• Title/Summary/Keyword: connected graph

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THE CONNECTED SUBGRAPH OF THE TORSION GRAPH OF A MODULE

  • Ghalandarzadeh, Shaban;Rad, Parastoo Malakooti;Shirinkam, Sara
    • Journal of the Korean Mathematical Society
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    • v.49 no.5
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    • pp.1031-1051
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    • 2012
  • In this paper, we will investigate the concept of the torsion-graph of an R-module M, in which the set $T(M)^*$ makes up the vertices of the corresponding torsion graph, ${\Gamma}(M)$, with any two distinct vertices forming an edge if $[x:M][y:M]M=0$. We prove that, if ${\Gamma}(M)$ contains a cycle, then $gr({\Gamma}(M)){\leq}4$ and ${\Gamma}(M)$ has a connected induced subgraph ${\overline{\Gamma}}(M)$ with vertex set $\{m{\in}T(M)^*{\mid}Ann(m)M{\neq}0\}$ and diam$({\overline{\Gamma}}(M)){\leq}3$. Moreover, if M is a multiplication R-module, then ${\overline{\Gamma}}(M)$ is a maximal connected subgraph of ${\Gamma}(M)$. Also ${\overline{\Gamma}}(M)$ and ${\overline{\Gamma}}(S^{-1}M)$ are isomorphic graphs, where $S=R{\backslash}Z(M)$. Furthermore, we show that, if ${\overline{\Gamma}}(M)$ is uniquely complemented, then $S^{-1}M$ is a von Neumann regular module or ${\overline{\Gamma}}(M)$ is a star graph.

ON THE MONOPHONIC NUMBER OF A GRAPH

  • Santhakumaran, A.P.;Titus, P.;Ganesamoorthy, K.
    • Journal of applied mathematics & informatics
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    • v.32 no.1_2
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    • pp.255-266
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    • 2014
  • For a connected graph G = (V,E) of order at least two, a set S of vertices of G is a monophonic set of G if each vertex v of G lies on an x - y monophonic path for some elements x and y in S. The minimum cardinality of a monophonic set of G is the monophonic number of G, denoted by m(G). Certain general properties satisfied by the monophonic sets are studied. Graphs G of order p with m(G) = 2 or p or p - 1 are characterized. For every pair a, b of positive integers with $2{\leq}a{\leq}b$, there is a connected graph G with m(G) = a and g(G) = b, where g(G) is the geodetic number of G. Also we study how the monophonic number of a graph is affected when pendant edges are added to the graph.

The Number of Maximal Independent sets of the Graph with joining Moon-Moser Graph and Complete Graph (Moon-Moser 그래프와 완전그래프를 결합한 그래프의 극대독립집합의 개수)

  • Chung, S.J.;Lee, C.S.
    • Journal of Korean Institute of Industrial Engineers
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    • v.20 no.4
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    • pp.65-72
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    • 1994
  • An independent set of nodes is a set of nodes no two of which are joined by an edge. An independent set is called maximal if no more nodes can be added to the set without destroying its independence. The greatest number of maximal independent set is the maximum possible number of maximal independent set of a graph. We consider the greatest number of maximal independent set in connected graphs with fixed numbers of edges and nodes. For arbitrary number of nodes with a certain class of number of edges, we present the connected graphs with the greatest number of maximal independent set. For a given class of number of edges, the structure of graphs with the greatest number of maximal independent set is that the two components are completely joined; one consists of disjoint triangles as many as possible and the other is the complete graph with remaining nodes.

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HAMILTONIAN INSERTED GRAPHS AND SQUARES

  • Pramanik, L.K.;Adhikari, M.R.
    • Journal of the Chungcheong Mathematical Society
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    • v.19 no.1
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    • pp.37-47
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    • 2006
  • In this paper we characterize the graphs whose inserted graphs are Hamiltonian, and we study the relationship between Hamiltonian graphs and inserted graphs. Also we prove that if a connected graph G contains at least 3 vertices then inserted graph of the square of G is Hamiltonian and if G contains at least 3 edges then the square of inserted graph of G is Hamiltonian.

