• Kyungpook Mathematical Journal
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• v.64 no.1
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• pp.171-184
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• 2024

In this paper, we show that if G is strongly regular then the Gallai graph Γ(G) and the anti-Gallai graph ∆(G) of G are edge-regular. We also identify conditions under which the Gallai and anti-Gallai graphs are themselves strongly regular, as well as conditions under which they are 2-connected. We include also a number of concrete examples and a discussion of spectral properties of the Gallai and anti-Gallai graphs.

• Bulletin of the Korean Mathematical Society
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• v.54 no.2
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• pp.497-505
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• 2017

The rank over a finite field of the adjacency matrix of a directed strongly regular graph is studied, with some applications to the construction of linear codes. Three techniques are used: code orthogonality, adjacency matrix determinant, and adjacency matrix spectrum.

• Bulletin of the Korean Mathematical Society
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• v.56 no.3
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• pp.729-743
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• 2019

In this paper, we introduce strongly quasi primary ideals which is an intermediate class of primary ideals and quasi primary ideals. Let R be a commutative ring with nonzero identity and Q a proper ideal of R. Then Q is called strongly quasi primary if $ab{\in}Q$ for $a,b{\in}R$ implies either $a^2{\in}Q$ or $b^n{\in}Q$ ($a^n{\in}Q$ or $b^2{\in}Q$) for some $n{\in}{\mathbb{N}}$. We give many properties of strongly quasi primary ideals and investigate the relations between strongly quasi primary ideals and other classical ideals such as primary, 2-prime and quasi primary ideals. Among other results, we give a characterization of divided rings in terms of strongly quasi primary ideals. Also, we construct a subgraph of ideal based zero divisor graph ${\Gamma}_I(R)$ and denote it by ${\Gamma}^*_I(R)$, where I is an ideal of R. We investigate the relations between ${\Gamma}^*_I(R)$ and ${\Gamma}_I(R)$. Further, we use strongly quasi primary ideals and ${\Gamma}^*_I(R)$ to characterize von Neumann regular rings.

• Korean Journal of Mathematics
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• v.20 no.2
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• pp.247-254
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• 2012

The Klee-Quaife problem is finding the minimum order ${\mu}(d,c,v)$ of the $(d,c,v)$ graph, which is a $c$-vertex connected $v$-regular graph with diameter $d$. Many authors contributed finding ${\mu}(d,c,v)$ and they also enumerated and classied the graphs in several cases. This problem is naturally extended to the case of digraphs. So we are interested in the extended Klee-Quaife problem. In this paper, we deal with an equivalent problem, finding the maximum diameter of digraphs with given order, focused on 2-regular case. We show that the maximum diameter of strongly connected 2-regular digraphs with order $n$ is $n-3$, and classify the digraphs which have diameter $n-3$. All 15 nonisomorphic extremal digraphs are listed.

• Bulletin of the Korean Mathematical Society
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• v.53 no.6
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• pp.1629-1643
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• 2016

Let R be a ring with identity, X be the set of all nonzero, nonunits of R and G be the group of all units of R. A ring R is called unit-duo ring if $[x]_{\ell}=[x]_r$ for all $x{\in}X$ where $[x]_{\ell}=\{ux{\mid}u{\in}G\}$ (resp. $[x]_r=\{xu{\mid}u{\in}G\}$) which are equivalence classes on X. It is shown that for a semisimple unit-duo ring R (for example, a strongly regular ring), there exist a finite number of equivalence classes on X if and only if R is artinian. By considering the zero divisor graph (denoted ${\tilde{\Gamma}}(R)$) determined by equivalence classes of zero divisors of a unit-duo ring R, it is shown that for a unit-duo ring R such that ${\tilde{\Gamma}}(R)$ is a finite graph, R is local if and only if diam(${\tilde{\Gamma}}(R)$) = 2.

• Journal of the Korean Mathematical Society
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• v.50 no.6
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• pp.1199-1211
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• 2013

We introduce a new class of graphs, called quasi $m$-Cayley graphs, having good symmetry properties, in the sense that they admit a group of automorphisms G that fixes a vertex of the graph and acts semiregularly on the other vertices. We determine when these graphs are strongly regular, and this leads us to define a new algebro-combinatorial structure, called quasi-partial difference family, or QPDF for short. We give several infinite families and sporadic examples of QPDFs. We also study several properties of QPDFs and determine, under several conditions, the form of the parameters of QPDFs when the group G is cyclic.

• Proceedings of the IEEK Conference
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• 2002.07c
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• pp.1371-1374
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• 2002

This paper compares four network con-figurations for using as the logical topology in multi- hop wavelength division multiplexing (WDM) networks. The regular network configurations studied in this paper axe ShuffleNet, de Bruijn graph, hypercube, and Man-hattan street network. Instead of using the weight mean hop distance of node placement problem for comparing optimum logical topology, we introduce a new objective function that includes h and the network cost. It can be seen that the network cost strongly depends on the logical topology selected for the implementation of the network. The objective of this paper is to find an optimum logical topology for WDM networks that gives low as well as low network cost.