### فهرست مطالب • Volume:8 Issue:1, 2019
• تاریخ انتشار: 1397/12/10
• تعداد عناوین: 5
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• Hasan Kharazi, Alireza Mosleh Tehrani * Pages 1-14
‎Let \$ 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.
Keywords: ‎Defensive alliance, Alliances in graphs, Edge cut
• Majid Arezoomand *, Alireza Abdollahi, Pablo Spiga Pages 15-40
Fixed-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.
Keywords: ‎‎Derangements‎, ‎Polycirculant Conjecture‎, ‎Transitive group
• Ebrahim Vatandoost *, Fatemeh Ramezani, Saeid Alikhani Pages 41-50
In 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.
Keywords: ‎Zero forcing number‎, ‎generalized Sierpi&#039, {n}ski graph‎, ‎Sierpi&#039, {n}ski graph‎, ‎path covering
• Hamidreza Maimani *, Zeinab Koushki Pages 51-59
A 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(Gsetminus 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(Gvee H)\$ and exact values of \$b(P_ntimes P_2)\$, and generalized corona product of graphs.
Keywords: bondage number, double domination, double bondage number
• Mostafa Momeni *, Ali Zaeembashi Pages 61-68
Let \$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 \$sgeq 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\$‎.
Keywords: ‎‎dominating function‎, ‎global dominating function‎, ‎\$s\$-dominating function‎, ‎\$gamma-\$function‎, ‎\$gamma, s-\$function