‎A set of vertices $S$ of a graph $G$ is called a fixing set of $G$‎, ‎if only the trivial automorphism of $G$ fixes every vertex in $S$‎. ‎The fixing number of a graph is the smallest cardinality of a fixing‎ ‎set‎. ‎The fixed number of a graph $G$ is the minimum $k$‎, ‎such that ‎every $k$-set of vertices of $G$ is a fixing set of $G$‎. ‎A graph $G$‎ ‎is called a $k$-fixed graph‎, ‎if its fix...

Let k, n be integers with 1 6 k 6 n/2 and select a permutation π ∈ Sn, that is to say a permutation of {1, . . . , n}, at random. What is i(n, k), the probability that π fixes some set of size k? Equivalently, what is the probability that the cycle decomposition of π contains disjoint cycles with lengths summing to k? Somewhat surprisingly, i(n, k) has only recently been at all well understood ...

A private set intersection protocol consists of two parties, a Sender and a Receiver, each with a secret input set. The protocol aims to have the Receiver output an intersection of the two sets while keeping the elements in the sets secret. This thesis thoroughly analyzes four recently published set intersection protocols, where it explains each protocol and checks whether it satisfies its corr...

A knockout (or single-elimination) tournament is a format of competition that very popular in practice (particularly sports, elections and decision making), which has been extensively intensively studied from theoretical point view for more than decade. Particular attention devoted to the Tournament Fixing problem, where, roughly speaking, objective determine whether we can conduct way makes ou...

The fixing number of a graph G is the smallest cardinality of a set of vertices S such that only the trivial automorphism of G fixes every vertex in S. The fixing set of a group Γ is the set of all fixing numbers of finite graphs with automorphism group Γ. Several authors have studied the distinguishing number of a graph, the smallest number of labels needed to label G so that the automorphism ...

The fixing number of a graph G is the minimum cardinality of a set S ⊂ V (G) such that every nonidentity automorphism of G moves at least one member of S, i.e., the automorphism group of the graph obtained from G by fixing every node in S is trivial. We provide a formula for the fixing number of a disconnected graph in terms of the fixing numbers of its components and make some observations abo...

This study set out to investigate the effect of bio-fertilizers containing nitrogen-fixing bacteria and phosphorus on grain yield and yield components of Faba bean. To this end a factorial experiment on the basis of randomized complete block design with four replications was conducted in Behbahan region at South eastern part of Khuzestan Province of Iran. The experimental treatments included Ni...

this study set out to investigate the effect of bio-fertilizers containing nitrogen-fixing bacteria and phosphorus on grain yield and yield components of faba bean. to this end a factorial experiment on the basis of randomized complete block design with four replications was conducted in behbahan region at south eastern part of khuzestan province of iran. the experimental treatments included ni...