eng
University of Isfahan
Transactions on Combinatorics
2251-8657
2251-8665
2018-06-01
7
2
1
16
10.22108/toc.2017.102359.1483
22243
Combinatorial parameters on bargraphs of permutations
Toufik Mansour
toufik@math.haifa.ac.il
1
Mark Shattuck
mark.shattuck@tdt.edu.vn
2
Department of Mathematics, University of Tennessee, Knoxville, TN, USA
Mathematics Department, University of Tennessee, Knoxville, TN, USA
In this paper, we consider statistics on permutations of length $n$ represented geometrically as bargraphs having the same number of horizontal steps. More precisely, we find the joint distribution of the descent and up step statistics on the bargraph representations, thereby obtaining a new refined count of permutations of a given length. To do so, we consider the distribution of the parameters on permutations of a more general multiset of which $mathcal{S}_n$ is a subset. In addition to finding an explicit formula for the joint distribution on this multiset, we provide counts for the total number of descents and up steps of all its members, supplying both algebraic and combinatorial proofs. Finally, we derive explicit expressions for the sign balance of these statistics, from which the comparable results on permutations follow as special cases.
http://toc.ui.ac.ir/article_22243_ee9a92039072d73f603a278c71ef4387.pdf
combinatorial statistic
$q$-generalization
bargraph
permutations
eng
University of Isfahan
Transactions on Combinatorics
2251-8657
2251-8665
2018-06-01
7
2
17
23
10.22108/toc.2017.104212.1496
21835
The log-convexity of the fubini numbers
Qing Zou
zou-qing@uiowa.edu
1
The University of Iowa
Let $f_n$ denotes the $n$th Fubini number. In this paper, first we give upper and lower bounds for the Fubini numbers $f_n$. Then the log-convexity of the Fubini numbers has been obtained. Furthermore we also give the monotonicity of the sequence ${sqrt[n]{f_n}}_{nge 1}$ by using the aforementioned bounds.
http://toc.ui.ac.ir/article_21835_8b52d6cf1daabf7e0e9be379112846e3.pdf
Fubini number
log-convexity
monotonicity
eng
University of Isfahan
Transactions on Combinatorics
2251-8657
2251-8665
2018-06-01
7
2
25
33
10.22108/toc.2017.101076.1462
22272
Solution to the minimum harmonic index of graphs with given minimum degree
Meili Liang
liangmeili2004@163.com
1
Bo Cheng
bocheng2006@126.com
2
Jianxi Liu
liujianxi2001@gmail.com
3
Guangdong University of Foreign Studies
Guangdong University of Foreign Studies
Guangdong University of Foreign Studies
The harmonic index of a graph $G$ is defined as $ H(G)=sumlimits_{uvin E(G)}frac{2}{d(u)+d(v)}$, where $d(u)$ denotes the degree of a vertex $u$ in $G$. Let $mathcal{G}(n,k)$ be the set of simple $n$-vertex graphs with minimum degree at least $k$. In this work we consider the problem of determining the minimum value of the harmonic index and the corresponding extremal graphs among $mathcal{G}(n,k)$. We solve the problem for each integer $k (1le kle n/2)$ and show the corresponding extremal graph is the complete split graph $K_{k,n-k}^*$. This result together with our previous result which solve the problem for each integer $k (n/2 le kle n-1)$ give a complete solution of the problem.
http://toc.ui.ac.ir/article_22272_28d4f6f37d2867d952c1398e234888f8.pdf
harmonic index
minimum degree
extremal graphs
eng
University of Isfahan
Transactions on Combinatorics
2251-8657
2251-8665
2018-06-01
7
2
35
46
10.22108/toc.2017.105701.1510
22320
On matrix and lattice ideals of digraphs
Hamid Damadi
hamid.damadi@aut.ac.ir
1
Farhad Rahmati
frahmati@aut.ac.ir
2
Department of Mathematics, Amirkabir University of Technology (Tehran Polytechnic) Tehran, Iran.
Amirkabir University of Technology
Let $textit{G}$ be a simple, oriented connected graph with $n$ vertices and $m$ edges. Let $I(textbf{B})$ be the binomial ideal associated to the incidence matrix textbf{B} of the graph $G$. Assume that $I_L$ is the lattice ideal associated to the rows of the matrix $textbf{B}$. Also let $textbf{B}_i$ be a submatrix of $textbf{B}$ after removing the $i$-th row. We introduce a graph theoretical criterion for $G$ which is a sufficient and necessary condition for $I(textbf{B})=I(textbf{B}_i)$ and $I(textbf{B}_i)=I_L$. After that we introduce another graph theoretical criterion for $G$ which is a sufficient and necessary condition for $I(textbf{B})=I_L$. It is shown that the heights of $I(textbf{B})$ and $I(textbf{B}_i)$ are equal to $n-1$ and the dimensions of $I(textbf{B})$ and $I(textbf{B}_i)$ are equal to $m-n+1$; then $I(textbf{B}_i)$ is a complete intersection ideal.
http://toc.ui.ac.ir/article_22320_b7155094bae6e4bfec0b32c67a2295ec.pdf
Directed graph
Binomial ideal
Matrix ideals
eng
University of Isfahan
Transactions on Combinatorics
2251-8657
2251-8665
2018-06-01
7
2
47
54
10.22108/toc.2018.55164.1417
22311
Reduced zero-divisor graphs of posets
Deiborlang Nongsiang
ndeiborlang@yahoo.in
1
Promode Saikia
promode4@gmail.com
2
North Eastern Hill University
North Eastern Hill University
This paper investigates properties of the reduced zero-divisor graph of a poset. We show that a vertex is an annihilator prime ideal if and only if it is adjacent to all other annihilator prime ideals and there are always two annihilator prime ideals which are not adjacent to a non-annihilator prime ideal. We also classify all posets whose reduced zero-divisor graph is planar or toroidal and the number of distinct annihilator prime ideals is four or seven.
http://toc.ui.ac.ir/article_22311_893dce7cc938e8e23dd5defcadb2c102.pdf
poset
reduced zero-divisor graph
annihilator prime ideal