2018
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The annihilator graph of a 0distributive lattice
2
2
In this article, for a lattice $mathcal L$, we define and investigate the annihilator graph $mathfrak {ag} (mathcal L)$ of $mathcal L$ which contains the zerodivisor graph of $mathcal L$ as a subgraph. Also, for a 0distributive lattice $mathcal L$, we study some properties of this graph such as regularity, connectedness, the diameter, the girth and its domination number. Moreover, for a distributive lattice $mathcal L$ with $Z(mathcal L)neqlbrace 0rbrace$, we show that $mathfrak {ag} (mathcal L) = Gamma(mathcal L)$ if and only if $mathcal L$ has exactly two minimal prime ideals. Among other things, we consider the annihilator graph $mathfrak {ag} (mathcal L)$ of the lattice $mathcal L=(mathcal D(n),)$ containing all positive divisors of a nonprime natural number $n$ and we compute some invariants such as the domination number, the clique number and the chromatic number of this graph. Also, for this lattice we investigate some special cases in which $mathfrak {ag} (mathcal D(n))$ or $Gamma(mathcal D(n))$ are planar, Eulerian or Hamiltonian.
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Saeid
Bagheri
Department of Mathematics, Faculty of Mathematical Sciences, Malayer University, Malayer, Iran
Department of Mathematics, Faculty of Mathematical
Iran
bagheri_saeid@yahoo.com


Mahtab
Koohi Kerahroodi
Department of Mathematics, Faculty of Mathematical Sciences, Malayer University, Malayer, Iran.
Department of Mathematics, Faculty of Mathematical
Iran
mahtabkh3@gmail.com
Distributive lattice
Annihilator graph
Zerodivisor graph
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A spectral excess theorem for digraphs with normal Laplacian matrices
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2
The spectral excess theorem, due to Fiol and Garriga in 1997, is an important result, because it gives a good characterization of distanceregularity in graphs. Up to now, some authors have given some variations of this theorem. Motivated by this, we give the corresponding result by using the Laplacian spectrum for digraphs. We also illustrate this Laplacian spectral excess theorem for digraphs with few Laplacian eigenvalues and we show that any strongly connected and regular digraph that has normal Laplacian matrix with three distinct eigenvalues, is distanceregular. Hence such a digraph is strongly regular with girth $g=2$ or $g=3$.
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19
27


Fateme
Shafiei
Isfahan University of Technology
Isfahan University of Technology
Iran
fatemeh.shafiei66@gmail.com
A Laplacian spectral excess theorem
Distanceregular digraphs
Strongly regular digraphs
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