algorithm - Traveling salesman (TSP): what is the Relation with number of vertices and length of the found route? -

i know there many algorithms (exact or approximate) implement traveling salesman problem.

i know relation between number of vertices (i.e., places visit) , length of route found these algorithms.

intuitively, less number of vertices, shorter route is. but, can 1 give me math relation between number of vertices , length of route found @ least 1 of existing traveling salesman algorithms?

thanks in advance.

in general given n nodes, let set of costs defined c = { c(i, j) = cost traverse edge node n(i) n(j) : 0 ≤ i, j < n , i, j integers}.

a naive bounding of closed circuit path total distance bounded below n*min(c) , above n*max(c) min(c) minimum cost traverse edge between 2 nodes, , max(c) maximum cost traverse edge between 2 nodes.

if looking shortest path not in circuit becomes (n-1)*min(c) , (n-1)*max(c) respectively.

beyond that, there variety of methods getting better upper bounds , better lower bounds.