| dc.description.abstract | All graphs in this paper are nontrivial and connected graph. For 𝑘-ordered set 𝑊 = {𝑠1, 𝑠2, … , 𝑠𝑘} of vertex set 𝐺, the multiset representation
of
a
vertex
𝑣
of
𝐺
with
respect
to
𝑊
is
𝑟𝑚(𝑣|𝑊)
= {𝑑(𝑣, 𝑠1), 𝑑(𝑣, 𝑠2), … , 𝑑(𝑣, 𝑠𝑘)} where 𝑑(𝑣, 𝑠𝑖) is a distance between of the vertex
𝑣
and
the
vertices
in
𝑊
together
with
their
multiplicities.
The
resolving
set
𝑊
is
a local
resolving
set
of
𝐺
if𝑟𝑚(𝑣|𝑊)
≠ 𝑟𝑚(𝑢|𝑊) for
every pair 𝑢, 𝑣 of adjacent vertices of 𝐺. The minimum local resolving set 𝑊 is a local multiset basis of 𝐺. If 𝐺 has a local multiset basis,
then its cardinality is called local multiset dimension,denoted by 𝜇𝑙(𝐺). If 𝐺 does not contain a local resolving set, then we write 𝜇𝑙(𝐺) =
∞. In our paper, we will investigate the establish sharp bounds of the local multiset dimension of 𝐺 and determine the exact value of
some family graphs. | en_US |
