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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.

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  • $\begingroup$ perhaps providing the actual question would be helpful. Literally, the join is the "graph with all the edges that connect the vertices of the first graph with the vertices of the second graph." $\endgroup$ – Andres Mejia Mar 10 '16 at 2:13
  • $\begingroup$ Let me know that when we draw join of two graphs, is that I should join every vertex of graph1 to every vertex of graph 2 by an edge? $\endgroup$ – Kavita Sahu Mar 10 '16 at 2:22
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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.

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A more Computer Science oriented solution: https://arxiv.org/pdf/1608.05594.pdf

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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.

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