• Title/Summary/Keyword: Complete Multipartite Graphs

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SIGNED A-POLYNOMIALS OF GRAPHS AND POINCARÉ POLYNOMIALS OF REAL TORIC MANIFOLDS

  • Seo, Seunghyun;Shin, Heesung
    • Bulletin of the Korean Mathematical Society
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    • v.52 no.2
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    • pp.467-481
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    • 2015
  • Choi and Park introduced an invariant of a finite simple graph, called signed a-number, arising from computing certain topological invariants of some specific kinds of real toric manifolds. They also found the signed a-numbers of path graphs, cycle graphs, complete graphs, and star graphs. We introduce a signed a-polynomial which is a generalization of the signed a-number and gives a-, b-, and c-numbers. The signed a-polynomial of a graph G is related to the $Poincar\acute{e}$ polynomial $P_{M(G)}(z)$, which is the generating function for the Betti numbers of the real toric manifold M(G). We give the generating functions for the signed a-polynomials of not only path graphs, cycle graphs, complete graphs, and star graphs, but also complete bipartite graphs and complete multipartite graphs. As a consequence, we find the Euler characteristic number and the Betti numbers of the real toric manifold M(G) for complete multipartite graphs G.

A DIFFERENCE SET METHOD FOR CIRCULANT DECOMPOSITIONS OF COMPLETE PARTITE GRAPHS INTO GREGARIOUS 4-CYCLES

  • Kim, Eun-Kyung;Cho, Young-Min;Cho, Jung-Rae
    • East Asian mathematical journal
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    • v.26 no.5
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    • pp.655-670
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    • 2010
  • The complete multipartite graph $K_{n(m)}$ with n $ {\geq}$ 4 partite sets of size m is shown to have a decomposition into 4-cycles in such a way that vertices of each cycle belong to distinct partite sets of $K_{n(m)}$, if 4 divides the number of edges. Such cycles are called gregarious, and were introduced by Billington and Hoffman ([2]) and redefined in [3]. We independently came up with the result of [3] by using a difference set method, and improved the result so that the composition is circulant, in the sense that it is invariant under the cyclic permutation of partite sets. The composition is then used to construct gregarious 4-cycle decompositions when one partite set of the graph has different cardinality than that of others. Some results on joins of decomposable complete multipartite graphs are also presented.

A REMARK ON CIRCULANT DECOMPOSITIONS OF COMPLETE MULTIPARTITE GRAPHS BY GREGARIOUS CYCLES

  • Cho, Jung Rae
    • East Asian mathematical journal
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    • v.33 no.1
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    • pp.67-74
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    • 2017
  • Let k, m and t be positive integers with $m{\geq}4$ and even. It is shown that $K_{km+1(2t)}$ has a decomposition into gregarious m-cycles. Also, it is shown that $K_{km(2t)}$ has a decomposition into gregarious m-cycles if ${\frac{(m-1)^2+3}{4m}}$ < k. In this article, we make a remark that the decompositions can be circulant in the sense that it is preserved by the cyclic permutation of the partite sets, and we will exhibit it by examples.

DECOMPOSITIONS OF COMPLETE MULTIPARTITE GRAPHS INTO GREGARIOUS 6-CYCLES USING COMPLETE DIFFERENCES

  • Cho, Jung-R.;Gould, Ronald J.
    • Journal of the Korean Mathematical Society
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    • v.45 no.6
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    • pp.1623-1634
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    • 2008
  • The complete multipartite graph $K_{n(2t)}$ having n partite sets of size 2t, with $n\;{\geq}\;6$ and $t\;{\geq}\;1$, is shown to have a decomposition into gregarious 6-cycles, that is, the cycles which have at most one vertex from any particular partite set. Complete sets of differences of numbers in ${\mathbb{Z}}_n$ are used to produce starter cycles and obtain other cycles by rotating the cycles around the n-gon of the partite sets.

