University of IsfahanTransactions on Combinatorics2251-86577420181201On the minimum stopping sets of product codes162251910.22108/toc.2017.101199.1465ENMortezaHivadiDepartment of mathematics, Institute for Advanced Studies in Basic Science,AkbarZare ChavoshiMalek ashtar university of technologyJournal Article20161221It is shown that the certain combinatorial structures called stopping sets have the important role in analysis of iterative decoding. In this paper, the number of minimum stopping sets of a product code is determined by the number of the minimum stopping sets of the corresponding component codes. As an example, the number of minimum stopping sets of the r-dimensional SPC product code is computed.https://toc.ui.ac.ir/article_22519_fbd0739ed66110173ab585dee89b4301.pdfUniversity of IsfahanTransactions on Combinatorics2251-86577420181201A note on $1$-factorizability of quartic supersolvable Cayley graphs7102270610.22108/toc.2018.104578.1500ENMiladAhanjidehDepartment of Mathematics, Tarbiat Modares University, Tehran, IranAliIranmaneshDepartment of Mathematics, Tarbiat Modares University, P. O. Box 14115-137, TehranJournal Article20170604Alspach et al. conjectured that every quartic Cayley graph on an even solvable group is $1$-factorizable. In this paper, we verify this conjecture for quartic Cayley graphs on supersolvable groups of even order.https://toc.ui.ac.ir/article_22706_8c2e42a9efd832e83505aa05a6b49a2c.pdfUniversity of IsfahanTransactions on Combinatorics2251-86577420181201Degree resistance distance of trees with some given parameters11242287610.22108/toc.2018.108656.1538ENFangguoHeCollege of Mathematics and Physics, Huanggang Normal University, Huanggang, ChinaXinnongJiangCollege of Life Science and Techonolgy, Huazhong University of Science and Technology, WuhanJournal Article20171218The degree resistance distance of a graph $G$ is defined as $D_R(G)=\sum_{i<j}(d(v_i)+d(v_j))R(v_i,v_j)$, where $d(v_i)$ is the degree of the vertex $v_i$, and $R(v_i,v_j)$ is the resistance distance between the vertices $v_i$ and $v_j$. Here we characterize the extremal graphs with respect to degree resistance distance among trees with given diameter, number of pendent vertices, independence number, covering number, and maximum degree, respectively.https://toc.ui.ac.ir/article_22876_656cf5d3e08fe0f15a55536167cbc556.pdfUniversity of IsfahanTransactions on Combinatorics2251-86577420181201Refinements of the Bell and Stirling numbers25422285910.22108/toc.2018.110171.1560ENTanayWakhareUniversity of Maryland0000-0003-2397-4185Journal Article20180315We introduce new refinements of the Bell, factorial, and unsigned Stirling numbers of the first and second kind that unite the derangement, involution, associated factorial, associated Bell, incomplete Stirling, restricted factorial, restricted Bell, and $r$-derangement numbers (and probably more!). By combining methods from analytic combinatorics, umbral calculus, and probability theory, we derive several recurrence relations and closed form expressions for these numbers. By specializing our results to the classical case, we recover explicit formulae for the Bell and Stirling numbers as sums over compositions.https://toc.ui.ac.ir/article_22859_cc19446462e7b703dcc4e65ff0d76cf2.pdfUniversity of IsfahanTransactions on Combinatorics2251-86577420181201Directed zero-divisor graph and skew power series rings43572300910.22108/toc.2018.109048.1543ENEbrahimHashemiDepartment of Mathematics, Shahrood University of Technology, Shahrood, IranMarziehYazdanfarDepartment of Mathematics, Shahrood University of Technology, Shahrood, IranAbdollahAlhevazDepartment of Mathematics, Shahrood University of Technology, Shahrood, Iran0000-0001-6167-607XJournal Article20180110Let $R$ be an associative ring with identity and $Z^{\ast}(R)$ be its set of non-zero zero-divisors. Zero-divisor graphs of rings are well represented in the literature of commutative and non-commutative rings. The directed zero-divisor graph of $R$, denoted by $\Gamma{(R)}$, is the directed graph whose vertices are the set of non-zero zero-divisors of $R$ and for distinct non-zero zero-divisors $x,y$, $x\rightarrow y$ is an directed edge if and only if $xy=0$. In this paper, we connect some graph-theoretic concepts with algebraic notions, and investigate the interplay between the ring-theoretical properties of a skew power series ring $R[[x;\alpha]]$ and the graph-theoretical properties of its directed zero-divisor graph $\Gamma(R[[x;\alpha]])$. In doing so, we give a characterization of the possible diameters of $\Gamma(R[[x;\alpha]])$ in terms of the diameter of $\Gamma(R)$, when the base ring $R$ is reversible and right Noetherian with an $\alpha$-condition, namely $\alpha$-compatible property. We also provide many examples for showing the necessity of our assumptions.https://toc.ui.ac.ir/article_23009_36558b6b0e7b2173c5ece6b5b1978c2e.pdf