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
University of Isfahan
Transactions on Combinatorics
2251-8657
2251-8665
7
3
2018
09
01
Sufficient conditions for triangle-free graphs to be super-$λ'$
29
36
EN
Huiwen
Cheng
Department of Mathematics, Zhongguancun Institute
15275013@qq.com
Yan-Jing
Li
Department of Mathematics, Beijing Jiaotong University, China
05118322@bjtu.edu.cn
10.22108/toc.2018.106623.1523
An edge-cut $F$ of a connected graph $G$ is called a restricted edge-cut if $G-F$ contains no isolated vertices. The minimum cardinality of all restricted edge-cuts is called the restricted edge-connectivity $λ'(G)$ of $G$. A graph $G$ is said to be $λ'$-optimal if $λ'(G)=xi(G)$, where $xi(G)$ is the minimum edge-degree of $G$. A graph is said to be super-$λ'$ if every minimum restricted edge-cut isolates an edge. In this paper, first, we provide a short proof of a previous theorem about the sufficient condition for $λ'$-optimality in triangle-free graphs, which was given in [J. Yuan and A. Liu, Sufficient conditions for $λ_k$-optimality in triangle-free graphs, Discrete Math., 310 (2010) 981--987]. Second, we generalize a known result about the sufficient condition for triangle-free graphs being super-$λ'$ which was given by Shang et al. in [L. Shang and H. P. Zhang, Sufficient conditions for graphs to be $λ'$-optimal and super-$λ'$, Network}, 309 (2009) 3336--3345].
Triangle-free,restricted edge-cut,super-$ld'$
http://toc.ui.ac.ir/article_22415.html
http://toc.ui.ac.ir/article_22415_297dcf89662c9125339a728e85ad39e3.pdf
University of Isfahan
Transactions on Combinatorics
2251-8657
2251-8665
7
3
2018
09
01
$mathcal{B}$-Partitions, determinant and permanent of graphs
37
54
EN
Ranveer
Singh
Department of Mathematics, Indian Institute of Technology Jodhpur, Jodhpur, India
pg201283008@iitj.ac.in
Ravindra
B.
Bapat
Stat-Math Unit, ISI Delhi
rbb@isid.ac.in
10.22108/toc.2017.105288.1508
Let $G$ be a graph (directed or undirected) having $k$ number of blocks $B_1, B_2,hdots,B_k$. A $mathcal{B}$-partition of $G$ is a partition consists of $k$ vertex-disjoint subgraph $(hat{B_1},hat{B_1},hdots,hat{B_k})$ such that $hat{B}_i$ is an induced subgraph of $B_i$ for $i=1,2,hdots,k.$ The terms $prod_{i=1}^{k}det(hat{B}_i), prod_{i=1}^{k}text{per}(hat{B}_i)$ represent the det-summands and the per-summands, respectively, corresponding to the $mathcal{B}$-partition $(hat{B_1},hat{B_1},hdots,hat{B_k})$. The determinant (permanent) of a graph having no loops on its cut-vertices is equal to the summation of the det-summands (per-summands), corresponding to all possible $mathcal{B}$-partitions. In this paper, we calculate the determinant and the permanent of classes of graphs such as block graph, block graph with negatives cliques, signed unicyclic graph, mixed complete graph, negative mixed complete graph, and star mixed block graphs.
$mathcal{B}$-partition,signed graph,mixed block graph
http://toc.ui.ac.ir/article_22426.html
http://toc.ui.ac.ir/article_22426_684cba1ff8383118f056e8041a6e743a.pdf
University of Isfahan
Transactions on Combinatorics
2251-8657
2251-8665
7
3
2018
09
01
Iota energy of weighted digraphs
55
73
EN
Sumaira
Hafeez
School of Natural Sciences, National university of sciences and Technology Islamabad, Pakistan
sumaira.hafeez123@gmail.com
Mehtab
Khan
Department of mathematics, school of Natural Sciences, National University of Sciences and Technology Islamabad, Pakistan
mehtabkhan85@gmail.com
10.22108/toc.2018.109248.1546
The eigenvalues of a digraph are the eigenvalues of its adjacency matrix. The iota energy of a digraph is recently defined as the sum of absolute values of imaginary part of its eigenvalues. In this paper, we extend the concept of iota energy of digraphs to weighted digraphs. We compute the iota energy formulae for the positive and negative weight directed cycles. We also characterize the unicyclic weighted digraphs with cycle weight $ r in [-1, 1]backslash {0}$ having minimum and maximum iota energy. We obtain well known McClelland upper bound for the iota energy of weighted digraphs. Finally, we find the class of noncospectral equienergetic weighted digraphs.
Weighted digraphs,Extremal energy,Equienergetic weighted digraphs
http://toc.ui.ac.ir/article_22707.html
http://toc.ui.ac.ir/article_22707_8eb3ec61aa18a4a5a2445c45716a4a23.pdf