Efficient Mod Sum Labeling Scheme of Generalized Friendship Graph

A graph is a mod sum graph if there is a labeling of the vertices with distinct positive integers so that an edge is present if and only if the sum of the labels of the vertices incident on the edge, modulo some positive integers, is the label of a vertex of the graph. Mod sum graph is the promotion of sum graph, and it’s very important in graph theory. This paper provides a mod sum labeling for the generalized friendship graph with the sequential numbering method. To develop our labeling scheme we first consider the case \( m = 3,4,5 \) before proving the result for the general case by considering both odd and even cycles. Finally, to prove all the generalized friendship graphs are mod sum graphs. This conclusion is further improved the theory of the labeled graph, and to provide a theoretical basis for later researchers.