• Bulletin of the Korean Mathematical Society
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• v.59 no.5
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• pp.1305-1315
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• 2022

Let u be a function on a connected finite graph G = (V, E). We consider the mean field equation (1) $-{\Delta}u={\rho}\({\frac{he^u}{\int_Vhe^ud{\mu}}}-{\frac{1}{{\mid}V{\mid}}}\),$ where ∆ is 𝜇-Laplacian on the graph, 𝜌 ∈ ℝ\{0}, h : V → ℝ⁺ is a function satisfying min_x∈V h(x) > 0. Following Sun and Wang [15], we use the method of Brouwer degree to prove the existence of solutions to the mean field equation (1). Firstly, we prove the compactness result and conclude that every solution to the equation (1) is uniformly bounded. Then the Brouwer degree can be well defined. Secondly, we calculate the Brouwer degree for the equation (1), say $$d_{{\rho},h}=\{{-1,\;{\rho}>0, \atop 1,\;{\rho}<0.}$$ Consequently, the equation (1) has at least one solution due to the Brouwer degree d_𝜌,h ≠ 0.

• Journal of the Korean Mathematical Society
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• v.51 no.1
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• pp.87-98
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• 2014

Let R be a commutative ring with identity and M an R-module. In this paper, we associate a graph to M, say ${\Gamma}(M)$, such that when M = R, ${\Gamma}(M)$ is exactly the classic zero-divisor graph. Many well-known results by D. F. Anderson and P. S. Livingston, in [5], and by D. F. Anderson and S. B. Mulay, in [6], have been generalized for ${\Gamma}(M)$ in the present article. We show that ${\Gamma}(M)$ is connected with $diam({\Gamma}(M)){\leq}3$. We also show that for a reduced module M with $Z(M)^*{\neq}M{\backslash}\{0\}$, $gr({\Gamma}(M))={\infty}$ if and only if ${\Gamma}(M)$ is a star graph. Furthermore, we show that for a finitely generated semisimple R-module M such that its homogeneous components are simple, $x,y{\in}M{\backslash}\{0\}$ are adjacent if and only if $xR{\cap}yR=(0)$. Among other things, it is also observed that ${\Gamma}(M)={\emptyset}$ if and only if M is uniform, ann(M) is a radical ideal, and $Z(M)^*{\neq}M{\backslash}\{0\}$, if and only if ann(M) is prime and $Z(M)^*{\neq}M{\backslash}\{0\}$.

• The Journal of the Korea Contents Association
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• v.18 no.2
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• pp.438-449
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• 2018

Recently, various graph data have been utilized in various fields such as social networks and citation networks. As the graph changes dynamically over time, it is necessary to manage the graph historical data for tracking changes and retrieving point-in-time graphs. Most historical data changes partially according to time, so unchanged data is stored redundantly when data is stored in units of time. In this paper, we propose a graph history storage management method to minimize the redundant storage of time graphs. The proposed method continuously detects the change of the graph and stores the overlapping subgraph in intersection snapshot. Intersection snapshots are connected by a number of delta snapshots to maintain change data over time. It improves space efficiency by collectively managing overlapping data stored in intersection snapshots. We also linked intersection snapshots and delta snapshots to retrieval the graph at that point in time. Various performance evaluations are performed to show the superiority of the proposed scheme.

• Journal of the Chungcheong Mathematical Society
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• v.19 no.4
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• pp.403-408
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• 2006

A pebbling move on a connected graph G is taking two pebbles off of one vertex and placing one of them on an adjacent vertex. The domination cover pebbling number ${\psi}(G)$ is the minimum number of pebbles required so that any initial configuration of pebbles can be transformed by a sequence of pebbling moves so that the set of vertices that contain pebbles forms a domination set of G. We determine the domination cover pebbling number for fans, fuses, and pseudo-star.

• Journal of the Korean Operations Research and Management Science Society
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• v.19 no.1
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• pp.189-199
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• 1994

This paper considers the covering network design problem (CNDP). In the CNDP, an undirected graph is given where nodes correspond to potential facility sites and arcs to potential links connecting facilities. The objective of the CNDP is to identify the least cost connected subgraph whose nodes cover the given demand points. The problem difines a demand point to be covered if some node in the selected graph is present within an appropriate distance from the demand point. We present an integer programming formulation for the problem and develop a dual-based solution procedure. The computational results for randomly generated test problems are also shown.

