Sharp Bounds on Vertex N-magic Total Labeling Graphs

A vertex N -magic total labeling is a bijective function that maps the vertices and edges of a graph G onto the successive integers from 1 to p + q. The labeling exhibits two distinct properties: First, the count of unique magic constants k i for i belonging to the set 1, 2,. . . , N is equivalent to the cardinality of N ; secondly, the magic constants k i must be arranged in a strictly ascending order. In the present context, the constant N is employed to represent different degrees of vertices. The term "magic constant values k i " for i ∈ 1, 2,. . . , N refers to specific numbers that exhibit unique and interesting properties and are employed in the context of this investigation. By adding up the weights of each vertex in V (G), we might receive a magical constant number k i for i ∈ 1, 2,. . . , N. Within the scope of this study, we discuss the sharp bounds of vertex N -magic total labeling graphs. In terms of magic constants k i for i ∈ 1, 2,. . . , N, we also found the requirement for vertex N -magic total labeling of trees. We investigated the potential for vertex N -magic total labeling at vertices in graphs with varying vertex degrees.