določene vrednosti ▫$D_{GL_n(K)}(K^n)$▫ za vsa polja ▫$K$▫ in vsa števila ▫$n$▫. Glede razlikovalnega števila grafov študiramo možne vrednosti razlikovalnega števila grafa glede na njegovo grupo avtomorfizmov, njegovo največjo stopnjo in druge strukturne lastnosti. Dokazano je, da če je ▫$mathrm{Aut}(G) = S_n$▫ in ima vsaka orbita v Aut▫$(G)$▫ velikost manj kot ▫$n choose n$▫, tedaj je ▫$D(G) = lceil n^{1/k} rceil$▫ za neko naravno število ▫$k$▫. Dokazan je izrek Brooks-ovega tipa za razlikovalno število: za vsak graf ▫$G$▫ velja ▫$D(G) le Delta(G)$▫, razen če je ▫$G$▫ polni graf, regularni polni dovodelni graf, ali pa ▫$C_5$▫. Vpeljemo tudi pojem enolično razlikovalnih grafov in proučujemo razlikovalno število nepovezanih grafov.
Keywords: matematika, teorija grafov, razlikovalno število, grupa, splošna linearna grupa, vektorski prostor, graf, avtomorfizem, razlikovalna množica, mathematics, graph theory, distinguishing number, group, general linear group, vector space, graph, graph automorphism, distinguishing set
Published in DKUM: 10.07.2015; Views: 1301; Downloads: 96
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10. Crossing numbers of Sierpiński-like graphsSandi Klavžar, Bojan Mohar, 2005, original scientific article Abstract: The crossing number of Sierpiński graphs ▫$S(n,k)$▫ and their regularizarions ▫$S^+(n,k)$▫ and ▫$S^{++}(n,k)$▫ are studied. Drawings of these graphs are presented and proved to be optimal for ▫$S^+(n,k)$▫ and ▫$S^{++}(n,k)$▫ for every ▫$n ge 1$▫ and ▫$k ge 1$▫. The crossing numbers of these graphs are expressed in terms of the crossing number of ▫$K_{k+1}$▫. These are the first nontrivial families of graphs of "fractal" type whose crossing number is known.
Keywords: matematika, teorija grafov, risanje grafov, prekrižno število, grafi Sierpińskega, avtomorfizmi grafov, mathematics, graf theory, graph drawing, crossing number, Sierpiński graphs, graph automorphism
Published in DKUM: 10.07.2015; Views: 1387; Downloads: 84
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