The annihilator graph of a 0-distributive lattice
Saeid
Bagheri
Department of Mathematics, Faculty of Mathematical Sciences, Malayer University, Malayer, Iran
author
Mahtab
Koohi Kerahroodi
Department of Mathematics, Faculty of Mathematical Sciences, Malayer University, Malayer, Iran.
author
text
article
2018
eng
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 $Z(\mathcal L)\neq\lbrace 0\rbrace$, 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.
Transactions on Combinatorics
University of Isfahan
2251-8657
7
v.
3
no.
2018
1
18
http://toc.ui.ac.ir/article_22285_719ab505eba5ec2cd4bf741957e5ce29.pdf
dx.doi.org/10.22108/toc.2017.104919.1507
A spectral excess theorem for digraphs with normal Laplacian matrices
Fateme
Shafiei
Isfahan University of Technology
author
text
article
2018
eng
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 $g=3$.
Transactions on Combinatorics
University of Isfahan
2251-8657
7
v.
3
no.
2018
19
27
http://toc.ui.ac.ir/article_22346_7909f36dcf76d9cf3fc6a7dff2c384b9.pdf
dx.doi.org/10.22108/toc.2018.105873.1513