• Journal of the Korean Mathematical Society
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• v.52 no.6
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• pp.1323-1336
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• 2015

Let S be a semigroup with 0 and R be a ring with 1. We extend the definition of the zero-divisor graphs of commutative semigroups to not necessarily commutative semigroups. We define an annihilating-ideal graph of a ring as a special type of zero-divisor graph of a semigroup. We introduce two ways to define the zero-divisor graphs of semigroups. The first definition gives a directed graph ${\Gamma}$(S), and the other definition yields an undirected graph ${\overline{\Gamma}}$(S). It is shown that ${\Gamma}$(S) is not necessarily connected, but ${\overline{\Gamma}}$(S) is always connected and diam$({\overline{\Gamma}}(S)){\leq}3$. For a ring R define a directed graph ${\mathbb{APOG}}(R)$ to be equal to ${\Gamma}({\mathbb{IPO}}(R))$, where ${\mathbb{IPO}}(R)$ is a semigroup consisting of all products of two one-sided ideals of R, and define an undirected graph ${\overline{\mathbb{APOG}}}(R)$ to be equal to ${\overline{\Gamma}}({\mathbb{IPO}}(R))$. We show that R is an Artinian (resp., Noetherian) ring if and only if ${\mathbb{APOG}}(R)$ has DCC (resp., ACC) on some special subset of its vertices. Also, it is shown that ${\overline{\mathbb{APOG}}}(R)$ is a complete graph if and only if either $(D(R))^2=0,R$ is a direct product of two division rings, or R is a local ring with maximal ideal m such that ${\mathbb{IPO}}(R)=\{0,m,m^2,R\}$. Finally, we investigate the diameter and the girth of square matrix rings over commutative rings $M_{n{\times}n}(R)$ where $n{\geq} 2$.

• 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.

• Journal of the Korean Mathematical Society
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• v.55 no.5
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• pp.1091-1101
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• 2018

We concern in this paper the graph Kazdan-Warner equation $${\Delta}f=g-he^f$$ on an infinite graph, the prototype of which comes from the smooth Kazdan-Warner equation on an open manifold. Different from the variational methods often used in the finite graph case, we use a heat flow method to study the graph Kazdan-Warner equation. We prove the existence of a solution to the graph Kazdan-Warner equation under the assumption that $h{\leq}0$ and some other integrability conditions or constrictions about the underlying infinite graphs.

• Honam Mathematical Journal
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• v.40 no.1
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• pp.75-82
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• 2018

Let P be a commutator poset with Z(P) its set of zero-divisors. The annihilator graph of P, denoted by AG(P), is the (undirected) graph with all elements of $Z(P){\setminus}\{0\}$ as vertices, and distinct vertices x, y are adjacent if and only if $ann(xy)\;{\neq}\;(x)\;{\cup}\;ann(y)$. In this paper, we study basic properties of AG(P).

• Bulletin of the Korean Mathematical Society
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• v.41 no.1
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• pp.125-135
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• 2004

One of the intriguing and fundamental algorithmic graph problems is the computation of the strongly connected components of a directed graph G. In this paper we first introduce a simple procedure for determining the location of the nonzero elements occurring in $B^{-1}$ without fully inverting B, where EB\;{\equiv}\;(b_{ij)\;and\;B^T$ are diagonally dominant matrices with $b_{ii}\;>\;0$ for all i and $b_{ij}\;{\leq}\;0$, for $i\;{\neq}\;j$, and then, as an application, show that all of the strongly connected components of a directed graph G can be recognized by the location of the nonzero elements occurring in the matrix $C(G)\;=\;(D\;-\;A(G))^{-1}$. Here A(G) is an adjacency matrix of G and D is an arbitrary scalar matrix such that (D - A(G)) becomes a diagonally dominant matrix.

• Journal of the Korean Ceramic Society
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• v.53 no.5
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• pp.521-528
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• 2016

Kinetic Monte Carlo (KMC) and graph searches show that proton conduction limiting barriers and trajectories in $BaZr_{0.875}Y_{0.125}O_3$ are affected by the presence of other protons. At 1000 K, KMC limiting conduction barriers increase from 0.39 eV to 0.45 eV as the proton number is increased. The proton-proton radial distribution begins to rise at $2{\AA}$ and peaks at $4{\AA}$, which is half the distance expected, based on the proton concentration. Density functional theory (DFT) calculations find proton/proton distances of 2.60 and $2.16{\AA}$ in the lowest energy two-proton configurations. A simple average of the limiting barriers for 7-10 step periodic long range paths found via graph theory at 1100 K shows an increase in activation barrier from 0.32 eV to 0.37 eV when a proton is added. Both KMC and graph theory show that protons can affect each other's pathways and raise the overall conduction barriers.

