Name of a specific graph with fixed distance and degree

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I was trying to come up with a graph that maximizes the number of nodes such that there exists a path between any two nodes that is length 2, and the degree of each node is 10. Then I tried to generalize the result and failed and now I want to learn if there is a graph type with for every two nodes there exists a path of length k that connects them and the degree of each node is equal to n. Is there a theory behind this class of graphs?