The study of the characteristics and topologies of networks, including social, computer and biological networks, is known as graph theory. Numerous disciplines, such as epidemiology, which is the ...study of the transmission and management of infectious diseases like Covid-19, can benefit from the study of graph theory. Like, the first case of COVID-19 (Coronavirus) was reported in November of 2019; by the third week of April, 1,116,643 verified positive cases had been reported, with around 59,158 fatalities. All the meditational structures are symmetric, if we label to change the labels of graphs in either way the graph becomes symmetric and will be no change to it. So, there are many graph theoretical parameters to study the symmetric meditational structures, like metric and edge metric dimension. A notion known as edge metric dimension or edge metric basis states that with a chosen subset from the vertex set, known as the edge resolving set, the whole edge set of a structure uniquely recognised. This idea of vertices resolvability allows for the distinct identification of COVID antiviral medication structures and aids in the research of the structure's structural characteristics. The chosen resolving set and the computed edge metric dimension will be invariant due to the symmetry of the graphs. Arbidol, malaria, carboxylic, thalidomide and theaflavin are the names of these compounds which are discussed in this work.
A big amount of vanadium is used in the industry. It is used as a vanadium carbide stabiliser in steel production. Only
$ 0.5\% $
0.5
%
vanadium in an iron alloy will double the tensile strength. ...Vanadium salts are used in leather, ink, dye manufacturing and yellow ceramic glaze. On the other hand, vanadium pentoxide is used as a catalyst, in the chemical industry, for the production of sulphuric acid and in the petroleum refinery industry. Resolvability parameter is the most discussed topic in graph theory. This helps to reorganise a network in a unique way, sometimes in terms of atoms (metric dimension) and sometimes bound (edge metric dimension). We studied resolvability parameters of vanadium carbide network in this article i.e. metric, edge metric and partition dimension. Among these metrics resolvability defined by parameters taking a subset from the set of vertices, if it has a unique representation with all the vertices, then it is called a resolving set. The number of elements in the smallest resolving set is called the metric dimension. While the number of elements of the minimal subset of vertices taking from vertex set is called the edge metric dimension, if the representations are unique of all edges and the set is called the edge resolving set. This study can have an effect in many ways in the research, particularly, with the help of some key points such as network structure analysis, material science via material properties, graph theory applications, predictive modelling, material testing and characterisation, as well as in industrial applications.
The p-type networks are designed with the help of CVNET at topo group Cluj and also given support by nano studio. Such networks develop new p-type surfaces and also represent the decorations of the ...surfaces. This patched network is designed by two repeated units. The first one is triphenylene having a Z-pen formula and the second one is triphenylene with A-phe. Furthermore, these decorations are acquired as the result of map operations represented in the CVNET software, while its assembling is conducted with the help of the nano studio program. In the literature, its topology is discussed by Omega polynomials which is an applied graph theory topic. Another most applied topic of graph theory is known as the resolvability parameter. So this article studied the resolvability parameters of patched networks, such as metric dimension, and edge metric dimension. These parameters are defined as a resolving set is a subset of vertices of a graph with a condition that each vertex of that graph has a unique code or representation with respect to the chosen subset. Its minimum cardinality is known as metric dimension, while the edge metric dimension is defined by the minimum count of members in the edge resolving set and this set is defined as according to a chosen subset each edge of a graph has unique representations, then this set is known as edge resolving set. A resolving set is a subset of vertices of a graph with a condition that each edge of that graph has a unique code or representation with respect to the chosen subset. It is minimum cardinality is known as the edge metric dimension.
The chemical graph theory helps in understanding the complex structure of molecules. Researchers can acquire a detailed grasp of the physical science, chemical properties, and bio-organic aspects of ...pharmaceuticals by computing the resolvability, and topological parameters of a drug design. The resolvability constraints for graph
are a relatively new advanced field in which the entire structure is constructed so that each vertex (atom) or edge (bond) indicates a distinct position. A topological index is a type of molecular descriptor that is effectively used in modelling many physicochemical properties in numerous quantitative structure-property/activity relationship (QSPR/QSAR) studies. In the QSPR investigation, topological indices have been used to forecast the bioactivity of chemical molecules. In the proposed study, the metric, edge metric resolvability, and topological indices of oseltamivir and zanamivir, two extensively used antiviral medicines, are computed. Further, we also discuss linear regression analysis between indices and physiochemical properties of drugs. This theoretical analysis may help the chemist and people working in the pharmaceutical industry to predict the properties of zanamivir and oseltamivir drugs without experimenting.
Let
G
=
(
V
,
E
)
be a connected graph. The distance between the edge
e
=
u
v
∈
E
and the vertex
x
∈
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
∈
E
, there exists a vertex
x
∈
S
E
such that
d
(
e
1
,
x
)
≠
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
μ
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
W
n
, and convex polytope
D
n
are bounded, while the prism related graph
D
n
∗
has unbounded edge metric dimension.
A vertex u ∈ V(G) resolves (distinguish or recognize) two elements (vertices or edges) v, w ∈ E(G)UV(G) if d G (u, v) ≠ d G (u, w) . A subset L m of vertices in a connected graph G is called a mixed ...metric generator for G if every two distinct elements (vertices and edges) of G are resolved by some vertex set of L m . The minimum cardinality of a mixed metric generator for G is called the mixed metric dimension and is denoted by dim m (G). In this paper, we studied the mixed metric dimension for three families of graphs D n , A n , and R n , known from the literature. We proved that, for D n the dim m (D n ) = dim e (D n ) = dim(D n ), when n is even, and for An the dim m (A n ) = dim e (A n ), when n is even and odd. The graph R n has mixed metric dimension 5.
A motion of a robot in space is represented by a graph. A robot change its position from point to point and its position can be determined itself by distinct labelled landmarks points. The problem is ...to determine the minimum number of landmarks to find the unique position of the robot, this phenomena is known as metric dimension. Motivated by this a new modification was introduced by Kelenc. In this paper, we computed the edge metric dimension of barycentric subdivision of Cayley graphs <inline-formula> <tex-math notation="LaTeX">Cay(\mathbb {Z}_{\alpha }\oplus \mathbb {Z}_{\beta }) </tex-math></inline-formula>, for every <inline-formula> <tex-math notation="LaTeX">\alpha \ge 6, \beta \ge 2 </tex-math></inline-formula> and an observation is made that it has constant edge metric dimension and only three carefully chosen vertices can appropriately suffice to resolve all the edges of barycentric subdivision of Cayley graphs <inline-formula> <tex-math notation="LaTeX">Cay(\mathbb {Z}_{\alpha }\oplus \mathbb {Z}_{\beta }) </tex-math></inline-formula>.
PDF
