Graceful labeling construction for some special tree graph using adjacency matrix

Abstract

In 1967, Rosa introduced β − labeling which was then popularized by Golomb under the name graceful. Graceful labeling on a graph G is an injective function f : V ( G ) → { 0 , 1 , 2 , . . . , | E ( G ) |} such that, when each edge uv ∈ E ( G ) is assigned the label | f ( u ) − f ( v ) | the resulting edge labels are distinct. If graph G has graceful labeling then G is called a graceful graph. Rosa also introduced α − labeling on graph G which is a graceful labeling f with an additional condition that there is λ ∈ { 1 , 2 , . . . , | E ( G ) |} so that for every edge uv ∈ E ( G ) where f ( u ) < f ( v ) then f ( u ) ≤ λ < f ( v ) . This paper gives a new approach to showing a graph is admitted α − labeling using an adjacency matrix. Then this construction will be used to construct graceful labeling for the superstar graph. Moreover, we give a graceful labeling construction for a super-rooted tree graph.