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In this paper, we use theory of rough set to study graphs using the concept of orbits. We investigate the indiscernibility partitions and approximations of graphs induced by orbits of graphs. We also study rough membership functions, essential sets, discernibility matrix and their relationships for graphs., Comment: 15 pages, 2 figures, 1 table

The fractional versions of graph theoretic-invariants multiply the range of applications in scheduling, assignment and operational research problems. In this paper, we introduce the fractional version of local metric dimension of graphs. The local resolving neighborhood $L(xy)$ of an edge $xy$ of a graph $G$ is the set of those vertices in $G$ which resolve the vertices $x$ and $y$. A function $f:V(G)\rightarrow[0, 1]$ is a local resolving function of $G$ if $f(L(xy))\geq1$ for all edges $xy$ in $G$. The minimum value of $f(V(G))$ among all local resolving functions $f$ of $G$ is the fractional local metric dimension of $G$. We study the properties and bounds of fractional local metric dimension of graphs and give some characterization results. We determine the fractional local metric dimension of strong and cartesian product of graphs., Comment: 16 pages, 0 figures

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

A locating-dominating set of a graph $G$ is a dominating set of $G$ such that every vertex of $G$ outside the dominating set is uniquely identified by its neighborhood within the dominating set. The location-domination number of $G$ is the minimum cardinality of a locating-dominating set in $G$. Let $G_{1}$ and $G_{2}$ be the disjoint copies of a graph $G$ and $f:V(G_{1})\rightarrow V(G_{2})$ be a function. A functigraph $F^f_{G}$ consists of the vertex set $V(G_{1})\cup V(G_{2})$ and the edge set $E(G_{1})\cup E(G_{2})\cup \{uv:v=f(u)\}$. In this paper, we study the variation of the location-domination number in passing from $G$ to $F^f_{G}$ and find its sharp lower and upper bounds. We also study the location-domination number of functigraphs of the complete graphs for all possible definitions of the function $f$. We also obtain the location-domination number of functigraphs of a family of spanning subgraph of the complete graphs., Comment: 14 pages, 3 figures

The fixing number of a graph $G$ is the smallest cardinality of a set of vertices $F\subseteq V(G)$ such that only the trivial automorphism of $G$ fixes every vertex in $F$. Let $\Pi$ $=$ $\{F_1,F_2,\ldots,F_k\}$ be an ordered $k$-partition of $V(G)$. Then $\Pi$ is called a {\it fixatic partition} if for all $i$; $1\leq i\leq k$, $F_i$ is a fixing set for $G$. The cardinality of a largest fixatic partition is called the {\it fixatic number} of $G$. In this paper, we study the fixatic numbers of graphs. Sharp bounds for the fixatic number of graphs in general and exact values with specified conditions are given. Some realizable results are also given in this paper., Comment: 10 pages, 1figure

An automorphism of a graph describes its structural symmetry and the concept of fixing number of a graph is used for breaking its symmetries (except the trivial one). In this paper, we evaluate automorphisms of the co-normal product graph $G_1\ast G_2$ of two simple graphs $G_1$ and $G_2$ and give sharp bounds on the order of its automorphism group. We study the fixing number of $G_1\ast G_2$ and prove sharp bounds on it. Moreover, we compute the fixing number of the co-normal product of some families of graphs., Comment: 15 pages

In this paper, we investigate the problem of covering the vertices of a graph associated to a finite vector space as introduced by Das \cite{Das}, such that we can uniquely identify any vertex by examining the vertices that cover it. We use locating-dominating sets and identifying codes, which are closely related concepts for this purpose. These sets consist of a dominating set of graph such that every vertex is uniquely identified by its neighborhood within the dominating sets. We find the location-domination number and the identifying number of the graph and study the exchange property for locating-dominating sets and identifying codes., Comment: 12 pages, 1 figure

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 $E(G_{1})\cup E(G_{2})\cup \{uv:g(u)=v\}$. In this paper, we extend the study of the distinguishing number of a graph to its functigraph. We discuss the behavior of the distinguishing number in passing from $G$ to $F_{G}$ and find its sharp lower and upper bounds. We also discuss the distinguishing number of functigraphs of complete graphs and join graphs., Comment: 10 pages, 1 figure. arXiv admin note: text overlap with arXiv:1611.03346

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