TOTAL DOMINATION NUMBER OF CENTRAL GRAPHS

  • Kazemnejad, Farshad;Moradi, Somayeh
    • Bulletin of the Korean Mathematical Society
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    • v.56 no.4
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    • pp.1059-1075
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    • 2019
  • Let G be a graph with no isolated vertex. A total dominating set, abbreviated TDS of G is a subset S of vertices of G such that every vertex of G is adjacent to a vertex in S. The total domination number of G is the minimum cardinality of a TDS of G. In this paper, we study the total domination number of central graphs. Indeed, we obtain some tight bounds for the total domination number of a central graph C(G) in terms of some invariants of the graph G. Also we characterize the total domination number of the central graph of some families of graphs such as path graphs, cycle graphs, wheel graphs, complete graphs and complete multipartite graphs, explicitly. Moreover, some Nordhaus-Gaddum-like relations are presented for the total domination number of central graphs.

Binary Locally Repairable Codes from Complete Multipartite Graphs (완전다분할그래프 기반 이진 부분접속복구 부호)

  • Kim, Jung-Hyun;Nam, Mi-Young;Song, Hong-Yeop
    • The Journal of Korean Institute of Communications and Information Sciences
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    • v.40 no.9
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    • pp.1734-1740
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    • 2015
  • This paper introduces a generalized notion, referred to as joint locality, of the usual locality in distributed storage systems and proposes a code construction of binary locally repairable codes with joint locality ($r_1$=2, $r_2$=3 or 4). Joint locality is a set of numbers of nodes for repairing various failure patterns of nodes. The proposed scheme simplifies the code design problem utilizing complete multipartite graphs. Moreover, our construction can generate binary locally repairable codes achieving (2,t)-availability for any positive integer t. It means that each node can be repaired by t disjoint repair sets of cardinality 2. This property is useful for distributed storage systems since it permits parallel access to hot data.

ON DECOMPOSITIONS OF THE COMPLETE EQUIPARTITE GRAPHS Kkm(2t) INTO GREGARIOUS m-CYCLES

  • Kim, Seong Kun
    • East Asian mathematical journal
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    • v.29 no.3
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    • pp.337-347
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    • 2013
  • For an even integer m at least 4 and any positive integer $t$, it is shown that the complete equipartite graph $K_{km(2t)}$ can be decomposed into edge-disjoint gregarious m-cycles for any positive integer ${\kappa}$ under the condition satisfying ${\frac{{(m-1)}^2+3}{4m}}$ < ${\kappa}$. Here it will be called a gregarious cycle if the cycle has at most one vertex from each partite set.

GENERALIZATION ON PRODUCT DEGREE DISTANCE OF TENSOR PRODUCT OF GRAPHS

  • PATTABIRAMAN, K.
    • Journal of applied mathematics & informatics
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    • v.34 no.3_4
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    • pp.341-354
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    • 2016
  • In this paper, the exact formulae for the generalized product degree distance, reciprocal product degree distance and product degree distance of tensor product of a connected graph and the complete multipartite graph with partite sets of sizes m0, m1, ⋯ , mr−1 are obtained.

A NOTE ON DECOMPOSITION OF COMPLETE EQUIPARTITE GRAPHS INTO GREGARIOUS 6-CYCLES

  • Cho, Jung-Rae
    • Bulletin of the Korean Mathematical Society
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    • v.44 no.4
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    • pp.709-719
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    • 2007
  • In [8], it is shown that the complete multipartite graph $K_{n(2t)}$ having n partite sets of size 2t, where $n{\geq}6\;and\;t{\geq}1$, has a decomposition into gregarious 6-cycles if $n{\equiv}0,1,3$ or 4 (mod 6). Here, a cycle is called gregarious if it has at most one vertex from any particular partite set. In this paper, when $n{\equiv}0$ or 3 (mod 6), another method using difference set is presented. Furthermore, when $n{\equiv}0$ (mod 6), the decomposition obtained in this paper is ${\infty}-circular$, in the sense that it is invariant under the mapping which keeps the partite set which is indexed by ${\infty}$ fixed and permutes the remaining partite sets cyclically.