On the extremal values of ratios of number of paths
In this paper, we analyze the ratios of the numbers of paths pi (G) and pj (G) of different length in graph G. Namely, we are interested in the extremal values of these ratios for acyclic and cyclic graphs with given maximal degree. The values of infinum and supremum for graphs with given maximal degree are obtained. Also, the infinum of these ratios for trees with given maximal degree are obtained. Suprema for trees of given maximal degree are given when ratios of paths of length 1 and 2 are observed, and when ratios of paths of lengths 1 and 3 are observed. As the main result, a linear algorithm (in terms of maximal degree) for finding suprema of the ratios of the numbers of paths of length 2 and 3 for trees with given maximal degree is presented.