I came across the theorem : A connected graph is regular if and only if the all-one vector is an eigenvector of A. I am not sure on how to prove every connected regular graph has an all-one eigenvector.
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Applying $A$ to the all-one vector produces a vector with the vertex degrees as entries. This is a multiple of the all-one vector (i.e., all entries are equal) iff the graph is regular.
Hagen von Eitzen
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