Metric Dimension

In this paper, we determine and show the proof of the metric dimension of a wheel graph and the partition dimension of graph F = Ks + Kt. The solution for the metric dimension is divided into four cases. If n = 3 or 6, the metric dimension of wheel graph Wn is 3. If n = 4 or 5, the metric dimension of wheel graph Wn is 2. If n = 7 or 8, the metric dimension of wheel graph Wn+5k is 3 + 2k for k is a whole number. If n = 9, 10, or 11, the metric dimension of wheel graph Wn+5k is 4 + 2k for k is a whole number. The solution for the partition dimension is divided into two cases. If s ≥ t, the partition dimension of graph F is s + 1. If s < t, the partition dimension of graph F is t.

The Hosoya polynomial was introduced by Hosoya in 1988 for a molecular graph as, where is the number of pairs of vertices of laying at distance from each other, to count the number of paths of different lengths in . The most interesting application of the Hosoya polynomial is that almost all distance-based topological indices can be recovered from it. In this article, we give the general closed form of the Hosoya polynomial of times linearly concatenated benzene molecule by partitioning the vertices of different lengths in Harary, and TSZ indices to predict physical, chemical and pharmacological properties of molecules containing .

The idea of metric dimension in graph theory was introduced by P J Slater in [2]. It has been found applications in optimization, navigation, network theory, image processing, pattern recognition etc.Several other authors have studied metric dimension of various standard graphs. In this paper we introduce a real valued function called generalized metric G X × X × X ® R+ d : where X = r(v /W) =
{(d(v,v1),d(v,v2 ),...,d(v,v ) / v V (G))} k Î , denoted d G and is used to study metric dimension of graphs. It has been proved that metric dimension of any connected finite simple graph remains constant if d G numbers of pendant edges are added to the non-basis vertices.

Let G be a connected graph with vertex set V(G) and edge set E(G). The distance
between vertices u and v in G is denoted by d(u, v), which serves as the shortest path
length from u to v. Let 𝑊 = {𝑤1, 𝑤2, … , 𝑤𝑘 } ⊆ 𝑉(𝐺) be an ordered set, and v is a vertex
in G. The representation of v with respect to W is an ordered set 𝑘 − 𝑡𝑢𝑝𝑙𝑒, 𝑟(𝑣|𝑊) =
(𝑑(𝑣, 𝑤1), 𝑑(𝑣, 𝑤2), … , 𝑑(𝑣, 𝑤𝑘 )). The set W is called a resolving set for G if each
vertex in G has a different representation with respect to W. A resolving set containing
minimum cardinality is called a basis for G. The number of vertices in a basis of G is
called metric dimension of G, which is denoted by𝑑𝑖𝑚(𝐺). The 𝑆 ⊆ 𝑉(𝐺) is a
complement resolving set of G if there are two vertices𝑢, 𝑣 ∈ 𝑉(𝐺) ∖ 𝑆, such
that𝑟(𝑢|𝑆) = 𝑟(𝑣|𝑆). A complement basis of G is the complement resolving set
containing maximum cardinality. The number of vertices in a complement basis of G is
called complement metric dimension of G, which is denoted by 𝑑̅̅𝑖̅𝑚̅̅(𝐺). In this paper,
we examined complement metric dimension of particular graphs and their
characteristics. Furthermore, we determined complement metric dimension of corona
and comb products graphs

Metric dimension in graph theory has many applications in the real world. It has been applied to the optimization problems in complex networks, analyzing electrical networks; show the business relations, robotics, control of production processes etc. This paper studies the metric dimension of graphs with respect to contraction and its bijection between them. Also an algorithm to avoid the overlapping between the robots in a network is introduced.

In this paper we consider the NP-hard problem of determining the metric dimension of graphs. We propose a genetic algorithm (GA) that uses the binary encoding and the standard genetic operators adapted to the problem. The feasibility is enforced by repairing the individuals. The overall performance of the GA implementation is improved by a caching technique. Since the metric dimension problem up to now has been considered only theoretically, standard test instances for this problem do not exist. For that reason, we present the results of the computational experience on several sets of test instances for other NP-hard problems: pseudo boolean, crew scheduling and graph coloring. Testing on instances with up to 1534 nodes shows that GA relatively quickly obtains approximate solutions. For smaller instances, GA solutions are compared with CPLEX results. We have also modified our implementation to handle theoretically challenging large-scale classes of hypercubes and Hamming graphs. In this case the presented approach reaches optimal or best known solutions for hypercubes up to 131072 nodes and Hamming graphs up to 4913 nodes.

A set $W\subseteq V(G)$ is called a resolving set, if for each two distinct vertices $u,v\in V(G)$ there exists $w\in W$ such that $d(u,w)\neq d(v,w)$, where $d(x,y)$ is the distance between the vertices $x$ and $y$. The minimum cardinality of a resolving set for $G$ is called the metric dimension of $G$, and denoted by $\beta(G)$. In this paper, we prove that in a connected graph $G$ of order $n$, $\beta(G)\leq n-\gamma(G)$, where $\gamma(G)$ is the domination number of $G$, and the equality holds if and only if $G$ is a complete graph or a complete bipartite graph $K_{s,t}$, $ s,t\geq 2$. Then, we obtain new bounds for $\beta(G)$ in terms of minimum and maximum degree of $G$.

