I am asked to find the join of two graphs in graph theory. But I cannot find the exact definition! I know that in lattice theory, we join every vertex of a graph to every vertex of another graph to find the join of graphs. Any expert advice is welcome.
The join of two graphs $G_1$ and $G_2$ , denoted by $G_1\nabla G_2 $, is a graph obtained from $G_1$ and $G_2$ by joining each vertex of $G_1$ to all vertices of $G_2$ . After joining the two graph the resultant graph will be of diameter at most 2.
A more Computer Science oriented solution: https://arxiv.org/pdf/1608.05594.pdf
Join of two graphs $G_1=(V_1, E_1)$ and $G_2=(V_2, E_2)$ is mathematically denoted and defined as $G_1\triangledown G_2=(V_1\cup V_2, E_1\cup E_2\cup\lbrace (a, b): a\in V_1, b\in V_2\rbrace)$
Note that in this process, self loops will be generated if $ G_1$ and $G_2$ contain atleast one common vertex and multiple edges may arise if $ G_1$ and $G_2$ contain atleast two common vertices or so.