2018-07-21T15:24:26Z
http://toc.ui.ac.ir/?_action=export&rf=summon&issue=4111
Transactions on Combinatorics
Trans. Comb.
2251-8657
2251-8657
2018
7
3
The annihilator graph of a 0-distributive lattice
Saeid
Bagheri
Mahtab
Koohi Kerahroodi
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
2018
09
01
1
18
http://toc.ui.ac.ir/article_22285_719ab505eba5ec2cd4bf741957e5ce29.pdf
Transactions on Combinatorics
Trans. Comb.
2251-8657
2251-8657
2018
7
3
A spectral excess theorem for digraphs with normal Laplacian matrices
Fateme
Shafiei
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
2018
09
01
19
28
http://toc.ui.ac.ir/article_22346_f0401337d3cc116dc87ace2c1fba2dc5.pdf