• Title/Summary/Keyword: square graph

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SQUARE CONGRUENCE GRAPHS

  • Janardhanan Suresh Kumar;Sarika M. Nair
    • Communications of the Korean Mathematical Society
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    • v.38 no.3
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    • pp.937-941
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    • 2023
  • For each positive integer n, a square congruence graph S(n) is the graph with vertex set H = {1, 2, 3,...., n} and two vertices a, b are adjacent if they are distinct and a2 ≡ b2 (mod n). In this paper we investigate some structural properties of square congruence graph and we obtain the relationship between clique number, chromatic number and maximum degree of square congruence graph. Also we study square congruence graph with p vertices or 2p vertices for any prime number p.

ON THE DOMINATION NUMBER OF A GRAPH AND ITS SQUARE GRAPH

  • Murugan, E.;Joseph, J. Paulraj
    • Korean Journal of Mathematics
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    • v.30 no.2
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    • pp.391-402
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    • 2022
  • For a given graph G = (V, E), a dominating set is a subset V' of the vertex set V so that each vertex in V \ V' is adjacent to a vertex in V'. The minimum cardinality of a dominating set of G is called the domination number of G and is denoted by γ(G). For an integer k ≥ 1, the k-th power Gk of a graph G with V (Gk) = V (G) for which uv ∈ E(Gk) if and only if 1 ≤ dG(u, v) ≤ k. Note that G2 is the square graph of a graph G. In this paper, we obtain some tight bounds for the sum of the domination numbers of a graph and its square graph in terms of the order, order and size, and maximum degree of the graph G. Also, we characterize such extremal graphs.

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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THE PROBABILISTIC METHOD MEETS GO

  • Farr, Graham
    • Journal of the Korean Mathematical Society
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    • v.54 no.4
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    • pp.1121-1148
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    • 2017
  • Go is an ancient game of great complexity and has a huge following in East Asia. It is also very rich mathematically, and can be played on any graph, although it is usually played on a square lattice. As with any game, one of the most fundamental problems is to determine the number of legal positions, or the probability that a random position is legal. A random Go position is generated using a model previously studied by the author, with each vertex being independently Black, White or Uncoloured with probabilities q, q, 1 - 2q respectively. In this paper we consider the probability of legality for two scenarios. Firstly, for an $N{\times}N$ square lattice graph, we show that, with $q=cN^{-{\alpha}}$ and c and ${\alpha}$ constant, as $N{\rightarrow}{\infty}$ the limiting probability of legality is 0, exp($-2c^5$), and 1 according as ${\alpha}$ < 2/5, ${\alpha}=2/5$ and ${\alpha}$ > 2/5 respectively. On the way, we investigate the behaviour of the number of captured chains (or chromons). Secondly, for a random graph on n vertices with edge probability p generated according to the classical $Gilbert-Erd{\ddot{o}}s-R{\acute{e}}nyi$ model ${\mathcal{G}}$(n; p), we classify the main situations according to their asymptotic almost sure legality or illegality. Our results draw on a variety of probabilistic and enumerative methods including linearity of expectation, second moment method, factorial moments, polyomino enumeration, giant components in random graphs, and typicality of random structures. We conclude with suggestions for further work.

The study on the Straightness forms of STD-11 Die-hole in wire-cut Electronic Discharge Machining Conditions (와이어 컷 방전가공 조건에 따른 STD-11 Die-hole 가공시 진직정밀도 고찰)

  • 조규재;김선진
    • Proceedings of the Korean Society of Precision Engineering Conference
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    • 2000.05a
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    • pp.739-742
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    • 2000
  • From the experimental study of W-EDM for alloyed steel, the characteristics such as Hand Drum Form and surface roughness have been observed and evaluated for various conditions. In square hole, the increase of If as to made condition the calculate high value of surface roughness. Also compare dimensionless square hole with circle hole' graph. In circle hole, if a value of surface roughness IP 6 in a side of circle it show a 0.4${\mu}{\textrm}{m}$ and in IP 8, 0.6${\mu}{\textrm}{m}$, in IP 10, 0.7${\mu}{\textrm}{m}$, in IP 12. 0.8${\mu}{\textrm}{m}$ higher than before. This figure show the surface roughness is higher than before, because a table move in either X-axis or Y-axis in square hole, on the contrary, in circle there table move in X-axis and Y-axis at the same time. hand drum form getting small when wire tension increase 1000gf to 1500gf, at the same working conditions. the smaller of off time, the mailer of hand drum form in same condition and same wire tension. but if you compare square hole with circle hole' graph hand drum form displayed in maintained term of working condision, on the contrary, in case of square hole variation of hand drum form is more increase than a grow of IP

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Hand drum forms of STD-11 Die-hole in Wire-cut Electronic discharge Machining Conditions (STD-11 Die-hole 와이어 컷 방전가공시 가공조건에 따른 북현상 고찰)

  • 조규재
    • Proceedings of the Korean Society of Machine Tool Engineers Conference
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    • 2000.04a
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    • pp.567-572
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    • 2000
  • From the experimental study of W-EDM for alloyed steel, the characteristics such as Hand Drum Form and surface roughness have been observed and evaluated for various conditions. In square hole, the increase of IP as to made condition, the calculate high value of surface roughness. Also compare dimensionless square hole with circle hole' graph, In circle hole, if a value of surface roughness IP 6 in a side of circle it show a 0.4${\mu}{\textrm}{m}$ and in IP 8, 0.6${\mu}{\textrm}{m}$, in IP 10, 0.7${\mu}{\textrm}{m}$, in IP 12, 0.8${\mu}{\textrm}{m}$ higher than before. This figure show the surface roughness is higher than before, because a table move in either X-axis or Y-axis in square hole, on the contrary, in circle there table move in X-axis and Y-axis at the same time. hand drum form getting small when wire tension increase 1000gf to 1500gf, at the same working conditions. The smaller of off time, the maller of hand drum form in same condition and same wire tension. but if you compare square hole with circle hole' graph, hand drum form displayed in maintained term of working condision, on the contrary, in case of square hole variation of hand drum form is more increase than a grow of IP

