University of IsfahanTransactions on Combinatorics2251-86578120190301On the defensive alliances in graph1142322710.22108/toc.2018.50156.1396ENHasanKharaziDepartment of Mathematics, Faculty of Science, Iran University of Science and Technology, Tehran, Iran.AlirezaMosleh TehraniDepartment of Mathematics, Faculty of Science, Iran University of Science and Technology, Tehran, Iran.Journal Article20160317Let $ G = (V,E) $ be a graph. We say that $ S \subseteq V $ is a defensive alliance if for every $ u \in S $, the number of neighbors $ u $ has in $ S $ plus one (counting $ u $) is at least as large as the number of neighbors it has outside $ S $. Then, for every vertex $ u $ in a defensive alliance $ S $, any attack on a single vertex by the neighbors of $ u $ in $ V-S $ can be thwarted by the neighbors of $ u $ in $ S $ and $ u $ itself. In this paper, we study alliances that are containing a given vertex $ u $ and study their mathematical properties.https://toc.ui.ac.ir/article_23227_49f13d333028b939ed2d096a98282fcf.pdfUniversity of IsfahanTransactions on Combinatorics2251-86578120190301On problems concerning fixed-point-free permutations and on the polycirculant conjecture-a survey15402316610.22108/toc.2018.112665.1585ENMajidArezoomandUniversity of LarestanAlirezaAbdollahiUniversity of IsfahanPabloSpigaDipartimento di Matematica e Applicazioni, University of Milano-Bicocca,Journal Article20180825Fixed-point-free permutations, also known as derangements, have been studied for centuries. In particular, depending on their applications, derangements of prime-power order and of prime order have always played a crucial role in a variety of different branches of mathematics: from number theory to algebraic graph theory. Substantial progress has been made on the study of derangements, many long-standing open problems have been solved, and many new research problems have arisen. The results obtained and the methods developed in this area have also effectively been used to solve other problems regarding finite vertex-transitive graphs. The methods used in this area range from deep group theory, including the classification of the finite simple groups, to combinatorial techniques. This article is devoted to surveying results, open problems and methods in this area.https://toc.ui.ac.ir/article_23166_1e1c1fe183cadcd86904ba2543084f1f.pdfUniversity of IsfahanTransactions on Combinatorics2251-86578120190301On the zero forcing number of generalized Sierpinski graphs41502326510.22108/toc.2018.101107.1463ENEbrahimVatandoostImam Khomeini International UniversityFatemehRamezaniYazd UniversitySaeidAlikhaniYazd UniversityJournal Article20161217In this article we study the Zero forcing number of Generalized Sierpi\'{n}ski graphs $S(G,t)$. More precisely, we obtain a general lower bound on the Zero forcing number of $S(G,t)$ and we show that this bound is tight. In particular, we consider the cases in which the base graph $G$ is a star, path, a cycle or a complete graph.https://toc.ui.ac.ir/article_23265_e768f70fa89d95c17111a4dc08270b06.pdfUniversity of IsfahanTransactions on Combinatorics2251-86578120190301On the double bondage number of graphs products51592316710.22108/toc.2018.114111.1605ENHamidrezaMaimani0000-0001-9020-0871ZeinabKoushkiMathematics, research and science, tehranJournal Article20181119A set $D$ of vertices of graph $G$ is called $double$ $dominating$ $set$ if for any vertex $v$, $|N[v]\cap D|\geq 2$. The minimum cardinality of $double$ $domination$ of $G$ is denoted by $\gamma_d(G)$. The minimum number of edges $E'$ such that $\gamma_d(G\setminus E)>\gamma_d(G)$ is called the double bondage number of $G$ and is denoted by $b_d(G)$. This paper determines that $b_d(G\vee H)$ and exact values of $b(P_n\times P_2)$, and generalized corona product of graphs.https://toc.ui.ac.ir/article_23167_c10b003a8aa01309879f4e72bf73d795.pdfUniversity of IsfahanTransactions on Combinatorics2251-86578120190301A generalization of global dominating function61682355010.22108/toc.2019.110404.1562ENMostafaMomeniDepartment of Mathematics, Shahid Rajaee Teacher Training University, P.O. Box 16785-163, Tehran, IranAliZaeembashiDepartment of math, Shahid Rajaee Teacher Training University, Tehran, IranJournal Article20180411Let $G$ be a graph. A function $f : V (G) \longrightarrow \{0,1\}$, satisfying the condition that every vertex $u$ with $f(u) = 0$ is adjacent with at least one vertex $v$ such that $f(v) = 1$, is called a dominating function $(DF)$. The weight of $f$ is defined as $wet(f)=\Sigma_{v \in V(G)} f(v)$. The minimum weight of a dominating function of $G$ is denoted by $\gamma (G)$, and is called the domination number of $G$. A dominating function $f$ is called a global dominating function $(GDF)$ if $f$ is also a $DF$ of $\overline{G}$. The minimum weight of a global dominating function is denoted by $\gamma_{g}(G)$ and is called global domination number of $G$. In this paper we introduce a generalization of global dominating function. Suppose $G$ is a graph and $s\geq 2$ and $K_n$\ is the complete graph on $V(G)$. A function $ f:V(G)\longrightarrow \{ 0,1\} $ on $G$ is $s$-dominating function $(s-DF)$, if there exists some factorization $\{G_1,\ldots,G_s \}$ of $K_n$, such that $G_1=G$ \ and $f$\ is dominating function of each $G_i$.https://toc.ui.ac.ir/article_23550_0c12856c79206ad93e86a934875ed0e3.pdf