• Journal of applied mathematics & informatics
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• v.11 no.1_2
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• pp.109-123
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• 2003

A tree is called starlike if it has exactly one vertex of degree greate. than two. In [4] it was proved that two starlike trees G and H are cospectral if and only if they are isomorphic. We prove here that there exist no two non-isomorphic Laplacian cospectral starlike trees. Further, let G be a simple graph of order n with vertex set V(G) : {1,2, …, n} and let H = {$H_1$, $H_2$, …, $H_{n}$} be a family of rooted graphs. According to [2], the rooted product G(H) is the graph obtained by identifying the root of $H_{i}$ with the i-th vertex of G. In particular, if H is the family of the paths $P_k_1,P_k_2,...P_k_2$ with the rooted vertices of degree one, in this paper the corresponding graph G(H) is called the sunlike graph and is denoted by G($k_1,k_2,...k_n$). For any $(x_1,x_2,...,x_n)\;\in\;{I_*}^n$, where $I_{*}$ = : {0,1}, let G$(x_1,x_2,...,x_n)$ be the subgraph of G which is obtained by deleting the vertices $i_1,i_2,...i_j\;\in\;V(G)\;(O\leq j\leq n)$, provided that $x_i_1=x_i_2=...=x_i_j=o.\;Let \;G[x_1,x_2,...x_n]$ be characteristic polynomial of G$(x_1,x_2,...,x_n)$, understanding that G[0,0,...,0] $\equiv$1. We prove that $G[k_1,k_2,...,k_n]-\sum_{x\in In}[{\prod_{\imath=1}}^n\;P_k_i+x_i-2(\lambda)](-1)...G[x_1,x_2,...,X_n]$ where x=($x_1,x_2,...,x_n$);G[$k_1,k_2,...,k_n$] and $P_n(\lambda)$ denote the characteristic polynomial of G($k_1,k_2,...,k_n$) and $P_n$, respectively. Besides, if G is a graph with $\lambda_1(G)\;\geq1$ we show that $\lambda_1(G)\;\leq\;\lambda_1(G(k_1,k_2,...,k_n))<\lambda_1(G)_{\lambda_1}^{-1}(G}$ for all positive integers $k_1,k_2,...,k_n$, where $\lambda_1$ denotes the largest eigenvalue.

• Proceedings of the Korean Institute of Information and Commucation Sciences Conference
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• 2016.05a
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• pp.819-822
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• 2016

This paper proposes an algorithm recognizing finger using a distance graph of a detected finger's contour in a depth image. The distance graph shows angles and Euclidean distances between the center of palm and the hand contour as x and y axis respectively. We can obtain hand gestures from the graph using the fact that the graph has local maximum at the positions of finger tips. After we find the center of mass of the wrist using the fingers is thinner than the palm, we make its angle the orienting angle $0^{\circ}$. The simulation results show that the proposed algorithm can detect hand gestures well regardless of the hand direction.

• Bulletin of the Korean Mathematical Society
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• v.53 no.4
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• pp.1017-1031
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• 2016

Let R be a commutative ring with the non-zero identity and n be a natural number. ${\Gamma}^n_R$ is a simple graph with $R^n{\setminus}\{0\}$ as the vertex set and two distinct vertices X and Y in $R^n$ are adjacent if and only if there exists an $n{\times}n$ lower triangular matrix A over R whose entries on the main diagonal are non-zero such that $AX^t=Y^t$ or $AY^t=X^t$, where, for a matrix B, $B^t$ is the matrix transpose of B. ${\Gamma}^n_R$ is a generalization of Cayley graph. Let $T_n(R)$ denote the $n{\times}n$ upper triangular matrix ring over R. In this paper, for an arbitrary ring R, we investigate the properties of the graph ${\Gamma}^n_{T_n(R)}$.

• Journal of Korean Elementary Science Education
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• v.31 no.3
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• pp.321-333
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• 2012

The purpose of this study was to examine elementary school students' characteristics and difficulties in drawing and interpreting a line graph, and to present educational implications. Twenty five students(4th grader: 6, 5th grader: 9, and 6th grader: 10) at an elementary school participated in this study. We used a student's task which was about graphing on a given data table and interpreting his/her graph. The data table was on heating 200mL and 500mL of water and measuring their temperature at regular time intervals. We collected multiple source of data, and data analyzed based on the sub-variables of TOGS. The some results of this study are as follows: First, five children (20.0%), especially two of 10 sixth graders (20.0%), could not construct a line graph about a given data table. Second, twenty students (80.0%) had the ability on 'Scaling axes' and on 'Assigning variables to the axes', however, only a student understood why the time is on the longitudinal axis and the temperature is on the vertical axis. Third, in the case of 'Plotting points', twelve children (48.0%) could drew two graphs on a coordinate. Fourth, in the case of 'Selecting the corresponding value for Y (or X)', twenty student had little difficulty. on 'Describing the relationship between variables', seventeen students (68.0%) understood the relationship between time and temperature of water, and the relationship between temperature and amount of water. Finally, eleven students (44%) had the ability on 'Interrelating and extrapolation graphs.' Educational implications are also presented in this paper.