• Honam Mathematical Journal
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• v.41 no.1
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• pp.67-87
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• 2019

The main aim of this study is to characterize new classes of multicone graphs which are determined by their adjacency spectra, their Laplacian spectra, their complement with respect to signless Laplacian spectra and their complement with respect to their adjacency spectra. A multicone graph is defined to be the join of a clique and a regular graph. If n is a positive integer, a friendship graph $F_n$ consists of n edge-disjoint triangles that all of them meet in one vertex. It is proved that any connected graph cospectral to a multicone graph $F_n{\nabla}F_n=K_2{\nabla}nK_2{\nabla}nK_2$ is determined by its adjacency spectra as well as its Laplacian spectra. In addition, we show that if $n{\neq}2$, the complement of these graphs are determined by their adjacency spectra. At the end of the paper, it is proved that multicone graphs $F_n{\nabla}F_n=K_2{\nabla}nK_2{\nabla}nK_2$ are determined by their signless Laplacian spectra and also we prove that any graph cospectral to one of multicone graphs $F_n{\nabla}F_n$ is perfect.

• 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.

• 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.

• 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.