In this paper, we investigated the edge even graceful labeling property of the join of two graphs. A function f is called an edge even graceful labeling of a graph G =( V ( G ), E ( G )) with p =| V ( G )| vertices and q =| E ( G )| edges if f : E ( G )→{2,4,...,2 q } is bijective and the induced function f ∗ : V ( G ) →{0,2,4,⋯,2 q −2 }, defined as f ∗ ( x ) = ( ∑ xy ∈ E ( G ) f ( xy ) ) mod ( 2 k ) $ f^{\ast }(x) = ({\sum \nolimits }_{xy \in E(G)} f(xy)~)~\mbox{{mod}}~(2k) $ , where k = m a x ( p , q ), is an injective function. Sufficient conditions for the complete bipartite graph K m , n = m K 1 + n K 1 to have an edge even graceful labeling are established. Also, we introduced an edge even graceful labeling of the join of the graph K 1 with the star graph K 1, n , the wheel graph W n and the sunflower graph s f n for all n ∈ ℕ $n \in \mathbb {N}$ . Finally, we proved that the join of the graph K ¯ 2 $\overline {K}_{2}~$ with the star graph K 1, n , the wheel graph W n and the cyclic graph C n are edge even graceful graphs.

中文翻译：

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对两个图的连接进行边缘甚至优雅标记的进一步结果

在本文中，我们研究了两个图的连接的边缘均匀标记特性。函数 f 被称为图 G =( V ( G ), E ( G )) 与 p =| 的边缘甚至优雅标记 V ( G )| 顶点和 q =| E ( G )| 边如果 f : E ( G )→{2,4,...,2 q } 是双射的并且诱导函数 f ∗ : V ( G ) →{0,2,4,⋯,2 q −2 },定义为 f ∗ ( x ) = ( ∑ xy ∈ E ( G ) f ( xy ) ) mod ( 2 k ) $ f^{\ast }(x) = ({\sum \nolimits }_{xy \in E (G)} f(xy)~)~\mbox{{mod}}~(2k) $ ，其中 k = max ( p , q )，是一个单射函数。建立完整的二部图 K m , n = m K 1 + n K 1 具有边缘甚至优雅标记的充分条件。此外，我们引入了一个边，甚至优雅地标记了图 K 1 与星图 K 1, n 的连接，轮图 W n 和向日葵图 sfn 对于所有 n ∈ ℕ $n \in \mathbb {N}$ 。最后，我们证明了图 K ¯ 2 $\overline {K}_{2}~$ 与星图 K 1, n 、轮图 W n 和循环图 C n 的连接是边偶优美图.