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For a finite commutative ring $ R $ with identity $ 1\neq 0 $, the zero divisor graph $ \Gamma(R) $ is a simple connected graph having vertex set as the set of nonzero zero divisors of $ R $, where two vertices $ x $ and $ y $ are adjacent if and only if $ xy=0$. We find the signless Laplacian spectrum of the zero divisor graphs $ \Gamma(\mathbb{Z}_{n}) $ for various values of $ n$. Also, we find signless Laplacian spectrum of $ \Gamma(\mathbb{Z}_{n}) $ for $ n=p^z , z\geq 2 $, in terms of signless Laplacian spectrum of its components and zeros of the characteristic polynomial of an auxiliary matrix. Further, we characterise $ n $ for which zero divisor graph $ \Gamma(\mathbb{Z}_{n}) $ are signless Laplacian integral.
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참고문헌 (19)
참고문헌 신청
S. Akbari / 2004 / On zero divisor graphs of a commutative ring / J. Algebra 274:847~855
D. F. Anderson / 1999 / The zero divisor graph of a commutative ring / J. Algebra 217:434~447
M. F. Atiyah / 1969 / Introduction to Commutative Algebra / Addison-Wesley
I. Beck / 1988 / Coloring of a commutative rings / J. Algebra 116:208~226
D. M. Cardoso / 2013 / Spectra of graphs obtained by a generalization of the join of graph operations / Discrete Math 313:733~741
D. F. Anderson / 1999 / The zero divisor graph of a commutative ring / J. Algebra 217:434~447
M. F. Atiyah / 1969 / Introduction to Commutative Algebra / Addison-Wesley
I. Beck / 1988 / Coloring of a commutative rings / J. Algebra 116:208~226
D. M. Cardoso / 2013 / Spectra of graphs obtained by a generalization of the join of graph operations / Discrete Math 313:733~741
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