If I want to study about graph.is it nessary read all topic in undergrated combinatorics course before studying? I want to study graph theory ,but I'm not sure how is nessary need to understand topic in elementary combinaric because some book in intoduction to combinatorics start with graph before enumerative combinatorics ,also in discrete math textbook I read some part of graph theory and found it does not need advanced counting technique like  generating function that is why I ask this question 
 A: Not always, but usually, the prerequisites for a book will be posted in the preface or somewhere similar. For instance, if you go to Dover's webstore and look for Trudeau's Introduction to Graph Theory what you'll see is this excerpt from a review:

A stimulating excursion into pure mathematics aimed at "the mathematically traumatized," but great fun for mathematical hobbyists and serious mathematicians as well. Requiring only high school algebra as mathematical background, the book leads the reader from simple graphs through planar graphs, Euler's formula, Platonic graphs, coloring, the genus of a graph, Euler walks, Hamilton walks, and a discussion of The Seven Bridges of Konigsberg. Exercises are included at the end of each chapter. "The topics are so well motivated, the exposition so lucid and delightful, that the book's appeal should be virtually universal . . . Every library should have several copies" — Choice. 1976 edition.

(Emphasis mine.) You can find a similar sentiment in the preface of the book which you can read through Google Books.
The preface to Diestel's book says

The  mathematical  prerequisites  for  this  book,  as  for  most  graph
  theory texts, are minimal:  a first grounding in linear algebra is assumed
  for  Chapter  1.9  and  once  in  Chapter  5.5,  some  basic  topological  concepts about the Euclidean plane and 3-space are used in Chapter 4, and
  a  previous  first  encounter  with  elementary  probability  will  help  with
  Chapter 11.  (Even here, all that is assumed formally is the knowledge
  of basic definitions:  the few probabilistic tools used are developed in the text.)  There are two areas of graph theory which I find both fascinating and important, especially from the perspective of pure mathematics adopted here, but which are not covered in this book: these are algebraic graph theory and infinite graphs.

