Shahrood University of TechnologyJournal of Algebraic Systems2345-51288220210101THE COST NUMBER AND THE DETERMINING NUMBER OF A GRAPH209217194810.22044/jas.2020.8343.1408ENS. AlikhaniDepartment of Mathematics, Yazd University, 89195-741, Yazd, Iran.0000-0002-1801-203XS. SoltaniDepartment of Mathematics, Yazd University, 89195-741, Yazd, IranJournal Article20190424The distinguishing number $D(G)$ of a graph $G$ is the least integer $d$ such that $G$ has an vertex labeling with $d$ labels that is preserved only by a trivial automorphism. The minimum size of a label class in such a labeling of $G$ with $D(G) = d$ is called the cost of $d$-distinguishing $G$ and is denoted by $\rho_d(G)$. A set of vertices $S\subseteq V(G)$ is a determining set for $G$ if every automorphism of $G$ is uniquely determined by its action on $S$. The determining number of $G$, ${\rm Det}(G)$, is the minimum cardinality of determining sets of $G$. In this paper we compute the cost and the determining number for the friendship graphs and corona product of two graphs.https://jas.shahroodut.ac.ir/article_1948_8ed554351047fe89af7fd04bb8a07ed1.pdf