Topological indices are numerical parameters of a graph that characterize its topology and are usually graph invariant. In a QSAR/QSPR study, physicochemical properties and topological indices such...

On topological indices of poly oxide, poly silicate, DOX, and DSL networks

Minimal doubly resolving sets and the strong metric dimension of some convex polytopes

We consider the problem of determining the cardinality ψ(H2,k) of minimal
 doubly resolving sets of Hamming graphs H2,k. We prove that for k ≥ 6 every
 minimal resolving set of H2,k is also a doubly resolving set, and,
 consequently, ψ(H2,k) is equal to the metric dimension of H2,k, which is
 known from the literature. Moreover, we find an explicit expression for the
 strong metric dimension of all Hamming graphs Hn,k.

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Minimal doubly resolving sets and the strong metric dimension of Hamming graphs

On the fault-tolerant metric dimension of convex polytopes

Let $$G=(V, E)$$ be a connected graph. The distance between the edge $$e=uv\in E$$ and the vertex $$x\in V$$ is given by $$d(e, x) = \min \{d(u, x), d(v, x)\}$$. A subset $$S_{E}$$ of vertices is called an edge metric generator for G if for every two distinct edges $$e_{1}, e_{2}\in E$$, there exists a vertex $$x\in S_{E}$$ such that $$d(e_{1}, x)\ne d(e_{2}, x)$$. An edge metric generator containing a minimum number of vertices is called an edge metric basis for G and the cardinality of an edge metric basis is called the edge metric dimension denoted by $$\mu _{E}(G)$$. In this paper, we study the edge metric dimension of some classes of plane graphs. It is shown that the edge metric dimension of convex polytope antiprism $$A_{n}$$, the web graph $${\mathbb {W}}_{n}$$, and convex polytope $${\mathbb {D}}_{n}$$ are bounded, while the prism related graph $$D^{*}_{n}$$ has unbounded edge metric dimension.

On the edge metric dimension of convex polytopes and its related graphs

On the metric dimension of circulant graphs

Let G be a connected graph and d(x,y) be the distance between the vertices x and y. A subset of vertices W={w"1,w"2,...,w"k} is called a resolving set for G if for every two distinct vertices x,y@?V(G), there is a vertex w"i@?W such that d(x,w"i) d(y,w"i). A resolving set containing a minimum number of vertices is called a metric basis for G and the number of vertices in a metric basis is its metric dimension dim(G). A family G of connected graphs is a family with constant metric dimension if dim(G) is finite and does not depend upon the choice of G in G. In this paper, we study the metric dimension of some classes of convex polytopes which are obtained by the combinations of two different graph of convex polytopes. It is shown that these classes of convex polytopes have the constant metric dimension and only three vertices chosen appropriately suffice to resolve all the vertices of these classes of convex polytopes.

On families of convex polytopes with constant metric dimension

On classes of regular graphs with constant metric dimension

In graph theory, a graph is given names—generally a whole number—to edges, vertices, or both in a chart. Formally, given a graph G = ( V , E ) , a vertex naming is a capacity from V to an arrangement of marks. A diagram with such a capacity characterized defined is known as a vertex-marked graph. Similarly, an edge naming is a mapping of an element of E to an arrangement of marks. In this case, the diagram is called an edge-marked graph. We consider an edge irregular reflexive k-labeling for the disjoint association of wheel-related diagrams and deduce the correct estimation of the reflexive edge strength for the disjoint association of m copies of some wheel-related graphs, specifically gear graphs and prism graphs.

https://www.mdpi.com/2227-7390/6/9/142/pdf

Edge Irregular Reflexive Labeling for the Disjoint Union of Gear Graphs and Prism Graphs

/pdf/on-molecular-topological-properties-of-dendrimers-q4uf5s6eo8.pdf

Syed Ahtsham Ul Haq Bokhary

Papers

On the metric dimension of circulant graphs

On families of convex polytopes with constant metric dimension

On classes of regular graphs with constant metric dimension

Edge Irregular Reflexive Labeling for the Disjoint Union of Gear Graphs and Prism Graphs

On molecular topological properties of dendrimers