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SPANNING 3-FORESTS IN BRIDGES OF A TIGHT SEMIRING IN AN LV-GRAPH

  • Jung, Hwan-Ok
    • Journal of applied mathematics & informatics
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    • v.27 no.5_6
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    • pp.1307-1318
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    • 2009
  • An infinite locally finite plane graph is an LV-graph if it is 3-connected and VAP-free. In this paper, as a preparatory work for solving the problem concerning the existence of a spanning 3-tree in an LV-graph, we investigate the existence of a spanning 3-forest in a bridge of type 0,1 or 2 of a tight semi ring in an LV-graph satisfying certain conditions.

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SEMI-SYMMETRIC CUBIC GRAPH OF ORDER 12p3

  • Amoli, Pooriya Majd;Darafsheh, Mohammad Reza;Tehranian, Abolfazl
    • Bulletin of the Korean Mathematical Society
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    • v.59 no.1
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    • pp.203-212
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    • 2022
  • A simple graph is called semi-symmetric if it is regular and edge transitive but not vertex transitive. In this paper we prove that there is no connected cubic semi-symmetric graph of order 12p3 for any prime number p.

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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SECURE DOMINATION PARAMETERS OF HALIN GRAPH WITH PERFECT K-ARY TREE

  • R. ARASU;N. PARVATHI
    • Journal of applied mathematics & informatics
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    • v.41 no.4
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    • pp.839-848
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    • 2023
  • Let G be a simple undirected graph. A planar graph known as a Halin graph(HG) is characterised by having three connected and pendent vertices of a tree that are connected by an outer cycle. A subset S of V is said to be a dominating set of the graph G if each vertex u that is part of V is dominated by at least one element v that is a part of S. The domination number of a graph is denoted by the γ(G), and it corresponds to the minimum size of a dominating set. A dominating set S is called a secure dominating set if for each v ∈ V\S there exists u ∈ S such that v is adjacent to u and S1 = (S\{v}) ∪ {u} is a dominating set. The minimum cardinality of a secure dominating set of G is equal to the secure domination number γs(G). In this article we found the secure domination number of Halin graph(HG) with perfet k-ary tree and also we determined secure domination of rooted product of special trees.

The Gallai and Anti-Gallai Graphs of Strongly Regular Graphs

  • Jeepamol J. Palathingal;Aparna Lakshmanan S.;Greg Markowsky
    • Kyungpook Mathematical Journal
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    • v.64 no.1
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    • pp.171-184
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    • 2024
  • In this paper, we show that if G is strongly regular then the Gallai graph Γ(G) and the anti-Gallai graph ∆(G) of G are edge-regular. We also identify conditions under which the Gallai and anti-Gallai graphs are themselves strongly regular, as well as conditions under which they are 2-connected. We include also a number of concrete examples and a discussion of spectral properties of the Gallai and anti-Gallai graphs.

Automated Segmentation of the Lateral Ventricle Based on Graph Cuts Algorithm and Morphological Operations

  • Park, Seongbeom;Yoon, Uicheul
    • Journal of Biomedical Engineering Research
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    • v.38 no.2
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    • pp.82-88
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    • 2017
  • Enlargement of the lateral ventricles have been identified as a surrogate marker of neurological disorders. Quantitative measure of the lateral ventricle from MRI would enable earlier and more accurate clinical diagnosis in monitoring disease progression. Even though it requires an automated or semi-automated segmentation method for objective quantification, it is difficult to define lateral ventricles due to insufficient contrast and brightness of structural imaging. In this study, we proposed a fully automated lateral ventricle segmentation method based on a graph cuts algorithm combined with atlas-based segmentation and connected component labeling. Initially, initial seeds for graph cuts were defined by atlas-based segmentation (ATS). They were adjusted by partial volume images in order to provide accurate a priori information on graph cuts. A graph cuts algorithm is to finds a global minimum of energy with minimum cut/maximum flow algorithm function on graph. In addition, connected component labeling used to remove false ventricle regions. The proposed method was validated with the well-known tools using the dice similarity index, recall and precision values. The proposed method was significantly higher dice similarity index ($0.860{\pm}0.036$, p < 0.001) and recall ($0.833{\pm}0.037$, p < 0.001) compared with other tools. Therefore, the proposed method yielded a robust and reliable segmentation result.