University of Isfahan
Transactions on Combinatorics
2251-8657
2251-8665
7
3
2018
09
01
The annihilator graph of a 0-distributive lattice
1
18
EN
Saeid
Bagheri
0000-0002-8394-1337
Department of Mathematics, Faculty of Mathematical Sciences, Malayer University, Malayer, Iran
bagheri_saeid@yahoo.com
Mahtab
Koohi Kerahroodi
Department of Mathematics, Faculty of Mathematical Sciences, Malayer University, Malayer, Iran.
mahtabkh3@gmail.com
10.22108/toc.2017.104919.1507
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 zero-divisor graph of $mathcal L$ as a subgraph. Also, for a 0-distributive 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 non-prime 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.
Distributive lattice,Annihilator graph,Zero-divisor graph
http://toc.ui.ac.ir/article_22285.html
http://toc.ui.ac.ir/article_22285_719ab505eba5ec2cd4bf741957e5ce29.pdf
University of Isfahan
Transactions on Combinatorics
2251-8657
2251-8665
7
3
2018
09
01
A spectral excess theorem for digraphs with normal Laplacian matrices
19
28
EN
Fateme
Shafiei
Isfahan University of Technology
fatemeh.shafiei66@gmail.com
10.22108/toc.2018.105873.1513
The spectral excess theorem, due to Fiol and Garriga in 1997, is an important result, because it gives a good characterization of distance-regularity 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 distance-regular. Hence such a digraph is strongly regular with girth $g=2$ or $g=3$.
A Laplacian spectral excess theorem,Distance-regular digraphs,Strongly regular digraphs
http://toc.ui.ac.ir/article_22346.html
http://toc.ui.ac.ir/article_22346_f0401337d3cc116dc87ace2c1fba2dc5.pdf