Let G(V, E) be a connected and simple graphs with vertex set V and edge set E. Define a coloring c : E(G) → {1, 2, 3, ..., k}, k ∈ N as the edges of G, where adjacent edges may be colored the same. ...If there are no two edges of path P are colored the same then a path P is a rainbow path. The graph G is rainbow connected if every two vertices in G has a rainbow path. A graph G is called antimagic if the vertex sum (i.e., sum of the labels assigned to edges incident to a vertex) has a different color. Since the vertex sum induce a coloring of their edges and there always exists a rainbow path between every pair of two vertices, we have a rainbow antimagic coloring. The rainbow antimagic connection number, denoted by rcA(G) is the smallest number of colors that are needed in order to make G rainbow connected under the assignment of vertex sum for every edge. We have found the exact value of the rainbow antimagic connection number of ladder graph, triangular ladder, and diamond.
Abstract
All graphs in this paper are connected and nontrivial graph. Let f : V(G) → {1,2,…,k} be a vertex coloring of a graph G where two adjacent vertices may be colored the same color. Consider ...the color classes Π =
{C
1
, C
2
,…, C
k
}.
For a vertex v of G, the representation color of v is the k-vektor
r(v
| Π) =
{d(v, C
1
),d(v, C
2
),…,
d(v, C
k
)}, where
d(v, Ci)
= min{
d(v,
c); c e C
i
}. If
r(u
| Π) = r(v | Π) for every two adjacent vertices u and v of G, then f is a metric coloring of
G
. The minimum
k
for which
G
has a metric
k
-coloring is called the metric chromatic number of G and is denoted by μ(G). The metric chromatic number on comb product of ladder graphs namely path graph, star graph, fan graph, cycle graph, and complete graph.
Let G be a simple, finite, and connected graph. A dominating set D is a set of vertices such that each vertex of G is either in D or has at least one neighbor in D. The minimum cardinality of such a ...set is called the domination number of G, denoted by γ(G). For an ordered set W = {w1, w2, ..., wk} of vertices and a vertex v in a connected graph G, the metric representation of v with respect to W is the k-vector r(v|W ) = (d(v, w1), d(v, w2), ..., d(v, wk)), where d(x, y) represents the distance between the vertices x and y. The set W is a resolving set for G if distinct vertices of G have distinct representations with respect to W. A resolving set of minimum cardinality is called a minimum resolving set or a basis and the cardinality of a basis for G, denoted by dim(G). A resolving domination number, denoted by γr(G), is the minimum cardinality of the resolving dominating set. In this paper, we study the existence of resolving domination number of special graph and its line graph L(G), middle graph M(G), total graph T(G), and central graph C(G) of Star graph and fan graph. We have found the minimum cardinality of those special graphs.
The aim of this research was to compare the HIC (Highly interactive classroom) system with a traditional method of teaching superposition of wave functions courses especially. In order to examine ...it's learning effectiveness across three dimensional, knowledge, comprehension, and application. The highly interactive classroom which collaborated with proof mathematics formulation, computer programming, and cloud classroom. The study followed a pretest, posttest, and delayed posttest with the quasi-experimental design. A total of 66 students (aged 18-19 years) divided into experimental group and control group. Experimental group 33 student (10 males and 23 females) was utilizing of cloud classroom and 33 students (12 males and 21 females) in the control group. Pretest, and posttest were evaluated by using inferential statistics t-test. The significant value of t-test score is -0.817 (p < 0.05) means there are significant of post-test score between control class and experiment class. The result indicated that student on an experimental group that teaching utilizing collaboration of computer programming, and highly interactive classroom better than control group was teaching by traditional method in post-test and delayed posttest across three learning dimensions. According to these result. The collaboration of mathematics formulation, computer programming, and HIC (highly interactive cloud classroom) is recommended to be a strategy for teaching on the combination of wave function topic.
Abstract
We use finite, connected, and undirected graph denoted by
G
. Let
V
(
G
) and
E
(
G
) be a vertex set and edge set respectively. A subset
D
of
V
(
G
) is an efficient dominating set of graph
...G
if each vertex in
G
is either in
D
or adjoining to a vertex in
D
. A subset
W
of
V
(
G
) is a resolving set of
G
if any vertex in
G
is differently distinguished by its representation respect of every vertex in an ordered set
W
. Let
W
= {
w
1
,
w
2
,
w
3
, …,
w
k
} be a subset of
V
(
G
). The representation of vertex
υ
∈
G
in respect of an ordered set
W
is
r
(
υ
|
W
) = (
d
(
υ, w
1
),
d
(
υ, w
2
), …,
d
(
υ, w
k
)). The set
W
is called a resolving set of
G
if
r
(
u
|
W
) ≠
r
(
υ
|
W
) ∀
u, υ
∈
G
. A subset
Z
of
V
(
G
) is called the resolving efficient dominating set of graph
G
if it is an efficient dominating set and
r
(
u
|
Z
) ≠
r
(
υ
|
Z
) ∀
u, υ
∈
G
. Suppose
γ
re
(
G
) denotes the minimum cardinality of the resolving efficient dominating set. In other word we call a resolving efficient domination number of graphs. We obtained
γ
re
G
of some comb product graphs in this paper, namely
P
m
⊲
P
n
,
S
m
⊲
P
n
, and
K
m
⊲
P
n
.