Let G be a connected graph with vertex set V(G) and edge set E(G). The distance
between vertices u and v in G is denoted by d(u, v), which serves as the shortest path
length from u to v. Let be an ordered set, and v is a
vertex in G. The representation of v with respect to W is an ordered set ,
| . The set W is called a resolving set for G
if each vertex in G has a different representation with respect to W. A resolving set
containing minimum cardinality is called a basis for G. The number of vertices in a
basis of G is called metric dimension of G, which is denoted by . The
is a complement resolving set of G if there are two vertices ,
such that | | . A complement basis of G is the complement resolving set
containing maximum cardinality. The number of vertices in a complement basis of G
is called complement metric dimension of G, which is denoted by ̅̅ ̅ ̅̅ . In this
paper, we examined complement metric dimension of particular graphs and their
characteristics. Furthermore, we determined complement metric dimension of corona
and comb products graphs

In this paper, we have introduced few Interconnection Networks, called David Derived Network DD(n) , Dominating David Derived Network DDD(n), Honeycomb cup Network HCC(n) and Kite Regular Trianguline Mesh KRrTM(n). We have given drawing algorithm for DDD(n) from Honeycomb network HC(n) and embedded poly–Honeycomb Networks, KRrTM(n) in to Dominating David Derived Networks. Also we have investigated the metric dimension of Star of David network SD(n) and lower bound of the metric dimension for DD(n).

Pointwise tangential dimensions are introduced for metric spaces. Under regularity conditions, the upper, resp. lower, tangential dimensions of X at x can be defined as the supremum, resp. infimum, of box dimensions of the tangent sets, a la Gromov, of X at x. Our main purpose is that of introducing a tool which is very sensitive to the &quot;multifractal behaviour at a point&quot; of a set, namely which is able to detect the &quot;oscillations&quot; of the dimension at a given point. In particular we exhibit examples where upper and lower tangential dimensions differ, even when the local upper and lower box dimensions coincide. Tangential dimensions can be considered as the classical analogue of the tangential dimensions for spectral triples introduced in [7], in the framework of Alain Connes&#39; noncommutative geometry [4]. © 2005 University of Houston.

In this paper, we determine and show the proof of the metric dimension of a wheel graph and the partition dimension of graph F = Ks + Kt. The solution for the metric dimension is divided into four cases. If n = 3 or 6, the metric dimension of wheel graph Wn is 3. If n = 4 or 5, the metric dimension of wheel graph Wn is 2. If n = 7 or 8, the metric dimension of wheel graph Wn+5k is 3 + 2k for k is a whole number. If n = 9, 10, or 11, the metric dimension of wheel graph Wn+5k is 4 + 2k for k is a whole number. The solution for the partition dimension is divided into two cases. If s ≥ t, the partition dimension of graph F is s + 1. If s &lt; t, the partition dimension of graph F is t.

A partition of the vertex set of a connected graph G is a locating partition of G if every vertex is uniquely determined by its vector of distances to the elements of . The partition dimension of G is the minimum cardinality of a locating partition of G . A pair of vertices u;v of a graph G are called twins if they have exactly the same set of neighbors other than u and v . A twin class is a maximal set of pairwise twin vertices. The twin number of a graph G is the maximum cardinality of a twin class of G . In this paper we undertake the study of the partition dimension of a graph by also considering its twin number. This approach allows us to obtain the set of connected graphs of order n 9 having partition dimension n

A k??quasiperfect dominating set of a connected graph G is a vertex subset S such that every vertex not in S is adjacent to at least one and at most k vertices in S. The cardinality of a minimum k-quasiperfect dominating set in G is denoted by ?1k(G). These graph parameters were first introduced by Chellali et al. (2013) as a generalization of both the perfect domination number ?11(G) and the domination number ?(G). The study of the so-called quasiperfect domination chain ?11(G) ? ?12(G)?... ? ?1?(G) = ?(G) enable us to analyze how far minimum dominating sets are from being perfect. In this paper, we provide, for any tree T and any positive integer k, a tight upper bound of ?1k(T). We also prove that there are trees satisfying all possible equalities and inequalities in this chain. Finally a linear algorithm for computing ?1k(T) in any tree T is presented.

For an ordered set {}kwwwW...,,, 21 = of vertices and a vertex v in a connected graph G, the representation of v with respect to W is the ordered k-tuple ( ) ( ) ( ) () (),,...,,,,, 21 kwvdwvdwvdWvr = | where ()yxd, represents the distance between the vertices x and y. The set W is called a resolving set for G if every vertex of G has a distinct representation. A resolving set containing a minimum number of vertices is called a basis for G. The dimension of G, denoted by (),dim G is the number of vertices ISWADI, BASKORO, SALMAN and SIMANJUNTAK 20 in a basis of G. Let {}iG be a finite collection of graphs and each iG has a fixed vertex oiv called a terminal. The amalgamation Amal {}oii vG, is formed by taking all of the iG ’s and identifying their terminals. In this paper, we determine the metric dimension of amalgamation of cycles. 1.