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ON THE MINIMUM ORDER OF 4-LAZY COPS-WIN GRAPHS

  • Sim, Kai An;Tan, Ta Sheng;Wong, Kok Bin
    • Bulletin of the Korean Mathematical Society
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    • v.55 no.6
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    • pp.1667-1690
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    • 2018
  • We consider the minimum order of a graph G with a given lazy cop number $c_L(G)$. Sullivan, Townsend and Werzanski [7] showed that the minimum order of a connected graph with lazy cop number 3 is 9 and $k_3{\square}k_3$ is the unique graph on nine vertices which requires three lazy cops. They conjectured that for a graph G on n vertices with ${\Delta}(G){\geq}n-k^2$, $c_L(G){\leq}k$. We proved that the conjecture is true for k = 4. Furthermore, we showed that the Petersen graph is the unique connected graph G on 10 vertices with ${\Delta}(G){\leq}3$ having lazy cop number 3 and the minimum order of a connected graph with lazy cop number 4 is 16.

ON DOMINATION NUMBERS OF GRAPH BUNDLES

  • Zmazek Blaz;Zerovnik Janez
    • Journal of applied mathematics & informatics
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    • v.22 no.1_2
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    • pp.39-48
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    • 2006
  • Let ${\gamma}$(G) be the domination number of a graph G. It is shown that for any $k {\ge} 0$ there exists a Cartesian graph bundle $B{\Box}_{\varphi}F$ such that ${\gamma}(B{\Box}_{\varphi}F) ={\gamma}(B){\gamma}(F)-2k$. The domination numbers of Cartesian bundles of two cycles are determined exactly when the fibre graph is a triangle or a square. A statement similar to Vizing's conjecture on strong graph bundles is shown not to be true by proving the inequality ${\gamma}(B{\bigotimes}_{\varphi}F){\le}{\gamma}(B){\gamma}(F)$ for strong graph bundles. Examples of graphs Band F with ${\gamma}(B{\bigotimes}_{\varphi}F) < {\gamma}(B){\gamma}(F)$ are given.

Edge Property of 2n-square Meshes as a Base Graphs of Pyramid Interconnection Networks (피라미드 상호연결망의 기반 그래프로서의 2n-정방형 메쉬 그래프의 간선 특성)

  • Chang, Jung-Hwan
    • The Journal of the Korea Contents Association
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    • v.9 no.12
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    • pp.582-591
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    • 2009
  • The pyramid graph is an interconnection network topology based on regular square mesh and tree structures. In this paper, we adopt a strategy of classification into two disjoint groups of edges in regular square mesh as a base sub-graph constituting of each layer in the pyramid graph. Edge set in the mesh can be divided into two disjoint sub-sets called as NPC(represents candidate edge for neighbor-parent) and SPC(represents candidate edge for shared-parent) whether the parents vertices adjacent to two end vertices of the corresponding edge have a relation of neighbor or shared in the upper layer of pyramid graph. In addition, we also introduce a notion of shrink graph to focus only on the NPC-edges by hiding SPC-edges in the original graph within the shrunk super-vertex on the resulting graph. In this paper, we analyze that the lower and upper bound on the number of NPC-edges in a Hamiltonian cycle constructed on $2^n\times2^n$ mesh is $2^{2n-2}$ and $3*(2^{2n-2}-2^{n-1})$ respectively. By expanding this result into the pyramid graph, we also prove that the maximum number of NPC-edges containable in a Hamiltonian cycle is $4^{n-1}-3*2^{n-1}$-2n+7 in the n-dimensional pyramid.

Cycle Extendability of Torus Sub-Graphs in the Enhanced Pyramid Network (개선된 피라미드 네트워크에서 토러스 부그래프의 사이클 확장성)

  • Chang, Jung-Hwan
    • Journal of Korea Multimedia Society
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    • v.13 no.8
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    • pp.1183-1193
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    • 2010
  • The pyramid graph is well known in parallel processing as a interconnection network topology based on regular square mesh and tree architectures. The enhanced pyramid graph is an alternative architecture by exchanging mesh into the corresponding torus on the base for upgrading performance than the pyramid. In this paper, we adopt a strategy of classification into two disjoint groups of edges in regular square torus as a basic sub-graph constituting of each layer in the enhanced pyramid graph. Edge set in the torus graph is considered as two disjoint sub-sets called NPC(represents candidate edge for neighbor-parent) and SPC(represents candidate edge for shared-parent) whether the parents vertices adjacent to two end vertices of the corresponding edge have a relation of neighbor or sharing in the upper layer of the enhanced pyramid graph. In addition, we also introduce a notion of shrink graph to focus only on the NPC-edges by hiding SPC-edges within the shrunk super-vertex on the resulting shrink graph. In this paper, we analyze that the lower and upper bounds on the number of NPC-edges in a Hamiltonian cycle constructed on $2^n{\times}2^n$ torus is $2^{2n-2}$ and $3{\cdot}2^{2n-2}$ respectively. By expanding this result into the enhanced pyramid graph, we also prove that the maximum number of NPC-edges containable in a Hamiltonian cycle is $4^{n-1}$-2n+1 in the n-dimensional enhanced pyramid.