The local edge metric dimension of graph Adawiyah, R; Dafik; Alfarisi, R ...
Journal of physics. Conference series,
05/2020, Letnik:
1543, Številka:
1
Journal Article
Recenzirano
Odprti dostop
In this paper, we introduce a new notion of graph theory study, namely a local edge metric dimension. It is a natural extension of metric dimension concept. dG(e,v) = min{d(x,v),d(y,v)} is the ...distance between the vertex v and the edge xy in graph G. A non empty set S⊂V is an edge metric generator for G if for any two edges e1,e2∈E there is a vertex k∈S such that dG(k,e1≠dG(k,e2)). The minimum cardinality of edge metric generator for G is called as edge metric dimension of G, denoted by dimE(G). The local edge metric dimension of G, denoted by dimE(G), is a local edge metric generator of G if r(xk|S)≠r(yk|S) for every pair xk,ky of adjacent edges of G. Our concern in this paper is investigating some results of local edge metric dimension on some graphs.
Abstract
Let G be a connected and simple graph. Proper vertex colouring c : V(G) — {1, 2, 3,…, k} where k → 2 that induces a proper edge colouring c’ :
E(G) —
{1, 2, 3,…, k — 1} define by ...c’(uv)=|c(u) — c(v)|, where uv in E(G) is called graceful k— colouring. Graceful colouring is a vertex colouring c of graph G if c is a graceful k-coloring for some k ∈ N. Graceful chromatic number of a graph G, denoted by
χ
β
(G) is the minimum k for which G has a graceful k—colouring. In our paper, we investigate the establish exact value of graceful chromatic number of comb product of ladder graph.
We assume that G = G(V, E) is a finite, simple, and undirected graph with V as set of vertices and E as set of edges. A bijection g : V (G) ∪ E(G) → {1, 2, 3, …|V(G)| + |E(G)} is called a super ...\({\mathscr{H}}\)-decomposition local antimagic total labeling for any two adjacent subgraph \({ {\mathcal H} }_{i}\) and \({ {\mathcal H} }_{j}\), \(w({ {\mathcal H} }_{i})\ne w({ {\mathcal H} }_{j})\), where \(w( {\mathcal H} )=\displaystyle {\sum }_{v\in V( {\mathcal H} )}f(v)+\displaystyle {\sum }_{e\in E( {\mathcal H} )}f(e)\). The chromatic number of super \({\mathscr{H}}\)-decomposition local antimagic total labeling is minimum number of colors in super \({\mathscr{H}}\)-decomposition local antimagic total labeling and denoted by \({\gamma }_{lat {\mathcal H} }(G)\). In this paper, we used some subdivision graph. Such as subdivision of star graph (S(Sn )), subdivision of sun graph (M(Mn )), subdivision of wheel graph (S(Wn )). The subdivision graph S(G) of a graph G is the graph obtained from G by replacing each of its edges by a path of length two.
On resolving total dominating set of sunlet graphs Ervani, R S R; Dafik; Tirta, I M ...
Journal of physics. Conference series,
03/2021, Letnik:
1832, Številka:
1
Journal Article, Conference Proceeding
Recenzirano
Odprti dostop
Abstract
The set
D
⊆
V
(
G
) is called dominating set on graph
G
so that every vertex not in
D
is adjacent to at least one vertex in
D
. The set
D
t
⊆
V
(
G
) is called total dominating set on graph
...G
so that the vertex in
D
t
are neighboring at least one dot in
D
t
. The smallest cardinality of the total dominating set is referred to as total domination number. The total domination number in G is shown by
γ
t
(
G
). The set of vertex
D
t
⊆
V
(
G
) is resolving total dominating set from
G
if the vertex representation
u,υ
∈
V
(
G
) with respect to
x
∈
D
t
is
r
(
υ
|
D
t
) so that
r
(
υ
|
D
t
) ≠
r
(
u
|
D
t
). The smallest cardinality of Resolving total dominating set in G is shown by
γ
rt
(
G
). In this article, we provide the results of the study for the differentiating number of total dominance from sunlet graphs.
A simple graph G=(V,E) is said to be an H-covering if every edge of G belongs to at least one subgraph isomorphic to H. A bijection f:V∪E→{1,2,3,…,V+E} is an (a,d)-H-antimagic total labeling of G if, ...for all subgraphs H′ isomorphic to H, the sum of labels of all vertices and edges in H′ form an arithmetic sequence {a,a+d,…,(k-1)d} where a>0, d≥0 are two fixed integers and k is the number of all subgraphs of G isomorphic to H. The labeling f is called super if the smallest possible labels appear on the vertices. A graph that admits (super) (a,d)-H-antimagic total labeling is called (super) (a,d)-H-antimagic. For a special d=0, the (super) (a,0)-H-antimagic total labeling is called H-(super)magic labeling. A graph that admits such a labeling is called H-(super)magic. The m-shadow of graph G, Dm(G), is a graph obtained by taking m copies of G, namely, G1,G2,…,Gm, and then joining every vertex u in Gi, i∈{1,2,…,m-1}, to the neighbors of the corresponding vertex v in Gi+1. In this paper we studied the H-supermagic labelings of Dm(G) where G are paths and cycles.
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