On the Non-commutative Neutrix Product of and
Brian Fisher, Fatma Al-Sirehy (1999)
Publications de l'Institut Mathématique
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Brian Fisher, Fatma Al-Sirehy (1999)
Publications de l'Institut Mathématique
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Tomasz Dzido, Krzysztof Krzywdziński (2015)
Czechoslovak Mathematical Journal
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In this paper, we investigate a measure of similarity of graphs similar to the Ramsey number. We present values and bounds for , the biggest number guaranteeing that there exist graphs on vertices, each two having edit distance at least . By edit distance of two graphs , we mean the number of edges needed to be added to or deleted from graph to obtain graph . This new extremal number is closely linked to the edit distance of graphs. Using probabilistic methods we show...
Mikhail Skopenkov (2003)
Fundamenta Mathematicae
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For any collection of graphs we find the minimal dimension d such that the product is embeddable into (see Theorem 1 below). In particular, we prove that (K₅)ⁿ and are not embeddable into , where K₅ and are the Kuratowski graphs. This is a solution of a problem of Menger from 1929. The idea of the proof is a reduction to a problem from so-called Ramsey link theory: we show that any embedding , where O is a vertex of (K₅)ⁿ, has a pair of linked (n-1)-spheres.
Pavel Híc, Milan Pokorný (2016)
Czechoslovak Mathematical Journal
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A graph is called distance integral (or -integral) if all eigenvalues of its distance matrix are integers. In their study of -integral complete multipartite graphs, Yang and Wang (2015) posed two questions on the existence of such graphs. We resolve these questions and present some further results on -integral complete multipartite graphs. We give the first known distance integral complete multipartite graphs with , and with , as well as the infinite classes of distance integral...
Xiaodan Chen, Guoliang Hao, Dequan Jin, Jingjian Li (2018)
Czechoslovak Mathematical Journal
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For a simple graph on vertices and an integer with , denote by the sum of largest signless Laplacian eigenvalues of . It was conjectured that , where is the number of edges of . This conjecture has been proved to be true for all graphs when , and for trees, unicyclic graphs, bicyclic graphs and regular graphs (for all ). In this note, this conjecture is proved to be true for all graphs when , and for some new classes of graphs.
Erica Flapan, Blake Mellor, Ramin Naimi (2008)
Fundamenta Mathematicae
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We show that, given any n and α, any embedding of any sufficiently large complete graph in ℝ³ contains an oriented link with components Q₁, ..., Qₙ such that for every i ≠ j, and , where denotes the second coefficient of the Conway polynomial of .
Yinkui Li, Yaping Mao, Zhao Wang, Zongtian Wei (2021)
Czechoslovak Mathematical Journal
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We study the generalized -connectivity as introduced by Hager in 1985, as well as the more recently introduced generalized -edge-connectivity . We determine the exact value of and for the line graphs and total graphs of trees, unicyclic graphs, and also for complete graphs for the case .
Werner Klöckl (2008)
Discussiones Mathematicae Graph Theory
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The distinguishing number D(G) of a graph G is the least integer d such that G has a labeling with d colors that is not preserved by any nontrivial automorphism. The restriction to proper labelings leads to the definition of the distinguishing chromatic number of G. Extending these concepts to infinite graphs we prove that and , where denotes the hypercube of countable dimension. We also show that , thereby completing the investigation of finite hypercubes with respect to . Our...
Yanan Chu, Lei Sun, Jun Yue (2019)
Czechoslovak Mathematical Journal
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A graph is called improperly -colorable if the vertex set can be partitioned into subsets such that the graph induced by the vertices of has maximum degree at most for all . In this paper, we mainly study the improper coloring of -planar graphs and show that -planar graphs with girth at least are -colorable.
Chun-Gil Park (2002)
Studia Mathematica
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The generalized non-commutative torus of rank n is defined by the crossed product , where the actions of ℤ on the fibre of a rational rotation algebra are trivial, and is a non-commutative torus . It is shown that is strongly Morita equivalent to , and that is isomorphic to if and only if the set of prime factors of k is a subset of the set of prime factors of p.
Shaohui Wang, Bing Wei (2017)
Czechoslovak Mathematical Journal
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Let and be the domination number and the independent domination number of , respectively. Rad and Volkmann posted a conjecture that for any graph , where is its maximum degree (see N. J. Rad, L. Volkmann (2013)). In this work, we verify the conjecture for bipartite graphs. Several graph classes attaining the extremal bound and graphs containing odd cycles with the ratio larger than are provided as well.
Péter P. Varjú (2012)
Journal of the European Mathematical Society
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Let be a fixed symmetric finite subset of that generates a Zariski dense subgroup of when we consider it as an algebraic group over by restriction of scalars. We prove that the Cayley graphs of with respect to the projections of is an expander family if ranges over square-free ideals of if and is an arbitrary numberfield, or if and .
Vladimír Lacko (2000)
Discussiones Mathematicae Graph Theory
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For the Traveling Salesman Problem (TSP) on Halin graphs with three types of cost functions: sum, bottleneck and balanced and with arbitrary real edge costs we compute in polynomial time the persistency partition , , of the edge set E, where: = e ∈ E, e belongs to all optimum solutions, = e ∈ E, e does not belong to any optimum solution and = e ∈ E, e belongs to some but not to all optimum solutions.
Jingru Yan (2023)
Czechoslovak Mathematical Journal
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A graph is -saturated if it contains no as a subgraph, but does contain after the addition of any edge in the complement of . The saturation number, , is the minimum number of edges of a graph in the set of all -saturated graphs of order . We determine the saturation number for and characterize the extremal graphs for .
Jing Lin (2022)
Czechoslovak Mathematical Journal
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Given a graph , let denote the maximum number of edges in a bipartite subgraph of . Given a fixed graph and a positive integer , let denote the minimum possible cardinality of , as ranges over all graphs on edges that contain no copy of . In this paper we prove that , which extends the results of N. Alon, M. Krivelevich, B. Sudakov. Write and for the subdivisions of and . We show that and , improving a result of Q. Zeng, J. Hou. We also give lower bounds on...
Demelash Ashagrie Mengesha, Tomáš Vetrík (2019)
Czechoslovak Mathematical Journal
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A directed Cayley graph is specified by a group and an identity-free generating set for this group. Vertices of are elements of and there is a directed edge from the vertex to the vertex in if and only if there is a generator such that . We study graphs for the direct product of two cyclic groups and , and the generating set . We present resolving sets which yield upper bounds on the metric dimension of these graphs for and .