Pelabelan Signed Product Cordial pada Graf Middle Path Union Jewel

Abstract

The cordial labeling of a graph 𝐺 is a function 𝑓:𝑉(𝐺) → {0,1} such that there exists a function 𝑓∗: 𝐸(𝐺) → {0,1} defined by 𝑓∗(𝑢𝑣) = |𝑓(𝑢) − 𝑓(𝑣)| under the condition that the difference in the number of nodes labeled 0 and 1 and edges labeled 0 and 1 at most be one. The signed product cordial labeling of a graph 𝐺 is a function 𝑔:𝑉(𝐺) → {1,−1} such that there exists a function 𝑔∗:𝐸(𝐺) → {1, −1} defined by 𝑔∗(𝑢𝑣) = 𝑔(𝑢)𝑔(𝑣) under the condition that the difference in the number of nodes labeled 1 and −1 and edges labeled 1 and −1 at most be one. This research discusses the signed product cordial labeling on the path union jewel graph 𝑃2(𝐽𝑛), the middle jewel graph 𝑀(𝐽𝑛), and the middle path union jewel graph

(𝑃2(𝐽𝑛)). After that, a comparison between cordial and signed product cordial labeling is also discussed, since they both have the same codomain cardinality of two and the same labeling requirements. The research process begins with labeling of nodes and edges, then continued by the formulation of nodes and edges labeling function formulas to prove that the graph 𝑃2(𝐽𝑛), graph 𝑀(𝐽𝑛), and graph

(𝑃2(𝐽𝑛)) have fulfilled the requirements of signed product cordial labeling. After that, a comparison of cordial and signed product cordial labeling is presented. The result of this research shows that the three graphs studied are signed product cordial graph and if a graph is a cordial graph then it is also a signed product cordial graph, and the converse also holds