Cardinality Of Such Set Research Articles

Mathematical analysis of the structure of one-heptagonal carbon nanocone in terms of its basis and dimension

For an undirected connected graph G = G(V, E) with vertex set V(G) and edge set E(G), a subset R of V is said to be a resolving in G, if each pair of vertices (say a and b; a ≠ b) in G satisfy the relation d(a, k) ≠ d(b, k), for at least one member k in R. The minimum set R with this resolving property is said to be a metric basis for G, and the cardinality of such set R, is referred to as the metric dimension of G, denoted by dim v (G). In this manuscript, we consider a complex molecular graph of one-heptagonal carbon nanocone (represented by HCN s ) and investigate its metric basis as well as metric dimension. We prove that just three specifically chosen vertices are enough to resolve the molecular graph of HCN s . Moreover, several theoretical as well as applicative properties including comparison have also been incorporated.

Geodetic domination integrity in fuzzy graphs

Let N = (V, E) be a simple graph and let X be a subset of V (N). If every node not in X lies on a geodesic path between two nodes from X then it is called a geodetic set. The geodetic number g (N) is the minimum cardinality of such set X. The subset X is called a dominating set if every node not in X has at least one neighbour in X. The minimum number of nodes of a dominating set is known as domination number γ (N). If the subset X is a geodetic set as well as a dominating set then it is called a geodetic dominating set. The minimum cardinality of a geodetic dominating set is known as geodetic domination number γg (N). The geodetic domination integrity of N is defined to be DIg (N) = min {|X| + m (N - X) : X is a geodetic dominating set of N}, where m (N - X) denotes the order of the largest component of N - X. Uncertain networks can be modelled using fuzzy graphs. In a graph, each vertex and each edge are equally significant. However, in fuzzy graphs, each vertex and each edge is important in terms of fuzziness in their own right. In this study, the concepts of geodetic dominating sets in fuzzy graphs and geodetic domination number are defined and bounds are obtained. Moreover, the vulnerability parameter Geodetic domination integrity is introduced in fuzzy graphs. Further, the geodetic domination integrity for complete fuzzy graphs, complete bipartite fuzzy graphs, Cartesian product of two strong fuzzy graphs and bounds are also discussed. The applications of this parameter are applied to a telecommunication network system model to identify the key persons in the system and applied in a fuzzy social network to find the most influential group within the network.

A Comparative Study of Three Resolving Parameters of Graphs

Graph theory is one of those subjects that is a vital part of the digital world. It is used to monitor the movement of robots on a network, to debug computer networks, to develop algorithms, and to analyze the structural properties of chemical structures, among other things. It is also useful in airplane scheduling and the study of diffusion mechanisms. The parameters computed in this article are very useful in pattern recognition and image processing. A number d f , w = min d w , t , d w , s is referred as distance between f = t s an edge and w a vertex. d w , f 1 ≠ d w , f 2 implies that two edges f 1 , f 2 ∈ E are resolved by node w ∈ V . A set of nodes A is referred to as an edge metric generator if every two links/edges of Γ are resolved by some nodes of A and least cardinality of such sets is termed as edge metric dimension, e dim Γ for a graph Γ . A set B of some nodes of Γ is a mixed metric generator if any two members of V ∪ E are resolved by some members of B . Such a set B with least cardinality is termed as mixed metric dimension, m dim Γ . In this paper, the metric dimension, edge metric dimension, and mixed metric dimension of dragon graph T n , m , line graph of dragon graph L T n , m , paraline graph of dragon graph L S T n , m , and line graph of line graph of dragon graph L L T n , m have been computed. It is shown that these parameters are constant, and a comparative analysis is also given for the said families of graphs.

Powerful and maximal rational metric dimension of a wheel

The rational distance from the vertex u to the vertex v in a graph G, denoted by d(v/u), is defined as the average distances from the vertex u to the closed neighbors of v if u 6= v, else it is 0. A subset S of vertices of G is called rational resolving set of G if for every pair u, v of distinct vertices in V−S, there is a w ∈ S such that d(u/w) ≠ d(v/w) in G. In this paper powerful and maximal rational resolving sets are introduced and minimum cardinality of such sets are computed for the wheel graphs.

Powerful and maximal rational metric dimension of a wheel

The rational distance from the vertex u to the vertex v in a graph G, denoted by d(v/u), is defined as the average distances from the vertex u to the closed neighbors of v if u 6= v, else it is 0. A subset S of vertices of G is called rational resolving set of G if for every pair u, v of distinct vertices in V−S, there is a w ∈ S such that d(u/w) ≠ d(v/w) in G. In this paper powerful and maximal rational resolving sets are introduced and minimum cardinality of such sets are computed for the wheel graphs.

Restrained geodetic domination in graphs

Let [Formula: see text] be a graph with edge set [Formula: see text] and vertex set [Formula: see text]. For a connected graph [Formula: see text], a vertex set [Formula: see text] of [Formula: see text] is said to be a geodetic set if every vertex in [Formula: see text] lies in a shortest path between any pair of vertices in [Formula: see text]. If the geodetic set [Formula: see text] is dominating, then [Formula: see text] is geodetic dominating set. A vertex set [Formula: see text] of [Formula: see text] is said to be a restrained geodetic dominating set if [Formula: see text] is geodetic, dominating and the subgraph induced by [Formula: see text] has no isolated vertex. The minimum cardinality of such set is called restrained geodetic domination (rgd) number. In this paper, rgd number of certain classes of graphs and 2-self-centered graphs was discussed. The restrained geodetic domination is discussed in graph operations such as Cartesian product and join of graphs. Restrained geodetic domination in corona product between a general connected graph and some classes of graphs is also discussed in this paper.

