Definition: G has radius at most r if there is a vertex v such that every vertex in G can be reached from v in at most r steps. The radius of G is the smallest possible radius of any vertex, known as...

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Extracted text: Definition: G has radius at most r if there is a vertex v such that every vertex in G can be reached from v in at most r steps. The radius of G is the smallest possible radius of any vertex, known as the center. Example: Radius of P is 4 with center being a middle vertex. Lemma: diameter < 2*radius="" pf:="" any="" two="" vertices="" are="" reachable="" from="" each="" other="" by="" walking="" through="" the="" center.="" п(п-1)="" lemma:="" the="" order="" of="" kn="" is="" (“)="" %3d="" 2="" pf:="" each="" vertex="" is="" incident="" on="" n-1="" edges,="" so="" n(n–="" 1)="" incidences="" overall.="" since="" each="" edge="" has="" two="" vertices,="" we="" need="" to="" divide="" by="" two.="" definition:="" the="" degree="" of="" a="" vertex="" v,="" degc(v)="" is="" the="" number="" of="" edges="" it="" is="" incident="" on.="" corollary="" if="" every="" vertex="" has="" degree="" at="" least="" |v(g)|/2,="" then="" the="" graph="" is="" hamiltonian.="" п="" proof:="" in="" that="" case="" deg(u)="" +="" deg(v)="">÷+ п = n for any two vertices, so Ore's theorem applies.