• Title/Summary/Keyword: Complete graph

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EVERY LINK IS A BOUNDARY OF A COMPLETE BIPARTITE GRAPH K2,n

  • Jang, Yongjun;Jeon, Sang-Min;Kim, Dongseok
    • Korean Journal of Mathematics
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    • v.20 no.4
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    • pp.403-414
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    • 2012
  • A voltage assignment on a graph was used to enumerate all possible 2-cell embeddings of a graph onto surfaces. The boundary of the surface which is obtained from 0 voltage on every edges of a very special diagram of a complete bipartite graph $K_{m,n}$ is surprisingly the ($m,n$) torus link. In the present article, we prove that every link is the boundary of a complete bipartite multi-graph $K_{m,n}$ for which voltage assignments are either -1 or 1 and that every link is the boundary of a complete bipartite graph $K_{2,n}$ for which voltage assignments are either -1, 0 or 1 where edges in the diagram of graphs may be linked but not knotted.

COMPLETE HAMACHER FUZZY GRAPHS

  • AL-HAWARY, TALAL ALI
    • Journal of applied mathematics & informatics
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    • v.40 no.5_6
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    • pp.1043-1052
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    • 2022
  • Our goal in this paper is to propose the advancement proposed by Alsina, Klement and other researchers in the area of fuzzy logic applied to the area of fuzzy graph theory. We introduce the notion of complete Hamacher fuzzy graphs and the notion of balanced Hamacher fuzzy graphs and discuss their properties. Moreover, several operations on these fuzzy graphs are explored via the complete notion.

Pebbling Numbers on Graphs (그래프 위에서의 Pebbling 수)

  • Chun, Kyung-Ah;Kim, Sung-Sook
    • The Journal of Natural Sciences
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    • v.12 no.1
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    • pp.1-9
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    • 2002
  • Let G be a connected graph on n vertices. The pebbling number of graph G, f(G), is the least m such that, however m pebbles are placed on the vertices of G, we can move a pebble to any vertex by a sequence of moves, each move taking two pebbles off one vertex and placing one on an adjacent vertex. In this paper, we compute the pebbling number of the Petersen Graph. We also show that the pebbling number of the categorical Product G.H is (m+n)h where G is the complete bipartite graph $K_{m,n}$ and H is the complete graph with $h(\geq4)$ vertices.

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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.

THE BASES OF PRIMITIVE NON-POWERFUL COMPLETE SIGNED GRAPHS

  • Song, Byung Chul;Kim, Byeong Moon
    • Korean Journal of Mathematics
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    • v.22 no.3
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    • pp.491-500
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    • 2014
  • The base of a signed digraph S is the minimum number k such that for any vertices u, v of S, there is a pair of walks of length k from u to v with different signs. Let K be a signed complete graph of order n, which is a signed digraph obtained by assigning +1 or -1 to each arc of the n-th order complete graph $K_n$ considered as a digraph. In this paper we show that for $n{\geq}3$ the base of a primitive non-powerful signed complete graph K of order n is 2, 3 or 4.

PEBBLING ON THE MIDDLE GRAPH OF A COMPLETE BINARY TREE

  • LOURDUSAMY, A.;NELLAINAYAKI, S. SARATHA;STEFFI, J. JENIFER
    • Journal of applied mathematics & informatics
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    • v.37 no.3_4
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    • pp.163-176
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    • 2019
  • Given a distribution of pebbles on the vertices of a connected graph G, a pebbling move is defined as the removal of two pebbles from some vertex and the placement of one of those pebbles at an adjacent vertex. The t-pebbling number, $f_t(G)$, of a connected graph G, is the smallest positive integer such that from every placement of $f_t(G)$ pebbles, t pebbles can be moved to any specified vertex by a sequence of pebbling moves. A graph G has the 2t-pebbling property if for any distribution with more than $2f_t(G)$ - q pebbles, where q is the number of vertices with at least one pebble, it is possible, using the sequence of pebbling moves, to put 2t pebbles on any vertex. In this paper, we determine the t-pebbling number for the middle graph of a complete binary tree $M(B_h)$ and we show that the middle graph of a complete binary tree $M(B_h)$ satisfies the 2t-pebbling property.

A NEW VERTEX-COLORING EDGE-WEIGHTING OF COMPLETE GRAPHS

  • Farahani, Mohammad Reza
    • Journal of applied mathematics & informatics
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    • v.32 no.1_2
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    • pp.1-6
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    • 2014
  • Let G = (V ; E) be a simple undirected graph without loops and multiple edges, the vertex and edge sets of it are represented by V = V (G) and E = E(G), respectively. A weighting w of the edges of a graph G induces a coloring of the vertices of G where the color of vertex v, denoted $S_v:={\Sigma}_{e{\ni}v}\;w(e)$. A k-edge-weighting of a graph G is an assignment of an integer weight, w(e) ${\in}${1,2,...,k} to each edge e, such that two vertex-color $S_v$, $S_u$ be distinct for every edge uv. In this paper we determine an exact 3-edge-weighting of complete graphs $k_{3q+1}\;{\forall}_q\;{\in}\;{\mathbb{N}}$. Several open questions are also included.

NONDEGENERATE AFFINE HOMOGENEOUS DOMAIN OVER A GRAPH

  • Choi, Yun-Cherl
    • Journal of the Korean Mathematical Society
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    • v.43 no.6
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    • pp.1301-1324
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    • 2006
  • The affine homogeneous hypersurface in ${\mathbb{R}}^{n+1}$, which is a graph of a function $F:{\mathbb{R}}^n{\rightarrow}{\mathbb{R}}$ with |det DdF|=1, corresponds to a complete unimodular left symmetric algebra with a nondegenerate Hessian type inner product. We will investigate the condition for the domain over the homogeneous hypersurface to be homogeneous through an extension of the complete unimodular left symmetric algebra, which is called the graph extension.

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.

THE BOUNDARIES OF DIPOLE GRAPHS AND THE COMPLETE BIPARTITE GRAPHS K2,n

  • Kim, Dongseok
    • Honam Mathematical Journal
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    • v.36 no.2
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    • pp.399-415
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    • 2014
  • We study the Seifert surfaces of a link by relating the embeddings of graphs with induced graphs. As applications, we prove that every link L is the boundary of an oriented surface which is obtained from a graph embedding of a complete bipartite graph $K_{2,n}$, where all voltage assignments on the edges of $K_{2,n}$ are 0. We also provide an algorithm to construct such a graph diagram of a given link and demonstrate the algorithm by dealing with the links $4^2_1$ and $5_2$.