• Bulletin of the Korean Mathematical Society
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• v.51 no.4
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• pp.1145-1153
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• 2014

Let G be a finite group and X be a union of conjugacy classes of G. Define C(G,X) to be the graph with vertex set X and $x,y{\in}X$ ($x{\neq}y$) joined by an edge whenever they commute. In the case that X = G, this graph is named commuting graph of G, denoted by ${\Delta}(G)$. The aim of this paper is to study the automorphism group of the commuting graph. It is proved that Aut(${\Delta}(G)$) is abelian if and only if ${\mid}G{\mid}{\leq}2$; ${\mid}Aut({\Delta}(G)){\mid}$ is of prime power if and only if ${\mid}G{\mid}{\leq}2$, and ${\mid}Aut({\Delta}(G)){\mid}$ is square-free if and only if ${\mid}G{\mid}{\leq}3$. Some new graphs that are useful in studying the automorphism group of ${\Delta}(G)$ are presented and their main properties are investigated.

• Communications of the Korean Mathematical Society
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• v.27 no.1
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• pp.57-68
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• 2012

The commuting graph of an arbitrary ring R, denoted by ${\Gamma}(R)$, is a graph whose vertices are all non-central elements of R, and two distinct vertices a and b are adjacent if and only if ab = ba. In this paper, we investigate the connectivity, the diameter, the maximum degree and the minimum degree of the commuting graph of group ring $Z_nQ_8$. The main result is that $\Gamma(Z_nQ_8)$ is connected if and only if n is not a prime. If $\Gamma(Z_nQ_8)$ is connected, then diam($Z_nQ_8$)= 3, while $\Gamma(Z_nQ_8)$ is disconnected then every connected component of $\Gamma(Z_nQ_8)$ must be a complete graph with a same size. Further, we obtain the degree of every vertex in $\Gamma(Z_nQ_8)$, the maximum degree and the minimum degree of $\Gamma(Z_nQ_8)$.

• Journal of KIISE:Computer Systems and Theory
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• v.29 no.9
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• pp.493-502
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• 2002

The connectivity and edge-connectivity have been the prime deterministic measure of fault tolerance in multicomputer networks. These parameters have a problem that they do not differentiate the different types of disconnected graphs which result from removing the disconnecting vertices or disconnecting edges. To compensate for this shortcoming, one can utilize generalized measures of connectedness such as superconnectivity, toughness, scattering number, vertex-integrity, binding number, and restricted connectivity. In this paper, we analyze such deterministic measures of fault tolerance in recursive circulants and hypercubes, and compare them in terms of fault tolerance.

• The Southeast Asian review
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• v.20 no.3
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• pp.41-64
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• 2010

The recent global financial crisis triggered by the sub-prime mortgage debacle in the United States hit hard most ASEAN countries that have just recovered from the unprecedented economic crisis ten years ago. This paper, using individual time-series and panel data from 1990 to 2009, intends to investigate and compare the impacts of the two aforementioned economic crises on trade in the four developing ASEAN countries that encompass Indonesia, Malaysia, the Philippines and Thailand. In doing so, the paper traces the behaviors of main macroeconomic variables before and after the crises on graphs, and then estimates classical export and import demand functions that include real exchange rate, home and foreign GDPs as explanatory variables. In the estimation functions, two dummy variables are added to consider the effects of the two economic crises separately. Individual country data analyses reveal that by and large the 1997 economic crisis seems hit those ASEAN countries' exports and imports harder than the recent global financial crisis. Surprisingly the recent financial crisis turns out more or less statistically insignificant for those countries' export and import performances. The fixed effect model estimation using panel data of those four ASEAN countries also shows that the 1997 economic crisis had affected exports and imports of those countries negatively while the recent global financial crisis was not statistically significant. These results indicate that overall the effect from the 1997 crisis was more devastating than that of the recent global crisis for those ASEAN countries.

