A Necessary And Sufficient Condition About Path Spectrum Of Order Four Of The Tree

The path spectrum of a graph is one of most important substance on graph structure theory. A path Pof a graph G is maximal if P is not a proper subpath of any other path of the graph G. And the path spectrum of G,denote by ps(G),is the set of length of all maximal paths in the graph. A set S of positive integers is called a path spectrum if there is a connected graph G such that ps(G) = S. The article[1] gave the path spectrum of a tree,characterize the path spectrum of order two and the path spectrum of order three.This paper study the path spectrum of a tree farther, give a necessary and sufficient condition for the path spectrum of order four.The main result are as follow:Theorem 4:Let a, b, c, d be four positive integers, a is even,b, c, d are odds and b < c < d,then there exists a tree , such that ps(T) = {a, b, c, d} if and only if one of the following conditions holds:1)a<2b or cb + c 3)a>b/2,d> b/2,a + d = b + c...