• Journal of the Korean Mathematical Society
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• v.60 no.6
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• pp.1171-1213
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• 2023

This work is a continuation of Crisp's work on automorphism groups of CLTTF Artin groups, where the defining graph of a CLTTF Artin group is connected, large-type, and triangle-free. More precisely, we provide an explicit presentation of the automorphism group of an edge-separated CLTTF Artin group whose defining graph has no separating vertices.

• Journal of applied mathematics & informatics
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• v.30 no.5_6
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• pp.793-807
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• 2012

If G is any connected graph of order p; then the thorn graph $G_p^*$ with code ($n_1$, $n_2$, ${\cdots}$, $n_p$) is obtained by adding $n_i$ pendent vertices to each $i^{th}$ vertex of G. By treating the pendent edge of a thorn graph as $P_2$, $K_2$, $K_{1,1}$, $K_1{\circ}K_1$ or $P_1{\circ}K_1$, we generalize a thorn graph by replacing $P_2$ by $P_m$, $K_2$ by $K_m$, $K_{1,1}$ by $K_{m,n}$, $K_1{\circ}K_1$ by $K_m{\circ}K_1$ and $P_1{\circ}K_1$ by $P_m{\circ}K_1$ and their respective generalized thorn graphs are denoted by $G_P$, $G_K$, $G_B$, $G_{KK}$ and $G_{PK}$ respectively. Many chemical compounds can be treated as $G_P$, $G_K$, $G_B$, $G_{KK}$ and $G_{PK}$ of some graphs in graph theory. In this paper, we obtain the bounds of the wiener index for these generalization of thorn graphs.

• Journal of applied mathematics & informatics
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• v.6 no.3
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• pp.727-734
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• 1999

Let G be a connected graph of n vertices. The problem of finding a depth-first spanning tree of G is to find a connected subgraph of G with the n vertices and n-1 edges by depth-first-search. in this paper we propose an O(n) time algorithm to solve this problem on permutation graphs.

• Journal of the Korean Mathematical Society
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• v.47 no.5
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• pp.925-934
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• 2010

A crystallization of a closed connected PL manifold M is a special edge-colored graph representing M via a contracted triangulation. The regular genus of M is the minimum genus of a closed connected surface into which a crystallization of M regularly embeds. We disprove a conjecture on the regular genus of $\mathbb{S}\;{\times}\;\mathbb{S}^n$, $n\;{\geq}\;3$, stated in [J. Korean Math. Soc. 41 (2004), no. 3, p. 420].

• Journal of applied mathematics & informatics
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• v.24 no.1_2
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• pp.357-366
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• 2007

Let G(V, E) be a simple graph, and let f be an integer function on V with $1{\leq}f(v){\leq}d(v)$ to each vertex $v{\in}V$. An f-edge cover-coloring of a graph G is a coloring of edge set E such that each color appears at each vertex $v{\in}V$ at least f(v) times. The f-edge cover chromatic index of G, denoted by ${\chi}'_{fc}(G)$, is the maximum number of colors such that an f-edge cover-coloring of G exists. Any simple graph G has an f-edge cover chromatic index equal to ${\delta}_f\;or\;{\delta}_f-1,\;where\;{\delta}_f{=}^{min}_{v{\in}V}\{\lfloor\frac{d(v)}{f(v)}\rfloor\}$. Let G be a connected and not complete graph with ${\chi}'_{fc}(G)={\delta}_f-1$, if for each $u,\;v{\in}V\;and\;e=uv{\nin}E$, we have ${\chi}'_{fc}(G+e)>{\chi}'_{fc}(G)$, then G is called an f-edge covered critical graph. In this paper, some properties on f-edge covered critical graph are discussed. It is proved that if G is an f-edge covered critical graph, then for each $u,\;v{\in}V\;and\;e=uv{\nin}E$ there exists $w{\in}\{u,v\}\;with\;d(w)\leq{\delta}_f(f(w)+1)-2$ such that w is adjacent to at least $d(w)-{\delta}_f+1$ vertices which are all ${\delta}_f-vertex$ in G.