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In this paper, we discuss automorphism related parameters of a graph associated to a finite vector space. The fixing neighborhood of a pair $(u,v)$ of vertices of a graph $G$ is the set of all those vertices $w$ of $G$, such that the orbits of $u$ and $v$ under the action of stabilizer of $w$ are not equal. The fixed number of a graph is the minimum number $k$ such that every subset of vertices of $G$ of cardinality $k$ is a fixing set of $G$. We study some properties of automorphisms of a graph associated to finite vector space and find the fixing neighborhood of pair of vertices of the graph. We also find the fixed number of the graph. It is shown that, for every positive integer $N$, there exists a graph $G$ with $fxd(G)-fix(G)\geq N$, where $fxd(G)$ is the fixed number and $fix(G)$ is the fixing number of $G$., Comment: 12 pages

The fixing number of a graph $G$ is the order of the smallest subset $S$ of its vertex set $V(G)$ such that stabilizer of $S$ in $G$, $\Gamma_{S}(G)$ is trivial. Let $G_{1}$ and $G_{2}$ be disjoint copies of a graph $G$, and let $g:V(G_{1})\rightarrow V(G_{2})$ be a function. A functigraph $F_{G}$ consists of the vertex set $V(G_{1})\cup V(G_{2})$ and the edge set $E(G_{1})\cup E(G_{2})\cup \{uv:v=g(u)\}$. In this paper, we study the behavior of the fixing number in passing from $G$ to $F_{G}$ and find its sharp lower and upper bounds. We also study the fixing number of functigraphs of some well known families of graphs like complete graphs, trees and join graphs., Comment: 10 pages, 1figure

An automorphism group of a graph $G$ is the set of all permutations of the vertex set of $G$ that preserve adjacency and non adjacency of vertices in a graph. A fixing set of a graph $G$ is a subset of vertices of $G$ such that only the trivial automorphism fixes every vertex in $S$. Minimum cardinality of a fixing set of $G$ is called the fixing number of $G$. In this article, we define a fractional version of the fixing number of a graph. We formulate the problem of finding the fixing number of a graph as an integer programming problem. It is shown that a relaxation of this problem leads to a linear programming problem and hence to a fractional version of the fixing number of a graph. We also characterize the graphs $G$ with the fractional fixing number $\frac{|V(G)|}{2}$ and the fractional fixing number of some families of graphs is also obtained., Comment: 19 Pages

In this paper, we deduce some properties of $f$-sets of connected graphs. Also, we introduce the concept of fixing share of each vertex of a fixing set $D$ to see the participation of each vertex when fixing a connected graph $G$. We define a parameter, called the fixing percentage, by using the concept of fixing share, which is helpful in determining the measure of the amount of fixing done by the elements of $D$ in $G$., Comment: 7 pages, 1 figure