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[There has been a flaw in my definition - as Sergei and Andreas pointed out. I hope I could fix it.] I want to understand how the concepts of directions, straight (or shortest) lines, and geodesics &...

A graph $G$ is connected if and only if the second-largest eigenvalue $\lambda_2$ of the Laplacian of $G$ is greater than zero. (See, e.g., the Wikipedia article on algebraic connectivity.) Is ...