On families of convex polytopes with constant metric dimension

AbstractLet G be a connected graph and d(x,y) be the distance between the vertices x and y. A subset of vertices W={w1,w2,…,wk} is called a resolving set for G if for every two distinct vertices x,y∈V(G), there is a vertex wi∈W such that d(x,wi)≠d(y,wi). 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

Game chromatic number of Cartesian and corona product graphs

The game chromatic number $\chi_g$ is investigated for Cartesian product $G\square H$ and corona product $G\circ H$ of two graphs $G$ and $H$. The exact values for the game chromatic number of Cartesian product graph of $S_{3}\square S_{n}$ is found, where $S_n$ is a star graph of order $n+1$. This extends previous results of Bartnicki et al. [1] and Sia [5] on the game chromatic number of Cartesian product graphs. Let $P_m$ be the path graph on $m$ vertices and $C_n$ be the cycle graph on $n$ vertices. We have determined the exact values for the game chromatic number of corona product graphs $P_{m}\circ K_{1}$ and $P_{m}\circ C_{n}$

On the metric dimension of rotationally-symmetric convex polytopes

Metric dimension is a~generalization of affine dimension to arbitrary metric spaces (provided a resolving set exists). Let $\mathcal{F}$ be a family of connected graphs $G_{n}$ : $\mathcal{F} = (G_{n})_{n}\geq 1$ depending on $n$ as follows: the order $|V(G)| = \varphi(n)$ and $\lim\limits_{n\rightarrow \infty}\varphi(n)=\infty$. If there exists a constant C > 0 such that $dim(G_{n}) \leq C$ for every $n \geq 1$ then we shall say that $\mathcal{F}$ has bounded metric dimension, otherwise $\mathcal{F}$ has unbounded metric dimension. If all graphs in $\mathcal{F}$ have the same metric dimension, then $\mathcal{F}$ is called a family of graphs with constant metric dimension.\\ In this paper, we study the metric dimension of some classes of convex polytopes which are rotationally-symmetric. It is shown that these classes of convex polytoes have the constant metric dimension and only three vertices chosen appropriately suffice to resolve all the vertices of these classes of convex polytopes. It is natural to ask for the characterization of classes of convex polytopes with constant metric dimension

Optimal Graphs in the Enhanced Mesh Networks

The degree diameter problem explores the biggest graph (in terms of number of nodes) subject to some restrictions on the valency and the diameter of the graph. The restriction on the valency of the graph does not impose any condition on the number of edges (apart from taking the graph simple), so the resulting graph may be thought of as being embedded in the complete graph. In a generality of the said problem, the graph is taken to be embedded in any connected host graph. In this article, host graph is considered as the enhanced mesh network constructed from the grid network. This article provides some exact values for the said problem and also gives some bounds for the optimal graphs

The K-Size Edge Metric Dimension of Graphs

In this paper, a new concept k-size edge resolving set for a connected graph G in the context of resolvability of graphs is defined. Some properties and realizable results on k-size edge resolvability of graphs are studied. The existence of this new parameter in different graphs is investigated, and the k-size edge metric dimension of path, cycle, and complete bipartite graph is computed. It is shown that these families have unbounded k-size edge metric dimension. Furthermore, the k-size edge metric dimension of the graphs Pm □ Pn, Pm □ Cn for m, n ≥ 3 and the generalized Petersen graph is determined. It is shown that these families of graphs have constant k-size edge metric dimension

On Molecular Topological Properties of Dendrimers

Topological indices are numerical parameters of a graph which characterize its topology and are usually graph invariant. In QSAR/QSPR study, physico-chemical properties and topological indices such as Randi\'{c}, atom-bond connectivity $(ABC)$ and geometric-arithmetic $(GA)$ index are used to predict the bioactivity of different chemical compounds. Graph theory has found a considerable use in this area of research. In this paper, we study the degree based molecular topological indices like $ABC_4$ and $GA_5$ for certain families of dendrimers. We derive the analytical closed formulae for these classes of dendrimers.The accepted manuscript in pdf format is listed with the files at the bottom of this page. The presentation of the authors' names and (or) special characters in the title of the manuscript may differ slightly between what is listed on this page and what is listed in the pdf file of the accepted manuscript; that in the pdf file of the accepted manuscript is what was submitted by the author

Resolvability in Subdivision of Circulant Networks Cn1,k

Circulant networks form a very important and widely explored class of graphs due to their interesting and wide-range applications in networking, facility location problems, and their symmetric properties. A resolving set is a subset of vertices of a connected graph such that each vertex of the graph is determined uniquely by its distances to that set. A resolving set of the graph that has the minimum cardinality is called the basis of the graph, and the number of elements in the basis is called the metric dimension of the graph. In this paper, the metric dimension is computed for the graph Gn1,k constructed from the circulant graph Cn1,k by subdividing its edges. We have shown that, for k=2, Gn1,k has an unbounded metric dimension, and for k=3 and 4, Gn1,k has a bounded metric dimension