Computing Edge-Weight Bounds of Antimagic Labeling on a Class of Trees

Graph labeling has wide applications in the field of computer science, such as coding theory, cryptography, software testing, database management systems, computer architecture, and networking. The computers connected in a network can now be converted in a graph and labels assigned to the graph so formed will help to regulate bandwidth, data traffic, in coding and decoding signals. Let
$\Lambda =(V(\Lambda),E(\Lambda))$
be a graph with
$|V(\Lambda)|=m$
and
$|V(\Lambda)|=n$
. A bijection from
$\zeta:V(\Lambda)\cup E(\Lambda)\rightarrow \{1,2,3, {\dots },m+n\}$
is called
$(a,d)$
-edge antimagic total labeling if the edge-weights
$\zeta (x)+\zeta (xy)+\zeta (y)$
for each
$xy \in E(\Lambda)$
form a sequence of consecutive positive integers with minimum edge-weight
$a$
and common difference
$d$
. In addition, it is called super
$(a,d)$
-edge antimagic total labeling if vertices receive the smallest labels. Enomoto
et al
. (2000) proposed the conjecture that every tree admits super
$(a,0)$
-EAT labeling. In this note, bounds of the minimum and maximum edge-weights for super
$(a,d)$
-EAT labeling on the more generalized class of subdivided caterpillars are obtained. Moreover, we have investigated the existence of super
$(a,d)$
-EAT labeling for the validation of the obtained bounds and the partial support of the aforesaid conjecture, where
$d\in \{0,1,2\}$
. In fact, the obtained results are a general extension of the results Akhlaq
et al
. [Utiltas Mathematica, 98 (2015), 227-249].