The radius of g denoted by rad g is the minimum

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The radius of G , denoted by rad( G ), is the minimum eccentricity (that is, min { ecc( v ) : v V ( G ) } ) if G is connected and otherwise. The diameter of G , denoted by diam( G ), is the maximum eccentricity (that is, max { ecc( v ) : v V ( G ) } ) if G is connected and otherwise. Alternatively, diam( G ) = max { dist( u, v ) : u, v V ( G ) } . Theorem 10. Let G be a connected graph. Then rad( G ) diam( G ) 2 rad( G ) .
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Proof. The left inequality follows from the definitions of rad( G ) and diam( G ). Let u, v V ( G ) be such that dist( u, v ) = diam( G ). Let w V ( G ) be such that ecc( w ) = rad( G ). Then diam( G ) = dist( u, v ) dist( u, w ) + dist( w, v ) 2 ecc( w ) = 2 rad( G ) . Diameter and eigenvalues Theorem 11. If G is connected with diameter d , then it has at least d + 1 distinct eigenvalues. Proof. Denote A = A ( G ). Suppose G has t d distinct eigenvalues. Then there exist v i , v j such that d ( v i , v j ) = t . Hence ( A t ) ij > 0 but ( A k ) ij = 0 for k = 1 , 2 , . . . , t - 1. So if p ( x ) is any polynomial of degree t , then p ( A ) ij 6 = 0. In particular, since the minimal polynomial m ( x ) of A has degree t , we have m ( A ) ij 6 = 0. Hence m ( A ) 6 = 0. This contradicts the assumption that m ( x ) is the minimal polynomial of A . 7 Shortest paths and Dijkstra’s algorithm Weighted graphs Definition 7. A weighted graph is a graph G in which each edge e is associated with a weight w ( e ) R ; that is, w : E ( G ) R . In this subject we always assume that w ( e ) is non-negative for all edges e . The (weighted) length of a path is the sum of the weights of the edges on the path. For u, v V ( G ), the (weighted) distance dist( u, v ) between u and v is the minimum length of all ( u, v )-paths in G , or if no ( u, v )-path exists. A ( u, v )-path in G with length dist( u, v ) is called a shortest path from u to v . Shortest path problem Problem 2. Given a weighted graph G with all edges having non-negative weights, and given a vertex u 0 of G , find a shortest path from u 0 to every other vertex of G (together with the distance). e.g. google maps direction finder Dijkstra’s algorithm Starting with u 0 , label the vertices of G . Initially label u 0 by 0 and all other vertices by . Vertices in S have “permanent” labels. Vertices in ¯ S = V ( G ) - S have temporary labels. A permanent label is the distance from u 0 to the vertex. A temporary label is an upper bound on the distance. Each time we choose a vertex of ¯ S with smallest temporary label, add it to S , and relabel the vertices in ¯ S adjacent to it. Repeat this for the enlarged S growing a tree along the way. In the end we obtain a “spanning tree of shortest paths”.
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Dijkstra’s algorithm 1. let ( u 0 ) := 0 , ‘ ( v ) := for all v 6 = u 0 , i := 0 , S := , ¯ S := V ( G ) [Initialisation] 2. while i < | V ( G ) | do (a) choose v ¯ S minimising ( v ) [Choose a vertex u i to scan] let u i := v (b) [Scan neighbours of u i , relabelling those in ¯ S ] for each edge u i v with v ¯ S , if ( v ) > ‘ ( u i ) + w ( u i v ) then let ( v ) := ( u i ) + w ( u i v ) and PARENT( v ) := u i [This makes v have a label of the length of the shortest path via u i , if that is shorter than all paths previously found.] (c) let S := S ∪ { u i } , ¯ S := ¯ S - { u i } , i := i + 1 3. for each vertex v V ( G ) output ( v )
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