• The Journal of the Petrological Society of Korea
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• v.26 no.3
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• pp.297-309
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• 2017

Jurassic granite from Geochang was analysed with respect to the characteristics of the rock cleavage. The comprehensive evaluation for rock cleavages was performed through the combination of the 16 parameters derived from the enlarged photomicrographs of the thin section and the spacing-cumulative frequency diagrams. The results of analysis for the representative values of these spacing parameters with respect to the rock cleavage are summarized as follows. First, the above parameters can be classified into group I (spacing frequency (N), total spacing ($1m{\geq}$), constant (a), exponent (${\lambda}$), slope of exponential straight line (${\theta}$), length of line (oa') and trigonometric ratios ($sin{\theta}$, $tan{\theta}$) and group II (mean spacing (Sm), difference value between mean spacing and median spacing (Sm-Sme), density (${\rho}$), lengths of lines (oa and aa'), area of a right-angled triangle (${\Delta}oaa^{\prime}$) and trigonometric ratio($cos{\theta}$). The values of the 8 parameters belonging to group I show an order of H(hardway, H1+H2)

• Kyungpook Mathematical Journal
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• v.54 no.1
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• pp.95-101
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• 2014

An n-tuple ($a_1,a_2,{\ldots},a_n$) is symmetric, if $a_k$ = $a_{n-k+1}$, $1{\leq}k{\leq}n$. Let $H_n$ = {$(a_1,a_2,{\ldots},a_n)$ ; $a_k$ ${\in}$ {+,-}, $a_k$ = $a_{n-k+1}$, $1{\leq}k{\leq}n$} be the set of all symmetric n-tuples. A symmetric n-sigraph (symmetric n-marked graph) is an ordered pair $S_n$ = (G,${\sigma}$) ($S_n$ = (G,${\mu}$)), where G = (V,E) is a graph called the underlying graph of $S_n$ and ${\sigma}$:E ${\rightarrow}H_n({\mu}:V{\rightarrow}H_n)$ is a function. The restricted super line graph of index r of a graph G, denoted by $\mathcal{R}\mathcal{L}_r$(G). The vertices of $\mathcal{R}\mathcal{L}_r$(G) are the r-subsets of E(G) and two vertices P = ${p_1,p_2,{\ldots},p_r}$ and Q = ${q_1,q_2,{\ldots},q_r}$ are adjacent if there exists exactly one pair of edges, say $p_i$ and $q_j$, where $1{\leq}i$, $j{\leq}r$, that are adjacent edges in G. Analogously, one can define the restricted super line symmetric n-sigraph of index r of a symmetric n-sigraph $S_n$ = (G,${\sigma}$) as a symmetric n-sigraph $\mathcal{R}\mathcal{L}_r$($S_n$) = ($\mathcal{R}\mathcal{L}_r(G)$, ${\sigma}$'), where $\mathcal{R}\mathcal{L}_r(G)$ is the underlying graph of $\mathcal{R}\mathcal{L}_r(S_n)$, where for any edge PQ in $\mathcal{R}\mathcal{L}_r(S_n)$, ${\sigma}^{\prime}(PQ)$=${\sigma}(P){\sigma}(Q)$. It is shown that for any symmetric n-sigraph $S_n$, its $\mathcal{R}\mathcal{L}_r(S_n)$ is i-balanced and we offer a structural characterization of super line symmetric n-sigraphs of index r. Further, we characterize symmetric n-sigraphs $S_n$ for which $\mathcal{R}\mathcal{L}_r(S_n)$~$\mathcal{L}_r(S_n)$ and $$\mathcal{R}\mathcal{L}_r(S_n){\sim_=}\mathcal{L}_r(S_n)$$, where ~ and $$\sim_=$$ denotes switching equivalence and isomorphism and $\mathcal{R}\mathcal{L}_r(S_n)$ and $\mathcal{L}_r(S_n)$ are denotes the restricted super line symmetric n-sigraph of index r and super line symmetric n-sigraph of index r of $S_n$ respectively.