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On inclusive 1-Distance Vertex Irregularity Strength of Firecracker,
Broom, and Banana Tree
Ikhsanul Halikin, Ade Rizky Savitri, and Kristiana Wijaya
Graphs, Combinatorics, and Algebra Research Group
Department of Mathematics, Faculty of Mathematics and Natural Sciences
University of Jember (UNEJ), Jl. Kalimantan 37 Jember, Indonesia 68121
Keywords: Inclusive 1-Distance Vertex Irregular Labelling, Inclusive 1-Distance Vertex Irregularity Strength,
Firecracker, Broom, Banana Tree.
Abstract: Let k be a natural number and G be a simple graph. An inclusive d-distance vertex irregular labelling of a
graph G is a function
so that the weights at each vertex are different. Let v be a
vertex of G. The weight of v V(G), denoted by wt(v), is the sum of the label of v and all vertex labels up to
distance 1 from v. An inclusive 1-distance vertex irregularity strength of G, denoted by
is the
minimum k for the existence of an inclusive 1-distance vertex irregular labelling of a G. Here, we find the
exact value of an inclusive 1-distance vertex irregularity strength of a firecracker, a broom, anda banana
tree.
1 INTRODUCTION
Suppose that is an undirected and finite graph
without loop and parallel edges. For a vertex v in a
graph G, the degree of v with notation d(v) is the
number of edges in G that are incident to v. For two
vertices u and v in a graph G (not necessarily
distinct), a u – v walk in G is defined as a sequence
of vertices and edges in G, starting with u and
ending at v such that consecutive vertices are
connected by an edge. A path defined as a u – v walk
with different vertices. The length of the shortest
path from vertex u to vertex v is said to be a distance
from u to v and denoted by d(u,v) (see Chartrand,
Lesniak & Zhang, (2011) for another terminology).
The labelling in graph is one of research topics
introduced in the 1960s. The labelling of a graph is a
function from a set of graph elements (vertices or
edges or both) onto a set of numbers (usually natural
numbers) with certain condition. There are many
kinds of graph labelling that have been introduced
(see Gallian (2016) for a complete survey).
Chartrand et al. suggested the concept of an irregular
labelling in 1988. The problem of this labelling is
how to assign natural numbers label to the edges of a
graph so that the sum of edge labels at each vertex is
different. In this labelling also introduced a notion,
called irregularity strength, i.e. the minimum largest
label among all of the possible irregular assignments
of a graph (Chartrand et al., 1988).
In 2007, Bačá et al. introduced the similar
assignment but apply to both edges and vertices of a
graph. This labelling is called the irregular total k-
labelling. A total k-labelling is a mapping from the
vertex set and edge set to the set of natural numbers
. The minimum k for such labelling is said
to be the total irregularity strength. Furthermore,
Mirka, Rodger & Simanjuntak (2003) introduced
another kind labelling, which is called distance
magic labelling.
Motivated by Mirka and Bačá, Slamin (2017)
introduced a distance vertex irregular labelling of
graphs. A distance vertex irregular labelling of a
graph G is a function
such that
the weight of every vertex v in G is different. The
weight of a vertex, denoted by wt(v),is the
sum of the labels of all the vertices of distance1
from v. Moreover, Bong, Lin & Slamin (2017),
generalized concept of a distance irregular vertex
labelling to inclusive vertex irregular d-distance
vertex labelling. Inclusive in this labelling means
that the weight of the vertex v included the label of a
vertex v. The minimum k for the existence of this
labelling is said to bea distance irregularity strength
of G and denoted by
. Furthermore, Bong,
Lin & Slamin (2017) obtained
, for G are a
path P
n
for n= 3k, , a star
and a double
228
Halikin, I., Savitri, A. and Wijaya, K.
On inclusive 1-Distance Vertex Irregularity Strength of Firecracker, Broom, and Banana Tree.
DOI: 10.5220/0008519802280232
In Proceedings of the International Conference on Mathematics and Islam (ICMIs 2018), pages 228-232
ISBN: 978-989-758-407-7
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