For $n \ge 2t+1$ where $t \ge 1$, the circulant graph $C_n (1, 2, \dots , t)$ consists of the vertices $v_0, v_1, v_2, \dots , v_{n-1}$ and the edges $v_i v_{i+1}$, $v_i v_{i+2}, \dots , v_i v_{i + t}$, where $i = 0, 1, 2, \dots , n-1$, and the subscripts are taken modulo $n$. We prove that the metric dimension ${\rm dim} (C_n (1, 2, \dots , t)) \ge \left\lceil \frac{2t}{3} \right\rceil + 1$ for $t \ge 5$, where the equality holds if and only if $t = 5$ and $n = 13$. Thus ${\rm dim} (C_n (1, 2, \dots , t)) \ge \left\lceil \frac{2t}{3} \right\rceil + 2$ for $t \ge 6$. This bound is sharp for every $t \ge 6$.

Topological indices are graph invariants most suitable for underlined structures of chemical compounds. Most of the topological indices are defined on the well-known graph concepts such as degree of a vertex, distances, eccentricity of a vertex etc. In this paper, new type of degree of a vertex is defined with the aid of Resolving property of the graph as the minimum cardinality of a Resolving set containing that vertex. The mathematical properties of this newly defined degree is established with the help of standard graphs and an attempt to analyze its applicability in chemical compounds are carried by taking silicate structures.

The unification process of Rough sets with Graphs is implemented in phenomenal applications in all the fields of Engineering.  With the rapid and exponential increase in the worldwide web, it is necessary to organize the data. The major part of the data like google links, the social networks can be represented in graphs. But in the case of uncertainty, the concepts of classical graph theory cannot handle complex networks. For Resolving these issues in 2006 Tong He introduced the concepts of Rough Graphs. In this paper, we have introduced metric dimensions in Rough graphs along with their Mathematical Properties.

A subset W of the vertices of a graph G is a Resolving set for G when for each pair of distinct vertices u; v 2 V (G) there exists w 2 W such that d(u; w) ̸ = d(v; w). The cardinality of a minimum Resolving set for G is the metric dimension of G. This concept has applications in many diverse areas including network discovery, robot navigation, image processing, combinatorial search and optimization. The problem of  nding metric dimension is NP-complete for general graphs but the metric dimension of trees can be obtained using a polynomial time algorithm. In this paper, we investigate the metric dimension of Cayley graphs on dihedral groups and we characterize a family of them.

‎For an ordered set $W=\{w_1,w_2,\ldots,w_k\}$ of vertices and a vertex $v$ in a connected graph $G$, the ordered $k$-vector $r(v|W):=(d(v,w_1),d(v,w_2),\ldots,d(v,w_k))$ is  called  the (metric) representation of $v$ with respect to $W$, where $d(x,y)$ is the distance between the vertices $x$ and $y$. The set $W$ is called  a Resolving set for $G$ if distinct vertices of $G$ have distinct representations with respect to $W$. The minimum cardinality of a Resolving set for $G$ is its metric dimension. In this paper, we investigate the metric dimension of the lexicographic product  of graphs $G$ and $H$, $G[H]$, for some known graphs.

‎For an ordered set $W=\{w_1‎, ‎w_2,\ldots,w_k\}$ of vertices and a‎ vertex $v$ in a connected graph $G$‎, ‎the ordered  $k$-vector‎ ‎$r(v|W)=(d(v,w_1),d(v,w_2),\ldots,d(v,w_k))$ is called the‎ ‎(metric) representation of $v$ with respect to $W$‎, ‎where $d(x,y)$‎ ‎is the distance between the vertices $x$ and $y$‎. ‎The set $W$ is‎ ‎called a Resolving set for $G$ if distinct vertices of $G$ have‎ ‎distinct representations with respect to $W$‎. ‎The minimum‎ ‎cardinality of a Resolving set for $G$ is its metric dimension‎, ‎and a Resolving set of minimum cardinality is a basis of $G$‎. ‎Lower bounds for metric dimension are important‎. ‎In this paper‎, ‎we investigate lower bounds for metric dimension‎. ‎Motivated by a lower bound for the metric dimension $k$ of a graph‎ ‎of order $n$ with diameter $d$ in [S‎. ‎Khuller‎, ‎B‎. ‎Raghavachari‎, ‎and‎ ‎A‎. ‎Rosenfeld‎, ‎Landmarks in graphs‎, ‎Discrete Applied Mathematics‎ ‎$70(3) (1996) 217-229$]‎, ‎which states that $k \geq n-d^k$‎, ‎we characterize‎ all graphs‎ ‎with this lower bound and obtain a new lower bound‎. ‎This new bound is better than the previous one‎, ‎for graphs with diameter more than $3$‎.

In this paper we present some properties of set-norm exhaustive set multi functions and also of atoms and pseudo-atoms of set multi functions taking values in the family of non-empty subsets of a commutative semi group with unity.