University of IsfahanTransactions on Combinatorics2251-86577220180601Combinatorial parameters on bargraphs of permutations1162224310.22108/toc.2017.102359.1483ENToufikMansourDepartment of Mathematics, University of Tennessee, Knoxville, TN, USAMarkShattuckMathematics Department, University of Tennessee, Knoxville, TN, USAJournal Article20170209In 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.https://toc.ui.ac.ir/article_22243_ee9a92039072d73f603a278c71ef4387.pdfUniversity of IsfahanTransactions on Combinatorics2251-86577220180601The log-convexity of the fubini numbers17232183510.22108/toc.2017.104212.1496ENQingZouThe University of IowaJournal Article20170519Let $f_n$ 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 $\{\sqrt[n]{f_n}\}_{n\ge 1}$ by using the aforementioned bounds.https://toc.ui.ac.ir/article_21835_8b52d6cf1daabf7e0e9be379112846e3.pdfUniversity of IsfahanTransactions on Combinatorics2251-86577220180601Solution to the minimum harmonic index of graphs with given minimum degree25332227210.22108/toc.2017.101076.1462ENMeiliLiangGuangdong University of Foreign StudiesBoChengGuangdong University of Foreign StudiesJianxiLiuGuangdong University of Foreign StudiesJournal Article20161215The harmonic index of a graph $G$ is defined as $ H(G)=\sum\limits_{uv\in 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 $k (1\le k\le n/2)$ 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 $k (n/2 \le k\le n-1)$ give a complete solution of the problem.https://toc.ui.ac.ir/article_22272_28d4f6f37d2867d952c1398e234888f8.pdfUniversity of IsfahanTransactions on Combinatorics2251-86577220180601On matrix and lattice ideals of digraphs35462232010.22108/toc.2017.105701.1510ENHamidDamadiDepartment of Mathematics, Amirkabir University of Technology (Tehran Polytechnic) Tehran, Iran.FarhadRahmatiAmirkabir University of TechnologyJournal Article20170802Let $\textit{G}$ 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.https://toc.ui.ac.ir/article_22320_b7155094bae6e4bfec0b32c67a2295ec.pdfUniversity of IsfahanTransactions on Combinatorics2251-86577220180601Reduced zero-divisor graphs of posets47542231110.22108/toc.2018.55164.1417ENDeiborlangNongsiangNorth Eastern Hill UniversityPromode KumarSaikiaNorth Eastern Hill UniversityJournal Article20160614This 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.https://toc.ui.ac.ir/article_22311_893dce7cc938e8e23dd5defcadb2c102.pdf