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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 consider three similar optimization problems: the fault-tolerant metric dimension problem, the local metric dimension problem and the strong metric dimension problem. These problems have applications in many diverse areas, including network discovery and verification, robot navigation and chemistry, etc. We give integer linear programming formulations of the fault-tolerant metric dimension problem and the local metric dimension problem. Also, we study local metric dimension and strong metric dimension of two convex polytopes $S_n$ and $U_n$., Comment: 19 pages, 2 figures

A set $W$ of vertices of $G$ is said to be a weak total resolving set for $G$ if $W$ is a resolving set for $G$ as well as for each $w\in W$, there is at least one element in $W-\{w\}$ that resolves $w$ and $v$ for every $v\in V(G)- W$. Weak total metric dimension of $G$ is the smallest order of a weak total resolving set for $G$. This paper includes the investigation of weak total metric dimension of trees. Also, weak total resolving number of a graph as well as randomly weak total $k$-dimensional graphs are defined and studied in this paper. Moreover, some characterizations and realizations regarding weak total resolving number and weak total metric dimension are given.

An atom $a$ of a molecular graph $G$ uniquely determines (resolves) a pair $(a_1,a_2)$ of atoms of $G$ if the distance between $a$ and $a_1$ is different from the distance between $a$ and $a_2$. In this paper, we quantify the involvement of each atom $a$ of $G$ in uniquely determining (resolving) a pair $(a_1,a_2)$ of atoms of $G$, which is called the resolving share of $a$ for the pair $(a_1,a_2)$. Using this quantity, we define a distance-based topological index of a molecular graph, which reflects the topology of that molecular graph according to the resolvability behavior of each of its atom, and is called the resolving topological index. Then we compute the resolving topological index of several molecular graphs., Comment: 16 pages, 2 figures