ПОЛИНОМИАЛЬНЫЙ ВАРИАНТ ЗАДАЧИ СУММ-ПРОИЗВЕДЕНИЙ

This work is about the generalization of sum-product problem. The general principle of it was formulated in the Erdos-Szemeredi’s hypothesis. Instead of the Minkowski sum in this hypothesis, the set of values f(x,y) of a homogeneous polynomial f lin two variables, where x and y belong to subgroup G of is considered. The lower bound on the cardinality of such set is obtained. This topic has an applied value in the theory of information and dynamics in calculating the probabilities of events, as well as in various methods of encoding and decoding information.

Every grid has an independent [1, 2]-set

It is well known that the Cartesian product of two paths, the grid Pm□Pn,m≤n, has an independent dominating set such that every vertex not in the set has exactly one neighbor in it, if and only if m=n=4 or m=2,n=2k+1. In this paper we prove that every grid Pm□Pn has an independent dominating set such that every vertex not in the set has at most two neighbors in it, and we also calculate the minimum cardinality of such sets.

2-Power Domination in Certain Interconnection Networks

Abstract The k-power domination problem generalizes domination and power domination problems. The k-power domination problem is to determine a minimum size vertex set S ⊆ V(G) such that after setting X = N[S] and iteratively adding to X vertices x that have a neighbour v in X such that at most k neighbours of v are not yet in X till we get X = V(G). The least cardinality of such set is called the k-power domination number of G and is denoted by γp,k (G). In this paper, we restrict our discussion to k = 2, referred to as 2-power domination. We compute 2-power domination number for certain interconnection networks such as hypertree, sibling tree, X-tree, Christmas tree, mesh, honeycomb mesh, hexagonal mesh, cylinder, generalized Petersen graph and subdivision of graphs.

Revisiting the vertex cover of graphs

A set U of vertices of graph G is a vertex cover of G if every edge in G is incident with a vertex in U. The minimum cardinality of such set is the vertex covering number of G and is denoted by (G). This paper characterizes bipartite graphs in terms of vertex cover. It provides the vertex covering number of (i) the complement of a nonempty bipartite graph, (ii) the powers of paths and cycles, and (iii) one supergraph of planar grid.

On the Olson and the Strong Davenport constants

Soit S un sous-ensemble d’un groupe abélien fini, noté additivement. Si 0 n’est pas une sous-somme (non vide) de S, on dit que S est un ensemble sans sous-somme nulle. Nous examinons la cardinalité maximale d’un ensemble sans sous-somme nulle, c’est-à-dire la (petite) constante d’Olson. Nous déterminons la cardinalité maximale d’un tel ensemble pour plusieurs types de groupes ; en particulier, les p-groupes dont le rang est suffisament grand relativement à l’exposant et plus particulièrement tous les groupes dont l’exposant est au plus 5. Nous obtenons ces résultats comme des cas particuliers de résultats plus généraux, donnant des bornes inférieures pour la cardinalité d’un ensemble sans sous-somme nulle pour des groupes variés. Nous examinons la qualité de ces bornes en considérant des cas explicites, avec l’aide d’un ordinateur. De plus, nous examinons une notion très proche de la constante d’Olson : la cardinalité maximale d’un ensemble minimal de somme nulle, c’est-à-dire la constante de Davenport forte. En particulier, nous déterminons la valeur de cette constante pour les p-groupes élémentaires dont le rang est au plus 2, en utilisant des résultats récents sur la constante d’Olson.

On Maximal Residue Difference Sets Modulo p

AbstractA residue difference set modulo p is a set A = {a1,a2,...,ak} of integers 1 ≤ ai ≤ p — 1 such that for all i and j with i ≠ j, where is the Legendre symbol. We give a lower and an upper bound for mp—the P maximal cardinality of such set A in the case of p ≡ 1 (mod 4).

On splitting sets in block designs and finding roots of polynomials

The general notion of t-splitting sets is introduced within the context of combinatorial block designs. A greatest lower bound on cardinality of such sets, and an upper bound on cardinality of the smallest such set in a given design are established. The abstraction of t-splitting sets is shown to provide a natural framework for the analysis of the problem of finding roots of polynomials over finite fields, and elementary concepts from design theory are applied to re-examine and extend some existing results in this area.

Minimal separating sets of maximum size

A connected graph G can be disconnected or reduced to a single vertex by removing an appropriate subset of the vertex set V( G), and can be disconnected by removing a suitable subset of the edge set E( G). Attention has usually been centered on separating sets having minimum cardinality, and parameters called the vertex connectivity and the edge connectivity defined. These classical concepts are generalized by using separating sets which are minimal. By considering the maximum as well as the minimum cardinality of such sets, one defines vertex and edge connectivity parameters. Sharp upper bounds are established for these numbers and their values computed for certain classes of graphs. An analogue of Whitney's theorem on connectivity is obtained. Parameters are also defined for minimal separating sets consisting of a mixture of vertices and edges, and these are shown to depend on the maximum and minimum values of the vertex and edge connectivity parameters.